internlm/Lean-Github · Datasets at Hugging Face

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Algebra/Lecture2.lean	Tutorial.homToPerm_injective	[346, 1]	[360, 15]	simp	G : Type inst✝ : Group G a : G h : (homToPerm G) a = 1 ⊢ a = a * 1	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Algebra/Lecture2.lean	Tutorial.homToPerm_injective	[346, 1]	[360, 15]	rfl	G : Type inst✝ : Group G a : G h : (homToPerm G) a = 1 ⊢ a * 1 = ((homToPerm G) a) 1	no goals
https://github.com/yuma-mizuno/lean-math-workshop.git	4a69b0130b276b45212e2b12b90032b146b56d67	Solution/Advanced/Algebra/Lecture2.lean	Tutorial.homToPerm_injective	[346, 1]	[360, 15]	simp [h]	G : Type inst✝ : Group G a : G h : (homToPerm G) a = 1 ⊢ ((homToPerm G) a) 1 = 1	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	Geometry.triangle_rotate	[28, 1]	[31, 36]	simp[collinear]	P L : Type p q r s : P inst✝ : Geometry P ⊢ ¬collinear p q r → ¬collinear q r p	P L : Type p q r s : P inst✝ : Geometry P ⊢ (∀ (x : line P), p ∈ x → q ∈ x → ¬r ∈ x) → ∀ (x : line P), q ∈ x → r ∈ x → ¬p ∈ x
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	Geometry.triangle_rotate	[28, 1]	[31, 36]	intro triangle_pqr l hq hr hp	P L : Type p q r s : P inst✝ : Geometry P ⊢ (∀ (x : line P), p ∈ x → q ∈ x → ¬r ∈ x) → ∀ (x : line P), q ∈ x → r ∈ x → ¬p ∈ x	P L : Type p q r s : P inst✝ : Geometry P triangle_pqr : ∀ (x : line P), p ∈ x → q ∈ x → ¬r ∈ x l : line P hq : q ∈ l hr : r ∈ l hp : p ∈ l ⊢ False
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	Geometry.triangle_rotate	[28, 1]	[31, 36]	apply triangle_pqr <;> assumption	P L : Type p q r s : P inst✝ : Geometry P triangle_pqr : ∀ (x : line P), p ∈ x → q ∈ x → ¬r ∈ x l : line P hq : q ∈ l hr : r ∈ l hp : p ∈ l ⊢ False	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	Geometry.quadrangle_rotate	[33, 1]	[35, 113]	intro ⟨triangle_pqr, triangle_pqs, triangle_prs, triangle_qrs⟩	P L : Type p q r s : P inst✝ : Geometry P ⊢ isQuadrangle p q r s → isQuadrangle q r s p	P L : Type p q r s : P inst✝ : Geometry P triangle_pqr : ¬collinear p q r triangle_pqs : ¬collinear p q s triangle_prs : ¬collinear p r s triangle_qrs : ¬collinear q r s ⊢ isQuadrangle q r s p
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	Geometry.quadrangle_rotate	[33, 1]	[35, 113]	exact ⟨triangle_qrs, triangle_rotate triangle_pqr, triangle_rotate triangle_pqs, triangle_rotate triangle_prs⟩	P L : Type p q r s : P inst✝ : Geometry P triangle_pqr : ¬collinear p q r triangle_pqs : ¬collinear p q s triangle_prs : ¬collinear p r s triangle_qrs : ¬collinear q r s ⊢ isQuadrangle q r s p	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_left	[55, 9]	[57, 65]	intro p q	P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ ∀ (p q : P), p ∈ p ⊔ q	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P ⊢ p ∈ p ⊔ q
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_left	[55, 9]	[57, 65]	exact (Exists.choose_spec (exists_connecting_line p q)).left	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P ⊢ p ∈ p ⊔ q	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_right	[59, 9]	[61, 66]	intro p q	P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ ∀ (p q : P), q ∈ p ⊔ q	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P ⊢ q ∈ p ⊔ q
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_right	[59, 9]	[61, 66]	exact (Exists.choose_spec (exists_connecting_line p q)).right	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P ⊢ q ∈ p ⊔ q	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_uniq	[63, 1]	[71, 25]	intro p q l hpq hpl hql	P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ ∀ {p q : P} {l : line P}, p ≠ q → p ∈ l → q ∈ l → l = p ⊔ q	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l ⊢ l = p ⊔ q
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_uniq	[63, 1]	[71, 25]	let m := p ⊔ q	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l ⊢ l = p ⊔ q	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q ⊢ l = p ⊔ q
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_uniq	[63, 1]	[71, 25]	have hpm : p ∈ m := by simp	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q ⊢ l = p ⊔ q	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m ⊢ l = p ⊔ q
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_uniq	[63, 1]	[71, 25]	have hqm : q ∈ m := by simp	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m ⊢ l = p ⊔ q	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m hqm : q ∈ m ⊢ l = p ⊔ q
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_uniq	[63, 1]	[71, 25]	have h : p = q ∨ l = m := point_line_uniq hpl hql hpm hqm	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m hqm : q ∈ m ⊢ l = p ⊔ q	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m hqm : q ∈ m h : p = q ∨ l = m ⊢ l = p ⊔ q
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_uniq	[63, 1]	[71, 25]	cases h with \| inl hpq' => contradiction \| inr h' => assumption	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m hqm : q ∈ m h : p = q ∨ l = m ⊢ l = p ⊔ q	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_uniq	[63, 1]	[71, 25]	simp	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q ⊢ p ∈ m	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_uniq	[63, 1]	[71, 25]	simp	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m ⊢ q ∈ m	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_uniq	[63, 1]	[71, 25]	contradiction	case inl P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m hqm : q ∈ m hpq' : p = q ⊢ l = p ⊔ q	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_uniq	[63, 1]	[71, 25]	assumption	case inr P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m hqm : q ∈ m h' : l = m ⊢ l = p ⊔ q	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_comm	[73, 1]	[78, 6]	rw[h]	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P h : q = p ⊢ p ⊔ q = q ⊔ p	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_comm	[73, 1]	[78, 6]	apply connectingLine_uniq <;> (simp ; try assumption)	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P hpq : q ≠ p ⊢ p ⊔ q = q ⊔ p	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_comm	[73, 1]	[78, 6]	simp	case a P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P hpq : q ≠ p ⊢ p ∈ p ⊔ q	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_comm	[73, 1]	[78, 6]	try assumption	case a P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P hpq : q ≠ p ⊢ ¬q = p	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.connectingLine_comm	[73, 1]	[78, 6]	assumption	case a P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P hpq : q ≠ p ⊢ ¬q = p	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_left	[84, 9]	[86, 68]	intro l m	P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ ∀ (l m : line P), l ⊓ m ∈ l	P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P ⊢ l ⊓ m ∈ l
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_left	[84, 9]	[86, 68]	exact (Exists.choose_spec (exists_intersection_point l m)).left	P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P ⊢ l ⊓ m ∈ l	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_right	[88, 9]	[90, 67]	intro l m	P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ ∀ (l m : line P), l ⊓ m ∈ m	P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P ⊢ l ⊓ m ∈ m
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_right	[88, 9]	[90, 67]	exact (Exists.choose_spec (exists_intersection_point l m)).right	P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P ⊢ l ⊓ m ∈ m	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_uniq	[92, 1]	[100, 30]	intro l m p hlm hpl hpm	P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ ∀ {l m : line P} {p : P}, l ≠ m → p ∈ l → p ∈ m → p = l ⊓ m	P L : Type p✝ q r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m ⊢ p = l ⊓ m
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_uniq	[92, 1]	[100, 30]	let q := l ⊓ m	P L : Type p✝ q r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m ⊢ p = l ⊓ m	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m ⊢ p = l ⊓ m
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_uniq	[92, 1]	[100, 30]	have hql : q ∈ l := by simp	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m ⊢ p = l ⊓ m	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l ⊢ p = l ⊓ m
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_uniq	[92, 1]	[100, 30]	have hqm : q ∈ m := by simp	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l ⊢ p = l ⊓ m	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l hqm : q ∈ m ⊢ p = l ⊓ m
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_uniq	[92, 1]	[100, 30]	have h : p = q ∨ l = m := point_line_uniq hpl hql hpm hqm	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l hqm : q ∈ m ⊢ p = l ⊓ m	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l hqm : q ∈ m h : p = q ∨ l = m ⊢ p = l ⊓ m
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_uniq	[92, 1]	[100, 30]	cases h with \| inl h' => exact h' \| inr hlm' => contradiction	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l hqm : q ∈ m h : p = q ∨ l = m ⊢ p = l ⊓ m	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_uniq	[92, 1]	[100, 30]	simp	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m ⊢ q ∈ l	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_uniq	[92, 1]	[100, 30]	simp	P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l ⊢ q ∈ m	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_uniq	[92, 1]	[100, 30]	exact h'	case inl P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l hqm : q ∈ m h' : p = q ⊢ p = l ⊓ m	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_uniq	[92, 1]	[100, 30]	contradiction	case inr P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l hqm : q ∈ m hlm' : l = m ⊢ p = l ⊓ m	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_comm	[102, 1]	[107, 6]	rw[h]	P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P h : m = l ⊢ l ⊓ m = m ⊓ l	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_comm	[102, 1]	[107, 6]	apply intersectionPoint_uniq <;> (simp ; try assumption)	P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P hpq : m ≠ l ⊢ l ⊓ m = m ⊓ l	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_comm	[102, 1]	[107, 6]	simp	case a P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P hpq : m ≠ l ⊢ l ⊓ m ∈ l	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_comm	[102, 1]	[107, 6]	try assumption	case a P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P hpq : m ≠ l ⊢ ¬m = l	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.intersectionPoint_comm	[102, 1]	[107, 6]	assumption	case a P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P hpq : m ≠ l ⊢ ¬m = l	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.triangle_lines_unequal	[109, 1]	[130, 16]	intro hnc	P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ ¬collinear p q r → p ⊔ q ≠ q ⊔ r ∧ p ⊔ q ≠ r ⊔ p ∧ q ⊔ r ≠ r ⊔ p	P L : Type p q r s : P inst✝ : ProjectivePlane P hnc : ¬collinear p q r ⊢ p ⊔ q ≠ q ⊔ r ∧ p ⊔ q ≠ r ⊔ p ∧ q ⊔ r ≠ r ⊔ p
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.triangle_lines_unequal	[109, 1]	[130, 16]	apply And.intro	P L : Type p q r s : P inst✝ : ProjectivePlane P hnc : ¬collinear p q r ⊢ p ⊔ q ≠ q ⊔ r ∧ p ⊔ q ≠ r ⊔ p ∧ q ⊔ r ≠ r ⊔ p	case left P L : Type p q r s : P inst✝ : ProjectivePlane P hnc : ¬collinear p q r ⊢ p ⊔ q ≠ q ⊔ r case right P L : Type p q r s : P inst✝ : ProjectivePlane P hnc : ¬collinear p q r ⊢ p ⊔ q ≠ r ⊔ p ∧ q ⊔ r ≠ r ⊔ p
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.triangle_lines_unequal	[109, 1]	[130, 16]	. intro h simp[collinear] at hnc apply hnc (p ⊔ q) . simp . simp . simp[h]	case left P L : Type p q r s : P inst✝ : ProjectivePlane P hnc : ¬collinear p q r ⊢ p ⊔ q ≠ q ⊔ r case right P L : Type p q r s : P inst✝ : ProjectivePlane P hnc : ¬collinear p q r ⊢ p ⊔ q ≠ r ⊔ p ∧ q ⊔ r ≠ r ⊔ p	case right P L : Type p q r s : P inst✝ : ProjectivePlane P hnc : ¬collinear p q r ⊢ p ⊔ q ≠ r ⊔ p ∧ q ⊔ r ≠ r ⊔ p
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.triangle_lines_unequal	[109, 1]	[130, 16]	. apply And.intro . intro h simp[collinear] at hnc apply (hnc (p ⊔ q)) . simp . simp . simp[h] . intro h simp[collinear] at hnc apply (hnc (q ⊔ r)) . simp[h] . simp . simp[h]	case right P L : Type p q r s : P inst✝ : ProjectivePlane P hnc : ¬collinear p q r ⊢ p ⊔ q ≠ r ⊔ p ∧ q ⊔ r ≠ r ⊔ p	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	let a := p ⊔ q	P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ ¬collinear p q r → ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ p)	P L : Type p q r s : P inst✝ : ProjectivePlane P a : line P := p ⊔ q ⊢ ¬collinear p q r → ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ p)
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	let b := q ⊔ r	P L : Type p q r s : P inst✝ : ProjectivePlane P a : line P := p ⊔ q ⊢ ¬collinear p q r → ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ p)	P L : Type p q r s : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r ⊢ ¬collinear p q r → ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ p)
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	intro hnc	P L : Type p q r s : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r ⊢ ¬collinear p q r → ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ p)	P L : Type p q r s : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ p)
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	simp[collinear]	P L : Type p q r s : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ p)	P L : Type p q r s : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r ⊢ ∀ (x : line (line P)), p ⊔ q ∈ x → q ⊔ r ∈ x → ¬r ⊔ p ∈ x
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	intro s hpq hqr hrp	P L : Type p q r s : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r ⊢ ∀ (x : line (line P)), p ⊔ q ∈ x → q ⊔ r ∈ x → ¬r ⊔ p ∈ x	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s ⊢ False
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	have hsq : s = q := by have diff_lines : a ≠ b := (triangle_lines_unequal hnc).left have hsq': s = q ∨ a = b := by apply point_line_uniq <;> (simp ; try assumption) cases hsq' with \| inl it => exact it \| inr not_it => contradiction	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s ⊢ False	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s hsq : s = q ⊢ False
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	have c : collinear p q r := by exists (r ⊔ p) rw[hsq] at hrp simp assumption	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s hsq : s = q ⊢ False	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s hsq : s = q c : collinear p q r ⊢ False
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	contradiction	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s hsq : s = q c : collinear p q r ⊢ False	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	have diff_lines : a ≠ b := (triangle_lines_unequal hnc).left	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s ⊢ s = q	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b ⊢ s = q
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	have hsq': s = q ∨ a = b := by apply point_line_uniq <;> (simp ; try assumption)	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b ⊢ s = q	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b hsq' : s = q ∨ a = b ⊢ s = q
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	cases hsq' with \| inl it => exact it \| inr not_it => contradiction	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b hsq' : s = q ∨ a = b ⊢ s = q	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	apply point_line_uniq <;> (simp ; try assumption)	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b ⊢ s = q ∨ a = b	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	simp	case a P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b ⊢ q ∈ b	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	try assumption	case a P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b ⊢ s ∈ q ⊔ r	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	assumption	case a P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b ⊢ s ∈ q ⊔ r	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	exact it	case inl P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b it : s = q ⊢ s = q	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	contradiction	case inr P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b not_it : a = b ⊢ s = q	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	exists (r ⊔ p)	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s hsq : s = q ⊢ collinear p q r	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s hsq : s = q ⊢ p ∈ r ⊔ p ∧ q ∈ r ⊔ p ∧ r ∈ r ⊔ p
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	rw[hsq] at hrp	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s hsq : s = q ⊢ p ∈ r ⊔ p ∧ q ∈ r ⊔ p ∧ r ∈ r ⊔ p	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ q hsq : s = q ⊢ p ∈ r ⊔ p ∧ q ∈ r ⊔ p ∧ r ∈ r ⊔ p
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	simp	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ q hsq : s = q ⊢ p ∈ r ⊔ p ∧ q ∈ r ⊔ p ∧ r ∈ r ⊔ p	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ q hsq : s = q ⊢ q ∈ r ⊔ p
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_triangle	[132, 1]	[156, 16]	assumption	P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ q hsq : s = q ⊢ q ∈ r ⊔ p	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	intro ⟨triangle_pqr, _, _, triangle_qrs⟩	P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ isQuadrangle p q r s → ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	let a := p ⊔ q	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	let b := q ⊔ r	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	let c := r ⊔ s	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	let t := a ⊓ c	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	have t_is_not_q : q ≠ t := by intro h have : collinear q r s := by simp[collinear] rw[h] exists c apply And.intro . simp . apply And.intro . simp . simp contradiction	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	have t_is_not_r : r ≠ t := by intro h have : collinear p q r := by simp[collinear] rw[h] exists a apply And.intro . simp . apply And.intro . simp . simp contradiction	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	have a_is_q_t : a = q ⊔ t := by apply connectingLine_uniq . assumption . simp . simp	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t a_is_q_t : a = q ⊔ t ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	have c_is_r_t : c = r ⊔ t := by apply connectingLine_uniq . assumption . simp . simp	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t a_is_q_t : a = q ⊔ t ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t a_is_q_t : a = q ⊔ t c_is_r_t : c = r ⊔ t ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	have triangle_tqr : ¬ collinear t q r := by simp[collinear] intro l htl hql hrl have l_is_a : l = a := by rw[a_is_q_t] apply connectingLine_uniq <;> assumption rw[l_is_a] at hrl have : collinear p q r := by simp[collinear] exists a apply And.intro . simp . apply And.intro . simp . assumption contradiction	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t a_is_q_t : a = q ⊔ t c_is_r_t : c = r ⊔ t ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t a_is_q_t : a = q ⊔ t c_is_r_t : c = r ⊔ t triangle_tqr : ¬collinear t q r ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	have : ¬ collinear a b c := by rw[a_is_q_t, c_is_r_t, connectingLine_comm] apply dual_triangle assumption	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t a_is_q_t : a = q ⊔ t c_is_r_t : c = r ⊔ t triangle_tqr : ¬collinear t q r ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t a_is_q_t : a = q ⊔ t c_is_r_t : c = r ⊔ t triangle_tqr : ¬collinear t q r this : ¬collinear a b c ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	assumption	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t a_is_q_t : a = q ⊔ t c_is_r_t : c = r ⊔ t triangle_tqr : ¬collinear t q r this : ¬collinear a b c ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	intro h	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c ⊢ q ≠ t	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ False
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	have : collinear q r s := by simp[collinear] rw[h] exists c apply And.intro . simp . apply And.intro . simp . simp	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ False	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t this : collinear q r s ⊢ False
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	contradiction	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t this : collinear q r s ⊢ False	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	simp[collinear]	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ collinear q r s	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ ∃ l, q ∈ l ∧ r ∈ l ∧ s ∈ l
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	rw[h]	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ ∃ l, q ∈ l ∧ r ∈ l ∧ s ∈ l	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ ∃ l, t ∈ l ∧ r ∈ l ∧ s ∈ l
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	exists c	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ ∃ l, t ∈ l ∧ r ∈ l ∧ s ∈ l	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ t ∈ c ∧ r ∈ c ∧ s ∈ c
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	apply And.intro	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ t ∈ c ∧ r ∈ c ∧ s ∈ c	case left P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ t ∈ c case right P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ r ∈ c ∧ s ∈ c
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	. simp	case left P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ t ∈ c case right P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ r ∈ c ∧ s ∈ c	case right P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ r ∈ c ∧ s ∈ c
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	. apply And.intro . simp . simp	case right P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ r ∈ c ∧ s ∈ c	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	intro h	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t ⊢ r ≠ t	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ False
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	have : collinear p q r := by simp[collinear] rw[h] exists a apply And.intro . simp . apply And.intro . simp . simp	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ False	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t this : collinear p q r ⊢ False
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	contradiction	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t this : collinear p q r ⊢ False	no goals
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	simp[collinear]	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ collinear p q r	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ ∃ l, p ∈ l ∧ q ∈ l ∧ r ∈ l
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	rw[h]	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ ∃ l, p ∈ l ∧ q ∈ l ∧ r ∈ l	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ ∃ l, p ∈ l ∧ q ∈ l ∧ t ∈ l
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	exists a	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ ∃ l, p ∈ l ∧ q ∈ l ∧ t ∈ l	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ p ∈ a ∧ q ∈ a ∧ t ∈ a
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	apply And.intro	P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ p ∈ a ∧ q ∈ a ∧ t ∈ a	case left P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ p ∈ a case right P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ q ∈ a ∧ t ∈ a
https://github.com/loewenheim/projective-plane.git	7ade52678ac3df537f38d04de46db9aca752f3ca	ProjectivePlane.lean	ProjectivePlane.dual_quadrangle_part	[159, 1]	[232, 13]	. simp	case left P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ p ∈ a case right P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ q ∈ a ∧ t ∈ a	case right P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ q ∈ a ∧ t ∈ a

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Algebra/Lecture2.lean

Tutorial.homToPerm_injective

[346, 1]

[360, 15]

simp

G : Type inst✝ : Group G a : G h : (homToPerm G) a = 1 ⊢ a = a * 1

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Algebra/Lecture2.lean

Tutorial.homToPerm_injective

[346, 1]

[360, 15]

rfl

G : Type inst✝ : Group G a : G h : (homToPerm G) a = 1 ⊢ a * 1 = ((homToPerm G) a) 1

no goals

https://github.com/yuma-mizuno/lean-math-workshop.git

4a69b0130b276b45212e2b12b90032b146b56d67

Solution/Advanced/Algebra/Lecture2.lean

Tutorial.homToPerm_injective

[346, 1]

[360, 15]

simp [h]

G : Type inst✝ : Group G a : G h : (homToPerm G) a = 1 ⊢ ((homToPerm G) a) 1 = 1

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

Geometry.triangle_rotate

[28, 1]

[31, 36]

simp[collinear]

P L : Type p q r s : P inst✝ : Geometry P ⊢ ¬collinear p q r → ¬collinear q r p

P L : Type p q r s : P inst✝ : Geometry P ⊢ (∀ (x : line P), p ∈ x → q ∈ x → ¬r ∈ x) → ∀ (x : line P), q ∈ x → r ∈ x → ¬p ∈ x

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

Geometry.triangle_rotate

[28, 1]

[31, 36]

intro triangle_pqr l hq hr hp

P L : Type p q r s : P inst✝ : Geometry P ⊢ (∀ (x : line P), p ∈ x → q ∈ x → ¬r ∈ x) → ∀ (x : line P), q ∈ x → r ∈ x → ¬p ∈ x

P L : Type p q r s : P inst✝ : Geometry P triangle_pqr : ∀ (x : line P), p ∈ x → q ∈ x → ¬r ∈ x l : line P hq : q ∈ l hr : r ∈ l hp : p ∈ l ⊢ False

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

Geometry.triangle_rotate

[28, 1]

[31, 36]

apply triangle_pqr <;> assumption

P L : Type p q r s : P inst✝ : Geometry P triangle_pqr : ∀ (x : line P), p ∈ x → q ∈ x → ¬r ∈ x l : line P hq : q ∈ l hr : r ∈ l hp : p ∈ l ⊢ False

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

Geometry.quadrangle_rotate

[33, 1]

[35, 113]

intro ⟨triangle_pqr, triangle_pqs, triangle_prs, triangle_qrs⟩

P L : Type p q r s : P inst✝ : Geometry P ⊢ isQuadrangle p q r s → isQuadrangle q r s p

P L : Type p q r s : P inst✝ : Geometry P triangle_pqr : ¬collinear p q r triangle_pqs : ¬collinear p q s triangle_prs : ¬collinear p r s triangle_qrs : ¬collinear q r s ⊢ isQuadrangle q r s p

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

Geometry.quadrangle_rotate

[33, 1]

[35, 113]

exact ⟨triangle_qrs, triangle_rotate triangle_pqr, triangle_rotate triangle_pqs, triangle_rotate triangle_prs⟩

P L : Type p q r s : P inst✝ : Geometry P triangle_pqr : ¬collinear p q r triangle_pqs : ¬collinear p q s triangle_prs : ¬collinear p r s triangle_qrs : ¬collinear q r s ⊢ isQuadrangle q r s p

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_left

[55, 9]

[57, 65]

intro p q

P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ ∀ (p q : P), p ∈ p ⊔ q

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P ⊢ p ∈ p ⊔ q

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_left

[55, 9]

[57, 65]

exact (Exists.choose_spec (exists_connecting_line p q)).left

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P ⊢ p ∈ p ⊔ q

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_right

[59, 9]

[61, 66]

intro p q

P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ ∀ (p q : P), q ∈ p ⊔ q

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P ⊢ q ∈ p ⊔ q

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_right

[59, 9]

[61, 66]

exact (Exists.choose_spec (exists_connecting_line p q)).right

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P ⊢ q ∈ p ⊔ q

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_uniq

[63, 1]

[71, 25]

intro p q l hpq hpl hql

P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ ∀ {p q : P} {l : line P}, p ≠ q → p ∈ l → q ∈ l → l = p ⊔ q

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l ⊢ l = p ⊔ q

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_uniq

[63, 1]

[71, 25]

let m := p ⊔ q

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l ⊢ l = p ⊔ q

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q ⊢ l = p ⊔ q

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_uniq

[63, 1]

[71, 25]

have hpm : p ∈ m := by simp

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q ⊢ l = p ⊔ q

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m ⊢ l = p ⊔ q

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_uniq

[63, 1]

[71, 25]

have hqm : q ∈ m := by simp

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m ⊢ l = p ⊔ q

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m hqm : q ∈ m ⊢ l = p ⊔ q

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_uniq

[63, 1]

[71, 25]

have h : p = q ∨ l = m := point_line_uniq hpl hql hpm hqm

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m hqm : q ∈ m ⊢ l = p ⊔ q

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m hqm : q ∈ m h : p = q ∨ l = m ⊢ l = p ⊔ q

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_uniq

[63, 1]

[71, 25]

cases h with | inl hpq' => contradiction | inr h' => assumption

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m hqm : q ∈ m h : p = q ∨ l = m ⊢ l = p ⊔ q

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_uniq

[63, 1]

[71, 25]

simp

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q ⊢ p ∈ m

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_uniq

[63, 1]

[71, 25]

simp

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m ⊢ q ∈ m

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_uniq

[63, 1]

[71, 25]

contradiction

case inl P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m hqm : q ∈ m hpq' : p = q ⊢ l = p ⊔ q

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_uniq

[63, 1]

[71, 25]

assumption

case inr P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P l : line P hpq : p ≠ q hpl : p ∈ l hql : q ∈ l m : line P := p ⊔ q hpm : p ∈ m hqm : q ∈ m h' : l = m ⊢ l = p ⊔ q

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_comm

[73, 1]

[78, 6]

rw[h]

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P h : q = p ⊢ p ⊔ q = q ⊔ p

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_comm

[73, 1]

[78, 6]

apply connectingLine_uniq <;> (simp ; try assumption)

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P hpq : q ≠ p ⊢ p ⊔ q = q ⊔ p

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_comm

[73, 1]

[78, 6]

simp

case a P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P hpq : q ≠ p ⊢ p ∈ p ⊔ q

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_comm

[73, 1]

[78, 6]

try assumption

case a P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P hpq : q ≠ p ⊢ ¬q = p

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.connectingLine_comm

[73, 1]

[78, 6]

assumption

case a P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P p q : P hpq : q ≠ p ⊢ ¬q = p

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_left

[84, 9]

[86, 68]

intro l m

P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ ∀ (l m : line P), l ⊓ m ∈ l

P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P ⊢ l ⊓ m ∈ l

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_left

[84, 9]

[86, 68]

exact (Exists.choose_spec (exists_intersection_point l m)).left

P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P ⊢ l ⊓ m ∈ l

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_right

[88, 9]

[90, 67]

intro l m

P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ ∀ (l m : line P), l ⊓ m ∈ m

P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P ⊢ l ⊓ m ∈ m

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_right

[88, 9]

[90, 67]

exact (Exists.choose_spec (exists_intersection_point l m)).right

P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P ⊢ l ⊓ m ∈ m

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_uniq

[92, 1]

[100, 30]

intro l m p hlm hpl hpm

P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ ∀ {l m : line P} {p : P}, l ≠ m → p ∈ l → p ∈ m → p = l ⊓ m

P L : Type p✝ q r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m ⊢ p = l ⊓ m

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_uniq

[92, 1]

[100, 30]

let q := l ⊓ m

P L : Type p✝ q r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m ⊢ p = l ⊓ m

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m ⊢ p = l ⊓ m

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_uniq

[92, 1]

[100, 30]

have hql : q ∈ l := by simp

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m ⊢ p = l ⊓ m

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l ⊢ p = l ⊓ m

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_uniq

[92, 1]

[100, 30]

have hqm : q ∈ m := by simp

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l ⊢ p = l ⊓ m

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l hqm : q ∈ m ⊢ p = l ⊓ m

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_uniq

[92, 1]

[100, 30]

have h : p = q ∨ l = m := point_line_uniq hpl hql hpm hqm

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l hqm : q ∈ m ⊢ p = l ⊓ m

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l hqm : q ∈ m h : p = q ∨ l = m ⊢ p = l ⊓ m

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_uniq

[92, 1]

[100, 30]

cases h with | inl h' => exact h' | inr hlm' => contradiction

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l hqm : q ∈ m h : p = q ∨ l = m ⊢ p = l ⊓ m

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_uniq

[92, 1]

[100, 30]

simp

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m ⊢ q ∈ l

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_uniq

[92, 1]

[100, 30]

simp

P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l ⊢ q ∈ m

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_uniq

[92, 1]

[100, 30]

exact h'

case inl P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l hqm : q ∈ m h' : p = q ⊢ p = l ⊓ m

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_uniq

[92, 1]

[100, 30]

contradiction

case inr P L : Type p✝ q✝ r s : P inst✝ : ProjectivePlane P l m : line P p : P hlm : l ≠ m hpl : p ∈ l hpm : p ∈ m q : P := l ⊓ m hql : q ∈ l hqm : q ∈ m hlm' : l = m ⊢ p = l ⊓ m

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_comm

[102, 1]

[107, 6]

rw[h]

P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P h : m = l ⊢ l ⊓ m = m ⊓ l

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_comm

[102, 1]

[107, 6]

apply intersectionPoint_uniq <;> (simp ; try assumption)

P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P hpq : m ≠ l ⊢ l ⊓ m = m ⊓ l

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_comm

[102, 1]

[107, 6]

simp

case a P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P hpq : m ≠ l ⊢ l ⊓ m ∈ l

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_comm

[102, 1]

[107, 6]

try assumption

case a P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P hpq : m ≠ l ⊢ ¬m = l

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.intersectionPoint_comm

[102, 1]

[107, 6]

assumption

case a P L : Type p q r s : P inst✝ : ProjectivePlane P l m : line P hpq : m ≠ l ⊢ ¬m = l

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.triangle_lines_unequal

[109, 1]

[130, 16]

intro hnc

P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ ¬collinear p q r → p ⊔ q ≠ q ⊔ r ∧ p ⊔ q ≠ r ⊔ p ∧ q ⊔ r ≠ r ⊔ p

P L : Type p q r s : P inst✝ : ProjectivePlane P hnc : ¬collinear p q r ⊢ p ⊔ q ≠ q ⊔ r ∧ p ⊔ q ≠ r ⊔ p ∧ q ⊔ r ≠ r ⊔ p

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.triangle_lines_unequal

[109, 1]

[130, 16]

apply And.intro

P L : Type p q r s : P inst✝ : ProjectivePlane P hnc : ¬collinear p q r ⊢ p ⊔ q ≠ q ⊔ r ∧ p ⊔ q ≠ r ⊔ p ∧ q ⊔ r ≠ r ⊔ p

case left P L : Type p q r s : P inst✝ : ProjectivePlane P hnc : ¬collinear p q r ⊢ p ⊔ q ≠ q ⊔ r case right P L : Type p q r s : P inst✝ : ProjectivePlane P hnc : ¬collinear p q r ⊢ p ⊔ q ≠ r ⊔ p ∧ q ⊔ r ≠ r ⊔ p

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.triangle_lines_unequal

[109, 1]

[130, 16]

. intro h simp[collinear] at hnc apply hnc (p ⊔ q) . simp . simp . simp[h]

case left P L : Type p q r s : P inst✝ : ProjectivePlane P hnc : ¬collinear p q r ⊢ p ⊔ q ≠ q ⊔ r case right P L : Type p q r s : P inst✝ : ProjectivePlane P hnc : ¬collinear p q r ⊢ p ⊔ q ≠ r ⊔ p ∧ q ⊔ r ≠ r ⊔ p

case right P L : Type p q r s : P inst✝ : ProjectivePlane P hnc : ¬collinear p q r ⊢ p ⊔ q ≠ r ⊔ p ∧ q ⊔ r ≠ r ⊔ p

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.triangle_lines_unequal

[109, 1]

[130, 16]

. apply And.intro . intro h simp[collinear] at hnc apply (hnc (p ⊔ q)) . simp . simp . simp[h] . intro h simp[collinear] at hnc apply (hnc (q ⊔ r)) . simp[h] . simp . simp[h]

case right P L : Type p q r s : P inst✝ : ProjectivePlane P hnc : ¬collinear p q r ⊢ p ⊔ q ≠ r ⊔ p ∧ q ⊔ r ≠ r ⊔ p

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

let a := p ⊔ q

P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ ¬collinear p q r → ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ p)

P L : Type p q r s : P inst✝ : ProjectivePlane P a : line P := p ⊔ q ⊢ ¬collinear p q r → ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ p)

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

let b := q ⊔ r

P L : Type p q r s : P inst✝ : ProjectivePlane P a : line P := p ⊔ q ⊢ ¬collinear p q r → ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ p)

P L : Type p q r s : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r ⊢ ¬collinear p q r → ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ p)

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

intro hnc

P L : Type p q r s : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r ⊢ ¬collinear p q r → ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ p)

P L : Type p q r s : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ p)

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

simp[collinear]

P L : Type p q r s : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ p)

P L : Type p q r s : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r ⊢ ∀ (x : line (line P)), p ⊔ q ∈ x → q ⊔ r ∈ x → ¬r ⊔ p ∈ x

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

intro s hpq hqr hrp

P L : Type p q r s : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r ⊢ ∀ (x : line (line P)), p ⊔ q ∈ x → q ⊔ r ∈ x → ¬r ⊔ p ∈ x

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s ⊢ False

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

have hsq : s = q := by have diff_lines : a ≠ b := (triangle_lines_unequal hnc).left have hsq': s = q ∨ a = b := by apply point_line_uniq <;> (simp ; try assumption) cases hsq' with | inl it => exact it | inr not_it => contradiction

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s ⊢ False

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s hsq : s = q ⊢ False

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

have c : collinear p q r := by exists (r ⊔ p) rw[hsq] at hrp simp assumption

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s hsq : s = q ⊢ False

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s hsq : s = q c : collinear p q r ⊢ False

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

contradiction

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s hsq : s = q c : collinear p q r ⊢ False

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

have diff_lines : a ≠ b := (triangle_lines_unequal hnc).left

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s ⊢ s = q

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b ⊢ s = q

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

have hsq': s = q ∨ a = b := by apply point_line_uniq <;> (simp ; try assumption)

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b ⊢ s = q

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b hsq' : s = q ∨ a = b ⊢ s = q

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

cases hsq' with | inl it => exact it | inr not_it => contradiction

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b hsq' : s = q ∨ a = b ⊢ s = q

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

apply point_line_uniq <;> (simp ; try assumption)

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b ⊢ s = q ∨ a = b

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

simp

case a P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b ⊢ q ∈ b

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

try assumption

case a P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b ⊢ s ∈ q ⊔ r

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

assumption

case a P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b ⊢ s ∈ q ⊔ r

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

exact it

case inl P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b it : s = q ⊢ s = q

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

contradiction

case inr P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s diff_lines : a ≠ b not_it : a = b ⊢ s = q

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

exists (r ⊔ p)

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s hsq : s = q ⊢ collinear p q r

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s hsq : s = q ⊢ p ∈ r ⊔ p ∧ q ∈ r ⊔ p ∧ r ∈ r ⊔ p

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

rw[hsq] at hrp

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ s hsq : s = q ⊢ p ∈ r ⊔ p ∧ q ∈ r ⊔ p ∧ r ∈ r ⊔ p

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ q hsq : s = q ⊢ p ∈ r ⊔ p ∧ q ∈ r ⊔ p ∧ r ∈ r ⊔ p

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

simp

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ q hsq : s = q ⊢ p ∈ r ⊔ p ∧ q ∈ r ⊔ p ∧ r ∈ r ⊔ p

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ q hsq : s = q ⊢ q ∈ r ⊔ p

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_triangle

[132, 1]

[156, 16]

assumption

P L : Type p q r s✝ : P inst✝ : ProjectivePlane P a : line P := p ⊔ q b : line P := q ⊔ r hnc : ¬collinear p q r s : line (line P) hpq : p ⊔ q ∈ s hqr : q ⊔ r ∈ s hrp : r ⊔ p ∈ q hsq : s = q ⊢ q ∈ r ⊔ p

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

intro ⟨triangle_pqr, _, _, triangle_qrs⟩

P L : Type p q r s : P inst✝ : ProjectivePlane P ⊢ isQuadrangle p q r s → ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

let a := p ⊔ q

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

let b := q ⊔ r

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

let c := r ⊔ s

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

let t := a ⊓ c

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

have t_is_not_q : q ≠ t := by intro h have : collinear q r s := by simp[collinear] rw[h] exists c apply And.intro . simp . apply And.intro . simp . simp contradiction

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

have t_is_not_r : r ≠ t := by intro h have : collinear p q r := by simp[collinear] rw[h] exists a apply And.intro . simp . apply And.intro . simp . simp contradiction

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

have a_is_q_t : a = q ⊔ t := by apply connectingLine_uniq . assumption . simp . simp

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t a_is_q_t : a = q ⊔ t ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

have c_is_r_t : c = r ⊔ t := by apply connectingLine_uniq . assumption . simp . simp

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t a_is_q_t : a = q ⊔ t ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t a_is_q_t : a = q ⊔ t c_is_r_t : c = r ⊔ t ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

have triangle_tqr : ¬ collinear t q r := by simp[collinear] intro l htl hql hrl have l_is_a : l = a := by rw[a_is_q_t] apply connectingLine_uniq <;> assumption rw[l_is_a] at hrl have : collinear p q r := by simp[collinear] exists a apply And.intro . simp . apply And.intro . simp . assumption contradiction

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t a_is_q_t : a = q ⊔ t c_is_r_t : c = r ⊔ t ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t a_is_q_t : a = q ⊔ t c_is_r_t : c = r ⊔ t triangle_tqr : ¬collinear t q r ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

have : ¬ collinear a b c := by rw[a_is_q_t, c_is_r_t, connectingLine_comm] apply dual_triangle assumption

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t a_is_q_t : a = q ⊔ t c_is_r_t : c = r ⊔ t triangle_tqr : ¬collinear t q r ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t a_is_q_t : a = q ⊔ t c_is_r_t : c = r ⊔ t triangle_tqr : ¬collinear t q r this : ¬collinear a b c ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

assumption

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t t_is_not_r : r ≠ t a_is_q_t : a = q ⊔ t c_is_r_t : c = r ⊔ t triangle_tqr : ¬collinear t q r this : ¬collinear a b c ⊢ ¬collinear (p ⊔ q) (q ⊔ r) (r ⊔ s)

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

intro h

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c ⊢ q ≠ t

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ False

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

have : collinear q r s := by simp[collinear] rw[h] exists c apply And.intro . simp . apply And.intro . simp . simp

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ False

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t this : collinear q r s ⊢ False

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

contradiction

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t this : collinear q r s ⊢ False

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

simp[collinear]

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ collinear q r s

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ ∃ l, q ∈ l ∧ r ∈ l ∧ s ∈ l

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

rw[h]

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ ∃ l, q ∈ l ∧ r ∈ l ∧ s ∈ l

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ ∃ l, t ∈ l ∧ r ∈ l ∧ s ∈ l

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

exists c

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ ∃ l, t ∈ l ∧ r ∈ l ∧ s ∈ l

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ t ∈ c ∧ r ∈ c ∧ s ∈ c

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

apply And.intro

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ t ∈ c ∧ r ∈ c ∧ s ∈ c

case left P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ t ∈ c case right P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ r ∈ c ∧ s ∈ c

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

. simp

case left P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ t ∈ c case right P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ r ∈ c ∧ s ∈ c

case right P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ r ∈ c ∧ s ∈ c

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

. apply And.intro . simp . simp

case right P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c h : q = t ⊢ r ∈ c ∧ s ∈ c

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

intro h

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t ⊢ r ≠ t

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ False

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

have : collinear p q r := by simp[collinear] rw[h] exists a apply And.intro . simp . apply And.intro . simp . simp

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ False

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t this : collinear p q r ⊢ False

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

contradiction

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t this : collinear p q r ⊢ False

no goals

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

simp[collinear]

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ collinear p q r

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ ∃ l, p ∈ l ∧ q ∈ l ∧ r ∈ l

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

rw[h]

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ ∃ l, p ∈ l ∧ q ∈ l ∧ r ∈ l

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ ∃ l, p ∈ l ∧ q ∈ l ∧ t ∈ l

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

exists a

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ ∃ l, p ∈ l ∧ q ∈ l ∧ t ∈ l

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ p ∈ a ∧ q ∈ a ∧ t ∈ a

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

apply And.intro

P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ p ∈ a ∧ q ∈ a ∧ t ∈ a

case left P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ p ∈ a case right P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ q ∈ a ∧ t ∈ a

https://github.com/loewenheim/projective-plane.git

7ade52678ac3df537f38d04de46db9aca752f3ca

ProjectivePlane.lean

ProjectivePlane.dual_quadrangle_part

[159, 1]

[232, 13]

. simp

case left P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ p ∈ a case right P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ q ∈ a ∧ t ∈ a

case right P L : Type p q r s : P inst✝ : ProjectivePlane P triangle_pqr : ¬collinear p q r left✝¹ : ¬collinear p q s left✝ : ¬collinear p r s triangle_qrs : ¬collinear q r s a : line P := p ⊔ q b : line P := q ⊔ r c : line P := r ⊔ s t : P := a ⊓ c t_is_not_q : q ≠ t h : r = t ⊢ q ∈ a ∧ t ∈ a