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internlm
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Lean-Github

Dataset card Data Studio Files Community
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https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Tutorial/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_rev
[192, 1]
[193, 8]
sorry
A : Type ?u.3654 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ (a * b)⁻¹ = b⁻¹ * a⁻¹
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Tutorial/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_eq_iff_eq_mul
[195, 1]
[197, 8]
sorry
A : Type ?u.3816 G : Type inst✝¹ : Ring A inst✝ : Group G a b c : G ⊢ a * b⁻¹ = c ↔ a = c * b
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Tutorial/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_eq_one
[199, 1]
[200, 8]
sorry
A : Type ?u.3960 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a * b⁻¹ = 1 ↔ a = b
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Tutorial/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_eq_one
[202, 1]
[203, 8]
sorry
A : Type ?u.4111 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a⁻¹ * b = 1 ↔ a = b
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Tutorial/Advanced/Algebra/Lecture1.lean
Tutorial.Subgroup.inv_mem_iff
[264, 1]
[265, 8]
sorry
G : Type inst✝ : Group G H : Subgroup G a : G ⊢ a⁻¹ ∈ H ↔ a ∈ H
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_cancel_left
[133, 1]
[145, 11]
calc a⁻¹ * (a * b) = (a⁻¹ * a) * b := by rw [mul_assoc] _ = 1 * b := by simp _ = b := by simp
A : Type ?u.2656 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a⁻¹ * (a * b) = b
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_cancel_left
[133, 1]
[145, 11]
rw [mul_assoc]
A : Type ?u.2656 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a⁻¹ * (a * b) = a⁻¹ * a * b
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_cancel_left
[133, 1]
[145, 11]
simp
A : Type ?u.2656 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a⁻¹ * a * b = 1 * b
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_cancel_left
[133, 1]
[145, 11]
simp
A : Type ?u.2656 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ 1 * b = b
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_left_cancel
[153, 1]
[165, 21]
intro h
A : Type ?u.4453 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G ⊢ a * x = a * y → x = y
A : Type ?u.4453 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : a * x = a * y ⊢ x = y
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_left_cancel
[153, 1]
[165, 21]
calc x = 1 * x := by simp _ = (a⁻¹ * a) * x := by simp _ = a⁻¹ * (a * x) := by rw [mul_assoc] _ = a⁻¹ * (a * y) := by rw [h] _ = (a⁻¹ * a) * y := by rw [← mul_assoc] _ = 1 * y := by simp _ = y := by simp
A : Type ?u.4453 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : a * x = a * y ⊢ x = y
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_left_cancel
[153, 1]
[165, 21]
simp
A : Type ?u.4453 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : a * x = a * y ⊢ x = 1 * x
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_left_cancel
[153, 1]
[165, 21]
simp
A : Type ?u.4453 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : a * x = a * y ⊢ 1 * x = a⁻¹ * a * x
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_left_cancel
[153, 1]
[165, 21]
rw [mul_assoc]
A : Type ?u.4453 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : a * x = a * y ⊢ a⁻¹ * a * x = a⁻¹ * (a * x)
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_left_cancel
[153, 1]
[165, 21]
rw [h]
A : Type ?u.4453 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : a * x = a * y ⊢ a⁻¹ * (a * x) = a⁻¹ * (a * y)
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_left_cancel
[153, 1]
[165, 21]
rw [← mul_assoc]
A : Type ?u.4453 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : a * x = a * y ⊢ a⁻¹ * (a * y) = a⁻¹ * a * y
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_left_cancel
[153, 1]
[165, 21]
simp
A : Type ?u.4453 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : a * x = a * y ⊢ a⁻¹ * a * y = 1 * y
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_left_cancel
[153, 1]
[165, 21]
simp
A : Type ?u.4453 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : a * x = a * y ⊢ 1 * y = y
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_one
[170, 1]
[180, 7]
apply mul_left_cancel (a := a⁻¹)
A : Type ?u.7038 G : Type inst✝¹ : Ring A inst✝ : Group G a : G ⊢ a * 1 = a
A : Type ?u.7038 G : Type inst✝¹ : Ring A inst✝ : Group G a : G ⊢ a⁻¹ * (a * 1) = a⁻¹ * a
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_one
[170, 1]
[180, 7]
simp
A : Type ?u.7038 G : Type inst✝¹ : Ring A inst✝ : Group G a : G ⊢ a⁻¹ * (a * 1) = a⁻¹ * a
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_self
[185, 1]
[188, 7]
apply mul_left_cancel (a := a⁻¹)
A : Type ?u.9053 G : Type inst✝¹ : Ring A inst✝ : Group G a : G ⊢ a * a⁻¹ = 1
A : Type ?u.9053 G : Type inst✝¹ : Ring A inst✝ : Group G a : G ⊢ a⁻¹ * (a * a⁻¹) = a⁻¹ * 1
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_self
[185, 1]
[188, 7]
simp
A : Type ?u.9053 G : Type inst✝¹ : Ring A inst✝ : Group G a : G ⊢ a⁻¹ * (a * a⁻¹) = a⁻¹ * 1
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_cancel_left
[193, 1]
[196, 7]
rw [← mul_assoc]
A : Type ?u.10914 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a * (a⁻¹ * b) = b
A : Type ?u.10914 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a * a⁻¹ * b = b
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_cancel_left
[193, 1]
[196, 7]
simp
A : Type ?u.10914 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a * a⁻¹ * b = b
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_cancel_right
[200, 1]
[202, 19]
simp [mul_assoc]
A : Type ?u.12143 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a * b * b⁻¹ = a
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_cancel_right
[206, 1]
[208, 19]
simp [mul_assoc]
A : Type ?u.13426 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a * b⁻¹ * b = a
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_right_cancel
[212, 1]
[222, 21]
intro h
A : Type ?u.14883 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G ⊢ x * a = y * a → x = y
A : Type ?u.14883 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : x * a = y * a ⊢ x = y
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_right_cancel
[212, 1]
[222, 21]
calc x = x * 1 := by simp _ = x * (a * a⁻¹) := by simp _ = (x * a) * a⁻¹ := by rw [← mul_assoc] _ = (y * a) * a⁻¹ := by rw [h] _ = y * (a * a⁻¹) := by rw [mul_assoc] _ = y * 1 := by simp _ = y := by simp
A : Type ?u.14883 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : x * a = y * a ⊢ x = y
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_right_cancel
[212, 1]
[222, 21]
simp
A : Type ?u.14883 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : x * a = y * a ⊢ x = x * 1
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_right_cancel
[212, 1]
[222, 21]
simp
A : Type ?u.14883 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : x * a = y * a ⊢ x * 1 = x * (a * a⁻¹)
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_right_cancel
[212, 1]
[222, 21]
rw [← mul_assoc]
A : Type ?u.14883 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : x * a = y * a ⊢ x * (a * a⁻¹) = x * a * a⁻¹
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_right_cancel
[212, 1]
[222, 21]
rw [h]
A : Type ?u.14883 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : x * a = y * a ⊢ x * a * a⁻¹ = y * a * a⁻¹
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_right_cancel
[212, 1]
[222, 21]
rw [mul_assoc]
A : Type ?u.14883 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : x * a = y * a ⊢ y * a * a⁻¹ = y * (a * a⁻¹)
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_right_cancel
[212, 1]
[222, 21]
simp
A : Type ?u.14883 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : x * a = y * a ⊢ y * (a * a⁻¹) = y * 1
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_right_cancel
[212, 1]
[222, 21]
simp
A : Type ?u.14883 G : Type inst✝¹ : Ring A inst✝ : Group G a x y : G h : x * a = y * a ⊢ y * 1 = y
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_eq_of_mul_eq_one_left
[226, 1]
[231, 7]
intro h
A : Type ?u.17097 G : Type inst✝¹ : Ring A inst✝ : Group G a x : G ⊢ x * a = 1 → a⁻¹ = x
A : Type ?u.17097 G : Type inst✝¹ : Ring A inst✝ : Group G a x : G h : x * a = 1 ⊢ a⁻¹ = x
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_eq_of_mul_eq_one_left
[226, 1]
[231, 7]
apply mul_right_cancel a
A : Type ?u.17097 G : Type inst✝¹ : Ring A inst✝ : Group G a x : G h : x * a = 1 ⊢ a⁻¹ = x
A : Type ?u.17097 G : Type inst✝¹ : Ring A inst✝ : Group G a x : G h : x * a = 1 ⊢ a⁻¹ * a = x * a
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_eq_of_mul_eq_one_left
[226, 1]
[231, 7]
rw [h]
A : Type ?u.17097 G : Type inst✝¹ : Ring A inst✝ : Group G a x : G h : x * a = 1 ⊢ a⁻¹ * a = x * a
A : Type ?u.17097 G : Type inst✝¹ : Ring A inst✝ : Group G a x : G h : x * a = 1 ⊢ a⁻¹ * a = 1
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_eq_of_mul_eq_one_left
[226, 1]
[231, 7]
simp
A : Type ?u.17097 G : Type inst✝¹ : Ring A inst✝ : Group G a x : G h : x * a = 1 ⊢ a⁻¹ * a = 1
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_one
[239, 1]
[242, 7]
apply inv_eq_of_mul_eq_one_left
A : Type ?u.18067 G : Type inst✝¹ : Ring A inst✝ : Group G ⊢ 1⁻¹ = 1
case a A : Type ?u.18067 G : Type inst✝¹ : Ring A inst✝ : Group G ⊢ 1 * 1 = 1
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_one
[239, 1]
[242, 7]
simp
case a A : Type ?u.18067 G : Type inst✝¹ : Ring A inst✝ : Group G ⊢ 1 * 1 = 1
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_inv
[246, 1]
[249, 7]
apply inv_eq_of_mul_eq_one_left
A : Type ?u.18884 G : Type inst✝¹ : Ring A inst✝ : Group G a : G ⊢ a⁻¹⁻¹ = a
case a A : Type ?u.18884 G : Type inst✝¹ : Ring A inst✝ : Group G a : G ⊢ a * a⁻¹ = 1
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_inv
[246, 1]
[249, 7]
simp
case a A : Type ?u.18884 G : Type inst✝¹ : Ring A inst✝ : Group G a : G ⊢ a * a⁻¹ = 1
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_rev
[254, 1]
[258, 7]
apply inv_eq_of_mul_eq_one_left
A : Type ?u.19458 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ (a * b)⁻¹ = b⁻¹ * a⁻¹
case a A : Type ?u.19458 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ b⁻¹ * a⁻¹ * (a * b) = 1
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_rev
[254, 1]
[258, 7]
rw [← mul_assoc]
case a A : Type ?u.19458 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ b⁻¹ * a⁻¹ * (a * b) = 1
case a A : Type ?u.19458 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ b⁻¹ * a⁻¹ * a * b = 1
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_rev
[254, 1]
[258, 7]
simp
case a A : Type ?u.19458 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ b⁻¹ * a⁻¹ * a * b = 1
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_eq_iff_eq_mul
[261, 1]
[270, 9]
constructor
A : Type ?u.20667 G : Type inst✝¹ : Ring A inst✝ : Group G a b c : G ⊢ a * b⁻¹ = c ↔ a = c * b
case mp A : Type ?u.20667 G : Type inst✝¹ : Ring A inst✝ : Group G a b c : G ⊢ a * b⁻¹ = c → a = c * b case mpr A : Type ?u.20667 G : Type inst✝¹ : Ring A inst✝ : Group G a b c : G ⊢ a = c * b → a * b⁻¹ = c
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_eq_iff_eq_mul
[261, 1]
[270, 9]
intro h
case mp A : Type ?u.20667 G : Type inst✝¹ : Ring A inst✝ : Group G a b c : G ⊢ a * b⁻¹ = c → a = c * b
case mp A : Type ?u.20667 G : Type inst✝¹ : Ring A inst✝ : Group G a b c : G h : a * b⁻¹ = c ⊢ a = c * b
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_eq_iff_eq_mul
[261, 1]
[270, 9]
apply mul_right_cancel (a := b⁻¹)
case mp A : Type ?u.20667 G : Type inst✝¹ : Ring A inst✝ : Group G a b c : G h : a * b⁻¹ = c ⊢ a = c * b
case mp A : Type ?u.20667 G : Type inst✝¹ : Ring A inst✝ : Group G a b c : G h : a * b⁻¹ = c ⊢ a * b⁻¹ = c * b * b⁻¹
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_eq_iff_eq_mul
[261, 1]
[270, 9]
simp [h]
case mp A : Type ?u.20667 G : Type inst✝¹ : Ring A inst✝ : Group G a b c : G h : a * b⁻¹ = c ⊢ a * b⁻¹ = c * b * b⁻¹
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_eq_iff_eq_mul
[261, 1]
[270, 9]
intro h
case mpr A : Type ?u.20667 G : Type inst✝¹ : Ring A inst✝ : Group G a b c : G ⊢ a = c * b → a * b⁻¹ = c
case mpr A : Type ?u.20667 G : Type inst✝¹ : Ring A inst✝ : Group G a b c : G h : a = c * b ⊢ a * b⁻¹ = c
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_eq_iff_eq_mul
[261, 1]
[270, 9]
rw [h]
case mpr A : Type ?u.20667 G : Type inst✝¹ : Ring A inst✝ : Group G a b c : G h : a = c * b ⊢ a * b⁻¹ = c
case mpr A : Type ?u.20667 G : Type inst✝¹ : Ring A inst✝ : Group G a b c : G h : a = c * b ⊢ c * b * b⁻¹ = c
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_eq_iff_eq_mul
[261, 1]
[270, 9]
simp
case mpr A : Type ?u.20667 G : Type inst✝¹ : Ring A inst✝ : Group G a b c : G h : a = c * b ⊢ c * b * b⁻¹ = c
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_eq_one
[273, 1]
[276, 30]
suffices a * b⁻¹ = 1 ↔ a = 1 * b by simpa using this
A : Type ?u.21692 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a * b⁻¹ = 1 ↔ a = b
A : Type ?u.21692 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a * b⁻¹ = 1 ↔ a = 1 * b
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_eq_one
[273, 1]
[276, 30]
exact mul_inv_eq_iff_eq_mul
A : Type ?u.21692 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a * b⁻¹ = 1 ↔ a = 1 * b
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.mul_inv_eq_one
[273, 1]
[276, 30]
simpa using this
A : Type ?u.21692 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G this : a * b⁻¹ = 1 ↔ a = 1 * b ⊢ a * b⁻¹ = 1 ↔ a = b
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_eq_one
[279, 1]
[289, 9]
constructor
A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a⁻¹ * b = 1 ↔ a = b
case mp A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a⁻¹ * b = 1 → a = b case mpr A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a = b → a⁻¹ * b = 1
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_eq_one
[279, 1]
[289, 9]
intro h
case mp A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a⁻¹ * b = 1 → a = b
case mp A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G h : a⁻¹ * b = 1 ⊢ a = b
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_eq_one
[279, 1]
[289, 9]
have h : a⁻¹ * b⁻¹⁻¹ = 1 := by simp [h]
case mp A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G h : a⁻¹ * b = 1 ⊢ a = b
case mp A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G h✝ : a⁻¹ * b = 1 h : a⁻¹ * b⁻¹⁻¹ = 1 ⊢ a = b
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_eq_one
[279, 1]
[289, 9]
rw [mul_inv_eq_one] at h
case mp A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G h✝ : a⁻¹ * b = 1 h : a⁻¹ * b⁻¹⁻¹ = 1 ⊢ a = b
case mp A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G h✝ : a⁻¹ * b = 1 h : a⁻¹ = b⁻¹ ⊢ a = b
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_eq_one
[279, 1]
[289, 9]
suffices a⁻¹⁻¹ = b⁻¹⁻¹ by simpa using this
case mp A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G h✝ : a⁻¹ * b = 1 h : a⁻¹ = b⁻¹ ⊢ a = b
case mp A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G h✝ : a⁻¹ * b = 1 h : a⁻¹ = b⁻¹ ⊢ a⁻¹⁻¹ = b⁻¹⁻¹
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_eq_one
[279, 1]
[289, 9]
rw [h]
case mp A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G h✝ : a⁻¹ * b = 1 h : a⁻¹ = b⁻¹ ⊢ a⁻¹⁻¹ = b⁻¹⁻¹
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_eq_one
[279, 1]
[289, 9]
simp [h]
A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G h : a⁻¹ * b = 1 ⊢ a⁻¹ * b⁻¹⁻¹ = 1
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_eq_one
[279, 1]
[289, 9]
simpa using this
A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G h✝ : a⁻¹ * b = 1 h : a⁻¹ = b⁻¹ this : a⁻¹⁻¹ = b⁻¹⁻¹ ⊢ a = b
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_eq_one
[279, 1]
[289, 9]
intro h
case mpr A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G ⊢ a = b → a⁻¹ * b = 1
case mpr A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G h : a = b ⊢ a⁻¹ * b = 1
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_eq_one
[279, 1]
[289, 9]
rw [h]
case mpr A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G h : a = b ⊢ a⁻¹ * b = 1
case mpr A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G h : a = b ⊢ b⁻¹ * b = 1
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.inv_mul_eq_one
[279, 1]
[289, 9]
simp
case mpr A : Type ?u.23889 G : Type inst✝¹ : Ring A inst✝ : Group G a b : G h : a = b ⊢ b⁻¹ * b = 1
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.Subgroup.inv_mem_iff
[351, 1]
[357, 18]
constructor
G : Type inst✝ : Group G H : Subgroup G a : G ⊢ a⁻¹ ∈ H ↔ a ∈ H
case mp G : Type inst✝ : Group G H : Subgroup G a : G ⊢ a⁻¹ ∈ H → a ∈ H case mpr G : Type inst✝ : Group G H : Subgroup G a : G ⊢ a ∈ H → a⁻¹ ∈ H
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.Subgroup.inv_mem_iff
[351, 1]
[357, 18]
intro h
case mp G : Type inst✝ : Group G H : Subgroup G a : G ⊢ a⁻¹ ∈ H → a ∈ H
case mp G : Type inst✝ : Group G H : Subgroup G a : G h : a⁻¹ ∈ H ⊢ a ∈ H
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.Subgroup.inv_mem_iff
[351, 1]
[357, 18]
have h := inv_mem h
case mp G : Type inst✝ : Group G H : Subgroup G a : G h : a⁻¹ ∈ H ⊢ a ∈ H
case mp G : Type inst✝ : Group G H : Subgroup G a : G h✝ : a⁻¹ ∈ H h : a⁻¹⁻¹ ∈ H ⊢ a ∈ H
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.Subgroup.inv_mem_iff
[351, 1]
[357, 18]
simpa using h
case mp G : Type inst✝ : Group G H : Subgroup G a : G h✝ : a⁻¹ ∈ H h : a⁻¹⁻¹ ∈ H ⊢ a ∈ H
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture1.lean
Tutorial.Subgroup.inv_mem_iff
[351, 1]
[357, 18]
exact inv_mem
case mpr G : Type inst✝ : Group G H : Subgroup G a : G ⊢ a ∈ H → a⁻¹ ∈ H
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.inv_smul_smul
[53, 1]
[56, 7]
rw [smul_smul]
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G x : X ⊢ a⁻¹ • a • x = x
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G x : X ⊢ (a⁻¹ * a) • x = x
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.inv_smul_smul
[53, 1]
[56, 7]
simp
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G x : X ⊢ (a⁻¹ * a) • x = x
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.smul_inv_smul
[60, 1]
[63, 7]
rw [smul_smul]
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G x : X ⊢ a • a⁻¹ • x = x
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G x : X ⊢ (a * a⁻¹) • x = x
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.smul_inv_smul
[60, 1]
[63, 7]
simp
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G x : X ⊢ (a * a⁻¹) • x = x
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.GroupAction.injective
[67, 1]
[73, 35]
intro x y (h : a • x = a • y)
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G ⊢ Function.Injective fun x => a • x
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G x y : X h : a • x = a • y ⊢ x = y
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.GroupAction.injective
[67, 1]
[73, 35]
calc x = a⁻¹ • a • x := by rw [inv_smul_smul] _ = a⁻¹ • a • y := by rw [h] _ = y := by rw [inv_smul_smul]
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G x y : X h : a • x = a • y ⊢ x = y
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.GroupAction.injective
[67, 1]
[73, 35]
rw [inv_smul_smul]
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G x y : X h : a • x = a • y ⊢ x = a⁻¹ • a • x
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.GroupAction.injective
[67, 1]
[73, 35]
rw [h]
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G x y : X h : a • x = a • y ⊢ a⁻¹ • a • x = a⁻¹ • a • y
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.GroupAction.injective
[67, 1]
[73, 35]
rw [inv_smul_smul]
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G x y : X h : a • x = a • y ⊢ a⁻¹ • a • y = y
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.GroupAction.surjective
[82, 1]
[86, 22]
intro y
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G ⊢ Function.Surjective fun x => a • x
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G y : X ⊢ ∃ a_1, (fun x => a • x) a_1 = y
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.GroupAction.surjective
[82, 1]
[86, 22]
use a⁻¹ • y
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G y : X ⊢ ∃ a_1, (fun x => a • x) a_1 = y
case h G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G y : X ⊢ (fun x => a • x) (a⁻¹ • y) = y
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.GroupAction.surjective
[82, 1]
[86, 22]
exact smul_inv_smul
case h G X : Type inst✝¹ : Group G inst✝ : GroupAction G X a : G y : X ⊢ (fun x => a • x) (a⁻¹ • y) = y
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.orbit_eq_orbit_iff_mem_orbit
[291, 1]
[308, 22]
constructor
G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X ⊢ orbit G x = orbit G y ↔ y ∈ orbit G x
case mp G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X ⊢ orbit G x = orbit G y → y ∈ orbit G x case mpr G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X ⊢ y ∈ orbit G x → orbit G x = orbit G y
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.orbit_eq_orbit_iff_mem_orbit
[291, 1]
[308, 22]
intro h
case mp G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X ⊢ orbit G x = orbit G y → y ∈ orbit G x
case mp G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X h : orbit G x = orbit G y ⊢ y ∈ orbit G x
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.orbit_eq_orbit_iff_mem_orbit
[291, 1]
[308, 22]
rw [h]
case mp G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X h : orbit G x = orbit G y ⊢ y ∈ orbit G x
case mp G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X h : orbit G x = orbit G y ⊢ y ∈ orbit G y
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.orbit_eq_orbit_iff_mem_orbit
[291, 1]
[308, 22]
exists 1
case mp G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X h : orbit G x = orbit G y ⊢ y ∈ orbit G y
case mp G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X h : orbit G x = orbit G y ⊢ 1 • y = y
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.orbit_eq_orbit_iff_mem_orbit
[291, 1]
[308, 22]
simp
case mp G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X h : orbit G x = orbit G y ⊢ 1 • y = y
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.orbit_eq_orbit_iff_mem_orbit
[291, 1]
[308, 22]
intro ⟨a, ha⟩
case mpr G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X ⊢ y ∈ orbit G x → orbit G x = orbit G y
case mpr G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y ⊢ orbit G x = orbit G y
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.orbit_eq_orbit_iff_mem_orbit
[291, 1]
[308, 22]
ext z
case mpr G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y ⊢ orbit G x = orbit G y
case mpr.h G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X ⊢ z ∈ orbit G x ↔ z ∈ orbit G y
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.orbit_eq_orbit_iff_mem_orbit
[291, 1]
[308, 22]
constructor
case mpr.h G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X ⊢ z ∈ orbit G x ↔ z ∈ orbit G y
case mpr.h.mp G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X ⊢ z ∈ orbit G x → z ∈ orbit G y case mpr.h.mpr G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X ⊢ z ∈ orbit G y → z ∈ orbit G x
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.orbit_eq_orbit_iff_mem_orbit
[291, 1]
[308, 22]
intro ⟨b, hb⟩
case mpr.h.mp G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X ⊢ z ∈ orbit G x → z ∈ orbit G y
case mpr.h.mp G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X b : G hb : b • x = z ⊢ z ∈ orbit G y
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.orbit_eq_orbit_iff_mem_orbit
[291, 1]
[308, 22]
rw [← hb, ← ha]
case mpr.h.mp G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X b : G hb : b • x = z ⊢ z ∈ orbit G y
case mpr.h.mp G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X b : G hb : b • x = z ⊢ b • x ∈ orbit G (a • x)
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.orbit_eq_orbit_iff_mem_orbit
[291, 1]
[308, 22]
exists b * a⁻¹
case mpr.h.mp G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X b : G hb : b • x = z ⊢ b • x ∈ orbit G (a • x)
case mpr.h.mp G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X b : G hb : b • x = z ⊢ (b * a⁻¹) • a • x = b • x
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.orbit_eq_orbit_iff_mem_orbit
[291, 1]
[308, 22]
simp [mul_smul]
case mpr.h.mp G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X b : G hb : b • x = z ⊢ (b * a⁻¹) • a • x = b • x
no goals
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.orbit_eq_orbit_iff_mem_orbit
[291, 1]
[308, 22]
intro ⟨b, hb⟩
case mpr.h.mpr G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X ⊢ z ∈ orbit G y → z ∈ orbit G x
case mpr.h.mpr G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X b : G hb : b • y = z ⊢ z ∈ orbit G x
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.orbit_eq_orbit_iff_mem_orbit
[291, 1]
[308, 22]
rw [← hb, ← ha]
case mpr.h.mpr G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X b : G hb : b • y = z ⊢ z ∈ orbit G x
case mpr.h.mpr G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X b : G hb : b • y = z ⊢ b • a • x ∈ orbit G x
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.orbit_eq_orbit_iff_mem_orbit
[291, 1]
[308, 22]
exists b * a
case mpr.h.mpr G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X b : G hb : b • y = z ⊢ b • a • x ∈ orbit G x
case mpr.h.mpr G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X b : G hb : b • y = z ⊢ (b * a) • x = b • a • x
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Algebra/Lecture3.lean
Tutorial.orbit_eq_orbit_iff_mem_orbit
[291, 1]
[308, 22]
simp [mul_smul]
case mpr.h.mpr G X : Type inst✝¹ : Group G inst✝ : GroupAction G X x y : X a : G ha : a • x = y z : X b : G hb : b • y = z ⊢ (b * a) • x = b • a • x
no goals