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stringlengths 6
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stringlengths 1
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https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.nestedIntervalSeq_isCauSeq | [214, 1] | [222, 28] | intro ε ε0 | ι : Type
U : ι → Set ℝ
⊢ IsCauSeq abs (nestedIntervalSeq U) | ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
⊢ ∃ i, ∀ j ≥ i, |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.nestedIntervalSeq_isCauSeq | [214, 1] | [222, 28] | have ⟨i, hi⟩ : ∃ i : ℕ, ε⁻¹ < 2 ^ i := pow_unbounded_of_one_lt ε⁻¹ (by linarith) | ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
⊢ ∃ i, ∀ j ≥ i, |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε | ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
i : ℕ
hi : ε⁻¹ < 2 ^ i
⊢ ∃ i, ∀ j ≥ i, |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.nestedIntervalSeq_isCauSeq | [214, 1] | [222, 28] | have hi : (2 ^ i : ℝ)⁻¹ < ε := inv_lt_of_inv_lt ε0 hi | ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
i : ℕ
hi : ε⁻¹ < 2 ^ i
⊢ ∃ i, ∀ j ≥ i, |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε | ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
i : ℕ
hi✝ : ε⁻¹ < 2 ^ i
hi : (2 ^ i)⁻¹ < ε
⊢ ∃ i, ∀ j ≥ i, |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.nestedIntervalSeq_isCauSeq | [214, 1] | [222, 28] | exists i | ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
i : ℕ
hi✝ : ε⁻¹ < 2 ^ i
hi : (2 ^ i)⁻¹ < ε
⊢ ∃ i, ∀ j ≥ i, |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε | ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
i : ℕ
hi✝ : ε⁻¹ < 2 ^ i
hi : (2 ^ i)⁻¹ < ε
⊢ ∀ j ≥ i, |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.nestedIntervalSeq_isCauSeq | [214, 1] | [222, 28] | intro j hj | ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
i : ℕ
hi✝ : ε⁻¹ < 2 ^ i
hi : (2 ^ i)⁻¹ < ε
⊢ ∀ j ≥ i, |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε | ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
i : ℕ
hi✝ : ε⁻¹ < 2 ^ i
hi : (2 ^ i)⁻¹ < ε
j : ℕ
hj : j ≥ i
⊢ |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.nestedIntervalSeq_isCauSeq | [214, 1] | [222, 28] | calc |nestedIntervalSeq U j - nestedIntervalSeq U i|
_ ≤ (2 ^ i : ℝ)⁻¹ := nestedIntervalSeq_isCauSeq_aux' U hj
_ < ε := hi | ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
i : ℕ
hi✝ : ε⁻¹ < 2 ^ i
hi : (2 ^ i)⁻¹ < ε
j : ℕ
hj : j ≥ i
⊢ |nestedIntervalSeq U j - nestedIntervalSeq U i| < ε | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.nestedIntervalSeq_isCauSeq | [214, 1] | [222, 28] | linarith | ι : Type
U : ι → Set ℝ
ε : ℝ
ε0 : ε > 0
⊢ 1 < 2 | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.nestedIntervalSeq_tendsto | [232, 1] | [234, 47] | apply (nestedIntervalCauSeq U).tendsto_limit | ι : Type
U : ι → Set ℝ
⊢ Tendsto (nestedIntervalSeq U) atTop (𝓝 (CauSeq.lim (nestedIntervalCauSeq U))) | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | by_contra H | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
⊢ HasFinSubCover U (Icc 0 1) | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
⊢ False |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | set c := (nestedIntervalCauSeq U).lim | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
⊢ False | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
⊢ False |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | rcases cover (nestedIntervalLim_mem U 0) with ⟨_, ⟨i, rfl⟩, hU' : c ∈ U i⟩ | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
⊢ False | case intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
⊢ False |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | rcases Metric.isOpen_iff.mp (hU i) c hU' with ⟨ε, ε0, hε⟩ | case intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
⊢ False | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
⊢ False |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | have ⟨n, hn⟩ : ∃ n : ℕ, (ε / 2)⁻¹ < 2 ^ n := by
apply pow_unbounded_of_one_lt (ε / 2)⁻¹ (by linarith) | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
⊢ False | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
⊢ False |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | suffices HasFinSubCover U I(n) by
apply nestedInterval_not_HasFinSubCover H n this | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
⊢ False | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
⊢ HasFinSubCover U (Icc (nestedInterval U n).1 (nestedInterval U n).2) |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | suffices I(n) ⊆ U i by
exists {i}
simpa using this | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
⊢ HasFinSubCover U (Icc (nestedInterval U n).1 (nestedInterval U n).2) | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
⊢ Icc (nestedInterval U n).1 (nestedInterval U n).2 ⊆ U i |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | suffices ∀ x, x ∈ I(n) → |x - c| < ε by
intro x hx
apply hε (this x hx) | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
⊢ Icc (nestedInterval U n).1 (nestedInterval U n).2 ⊆ U i | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
⊢ ∀ x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2, |x - c| < ε |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | intro x hx | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
⊢ ∀ x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2, |x - c| < ε | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
⊢ |x - c| < ε |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | have hba : β n - α n = (2 ^ n : ℝ)⁻¹ := nestedInterval_len U n | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
⊢ |x - c| < ε | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
⊢ |x - c| < ε |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | have hn := inv_lt_of_inv_lt (by linarith) hn | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
⊢ |x - c| < ε | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
⊢ |x - c| < ε |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | have hc : α n ≤ c ∧ c ≤ β n := nestedIntervalLim_mem U n | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
⊢ |x - c| < ε | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
⊢ |x - c| < ε |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | have hx : α n ≤ x ∧ x ≤ β n := hx | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
⊢ |x - c| < ε | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx✝ : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
hx : (nestedInterval U n).1 ≤ x ∧ x ≤ (nestedInterval U n).2
⊢ |x - c| < ε |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | calc |x - c|
_ = |(x - α n) - (c - α n)| := by simp
_ ≤ |x - α n| + |c - α n| := by apply abs_sub
_ = (x - α n) + (c - α n) := by
apply congrArg₂ <;> rw [abs_eq_self] <;> linarith
_ < ε / 2 + ε / 2 := by linarith
_ = ε := by ring | case intro.intro.intro.intro.intro
ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx✝ : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
hx : (nestedInterval U n).1 ≤ x ∧ x ≤ (nestedInterval U n).2
⊢ |x - c| < ε | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | apply pow_unbounded_of_one_lt (ε / 2)⁻¹ (by linarith) | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
⊢ ∃ n, (ε / 2)⁻¹ < 2 ^ n | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | linarith | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
⊢ 1 < 2 | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | apply nestedInterval_not_HasFinSubCover H n this | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
this : HasFinSubCover U (Icc (nestedInterval U n).1 (nestedInterval U n).2)
⊢ False | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | exists {i} | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
this : Icc (nestedInterval U n).1 (nestedInterval U n).2 ⊆ U i
⊢ HasFinSubCover U (Icc (nestedInterval U n).1 (nestedInterval U n).2) | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
this : Icc (nestedInterval U n).1 (nestedInterval U n).2 ⊆ U i
⊢ Icc (nestedInterval U n).1 (nestedInterval U n).2 ⊆ ⋃ i_1 ∈ {i}, U i_1 |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | simpa using this | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
this : Icc (nestedInterval U n).1 (nestedInterval U n).2 ⊆ U i
⊢ Icc (nestedInterval U n).1 (nestedInterval U n).2 ⊆ ⋃ i_1 ∈ {i}, U i_1 | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | intro x hx | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
this : ∀ x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2, |x - c| < ε
⊢ Icc (nestedInterval U n).1 (nestedInterval U n).2 ⊆ U i | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
this : ∀ x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2, |x - c| < ε
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
⊢ x ∈ U i |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | apply hε (this x hx) | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
this : ∀ x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2, |x - c| < ε
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
⊢ x ∈ U i | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | linarith | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
⊢ 0 < ε / 2 | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | simp | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx✝ : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
hx : (nestedInterval U n).1 ≤ x ∧ x ≤ (nestedInterval U n).2
⊢ |x - c| = |x - (nestedInterval U n).1 - (c - (nestedInterval U n).1)| | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | apply abs_sub | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx✝ : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
hx : (nestedInterval U n).1 ≤ x ∧ x ≤ (nestedInterval U n).2
⊢ |x - (nestedInterval U n).1 - (c - (nestedInterval U n).1)| ≤
|x - (nestedInterval U n).1| + |c - (nestedInterval U n).1| | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | apply congrArg₂ <;> rw [abs_eq_self] <;> linarith | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx✝ : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
hx : (nestedInterval U n).1 ≤ x ∧ x ≤ (nestedInterval U n).2
⊢ |x - (nestedInterval U n).1| + |c - (nestedInterval U n).1| =
x - (nestedInterval U n).1 + (c - (nestedInterval U n).1) | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | linarith | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx✝ : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
hx : (nestedInterval U n).1 ≤ x ∧ x ≤ (nestedInterval U n).2
⊢ x - (nestedInterval U n).1 + (c - (nestedInterval U n).1) < ε / 2 + ε / 2 | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Analysis/Lecture3.lean | Tutorial.HasFinSubCover_of_Icc | [254, 1] | [291, 21] | ring | ι : Type
U : ι → Set ℝ
hU : ∀ (i : ι), IsOpen (U i)
cover : Icc 0 1 ⊆ ⋃ i, U i
H : ¬HasFinSubCover U (Icc 0 1)
c : ℝ := CauSeq.lim (nestedIntervalCauSeq U)
i : ι
hU' : c ∈ U i
ε : ℝ
ε0 : ε > 0
hε : Metric.ball c ε ⊆ U i
n : ℕ
hn✝ : (ε / 2)⁻¹ < 2 ^ n
x : ℝ
hx✝ : x ∈ Icc (nestedInterval U n).1 (nestedInterval U n).2
hba : (nestedInterval U n).2 - (nestedInterval U n).1 = (2 ^ n)⁻¹
hn : (2 ^ n)⁻¹ < ε / 2
hc : (nestedInterval U n).1 ≤ c ∧ c ≤ (nestedInterval U n).2
hx : (nestedInterval U n).1 ≤ x ∧ x ≤ (nestedInterval U n).2
⊢ ε / 2 + ε / 2 = ε | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Tutorial/Advanced/Category/Lecture2.lean | Tutorial.Category.Initial.uniq' | [28, 1] | [30, 25] | sorry | C : Type u
inst✝ : Category C
a : C
h : Initial a
b : C
f g : Hom a b
⊢ f = h.fromInitial b | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Tutorial/Advanced/Category/Lecture2.lean | Tutorial.Category.Initial.uniq' | [28, 1] | [30, 25] | sorry | C : Type u
inst✝ : Category C
a : C
h : Initial a
b : C
f g : Hom a b
⊢ h.fromInitial b = g | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Tutorial/Advanced/Category/Lecture2.lean | Tutorial.Coequalizer.hom_id | [309, 1] | [309, 71] | cases i <;> rfl | J : Type u₁
inst✝¹ : Category J
C : Type u₂
inst✝ : Category C
F : Functor J C
i : Shape
⊢ ShapeHom.id i = 𝟙 i | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Injective.comp | [38, 1] | [48, 11] | rw [Injective] | X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
⊢ Injective (g ∘ f) | X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
⊢ ∀ {x₁ x₂ : X}, (g ∘ f) x₁ = (g ∘ f) x₂ → x₁ = x₂ |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Injective.comp | [38, 1] | [48, 11] | intro x₁ x₂ hgf | X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
⊢ ∀ {x₁ x₂ : X}, (g ∘ f) x₁ = (g ∘ f) x₂ → x₁ = x₂ | X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
x₁ x₂ : X
hgf : (g ∘ f) x₁ = (g ∘ f) x₂
⊢ x₁ = x₂ |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Injective.comp | [38, 1] | [48, 11] | have hf := hginj hgf | X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
x₁ x₂ : X
hgf : (g ∘ f) x₁ = (g ∘ f) x₂
⊢ x₁ = x₂ | X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
x₁ x₂ : X
hgf : (g ∘ f) x₁ = (g ∘ f) x₂
hf : f x₁ = f x₂
⊢ x₁ = x₂ |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Injective.comp | [38, 1] | [48, 11] | apply hfinj | X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
x₁ x₂ : X
hgf : (g ∘ f) x₁ = (g ∘ f) x₂
hf : f x₁ = f x₂
⊢ x₁ = x₂ | case a
X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
x₁ x₂ : X
hgf : (g ∘ f) x₁ = (g ∘ f) x₂
hf : f x₁ = f x₂
⊢ f x₁ = f x₂ |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Injective.comp | [38, 1] | [48, 11] | apply hf | case a
X Y Z : Type
f : X → Y
g : Y → Z
hfinj : Injective f
hginj : Injective g
x₁ x₂ : X
hgf : (g ∘ f) x₁ = (g ∘ f) x₂
hf : f x₁ = f x₂
⊢ f x₁ = f x₂ | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Injective.of_comp | [78, 1] | [89, 10] | rw [Injective] | X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
⊢ Injective f | X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
⊢ ∀ {x₁ x₂ : X}, f x₁ = f x₂ → x₁ = x₂ |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Injective.of_comp | [78, 1] | [89, 10] | intro x₁ x₂ hf | X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
⊢ ∀ {x₁ x₂ : X}, f x₁ = f x₂ → x₁ = x₂ | X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
⊢ x₁ = x₂ |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Injective.of_comp | [78, 1] | [89, 10] | have h : g (f x₁) = g (f x₂) := by
rw [hf] | X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
⊢ x₁ = x₂ | X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
h : g (f x₁) = g (f x₂)
⊢ x₁ = x₂ |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Injective.of_comp | [78, 1] | [89, 10] | apply hgfinj | X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
h : g (f x₁) = g (f x₂)
⊢ x₁ = x₂ | case a
X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
h : g (f x₁) = g (f x₂)
⊢ (g ∘ f) x₁ = (g ∘ f) x₂ |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Injective.of_comp | [78, 1] | [89, 10] | simp | case a
X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
h : g (f x₁) = g (f x₂)
⊢ (g ∘ f) x₁ = (g ∘ f) x₂ | case a
X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
h : g (f x₁) = g (f x₂)
⊢ g (f x₁) = g (f x₂) |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Injective.of_comp | [78, 1] | [89, 10] | apply h | case a
X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
h : g (f x₁) = g (f x₂)
⊢ g (f x₁) = g (f x₂) | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Injective.of_comp | [78, 1] | [89, 10] | rw [hf] | X Y Z : Type
f : X → Y
g : Y → Z
hgfinj : Injective (g ∘ f)
x₁ x₂ : X
hf : f x₁ = f x₂
⊢ g (f x₁) = g (f x₂) | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Surjective.comp | [137, 1] | [148, 14] | rw [Surjective] | X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
⊢ Surjective (g ∘ f) | X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
⊢ ∀ (y : Z), ∃ x, (g ∘ f) x = y |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Surjective.comp | [137, 1] | [148, 14] | intro z | X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
⊢ ∀ (y : Z), ∃ x, (g ∘ f) x = y | X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
⊢ ∃ x, (g ∘ f) x = z |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Surjective.comp | [137, 1] | [148, 14] | have ⟨y, hy⟩ := hgsurj z | X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
⊢ ∃ x, (g ∘ f) x = z | X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
y : Y
hy : g y = z
⊢ ∃ x, (g ∘ f) x = z |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Surjective.comp | [137, 1] | [148, 14] | have ⟨x, hx⟩ := hfsurj y | X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
y : Y
hy : g y = z
⊢ ∃ x, (g ∘ f) x = z | X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
y : Y
hy : g y = z
x : X
hx : f x = y
⊢ ∃ x, (g ∘ f) x = z |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Surjective.comp | [137, 1] | [148, 14] | exists x | X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
y : Y
hy : g y = z
x : X
hx : f x = y
⊢ ∃ x, (g ∘ f) x = z | X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
y : Y
hy : g y = z
x : X
hx : f x = y
⊢ (g ∘ f) x = z |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Surjective.comp | [137, 1] | [148, 14] | simp | X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
y : Y
hy : g y = z
x : X
hx : f x = y
⊢ (g ∘ f) x = z | X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
y : Y
hy : g y = z
x : X
hx : f x = y
⊢ g (f x) = z |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Surjective.comp | [137, 1] | [148, 14] | rw [hx, hy] | X Y Z : Type
f : X → Y
g : Y → Z
hfsurj : Surjective f
hgsurj : Surjective g
z : Z
y : Y
hy : g y = z
x : X
hx : f x = y
⊢ g (f x) = z | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Surjective.of_comp | [160, 1] | [164, 13] | intro z | X Y Z : Type
f : X → Y
g : Y → Z
h : Surjective (g ∘ f)
⊢ Surjective g | X Y Z : Type
f : X → Y
g : Y → Z
h : Surjective (g ∘ f)
z : Z
⊢ ∃ x, g x = z |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Surjective.of_comp | [160, 1] | [164, 13] | have ⟨x, hx⟩ := h z | X Y Z : Type
f : X → Y
g : Y → Z
h : Surjective (g ∘ f)
z : Z
⊢ ∃ x, g x = z | X Y Z : Type
f : X → Y
g : Y → Z
h : Surjective (g ∘ f)
z : Z
x : X
hx : (g ∘ f) x = z
⊢ ∃ x, g x = z |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Basic/Lecture4.lean | Tutorial.Surjective.of_comp | [160, 1] | [164, 13] | exists f x | X Y Z : Type
f : X → Y
g : Y → Z
h : Surjective (g ∘ f)
z : Z
x : X
hx : (g ∘ f) x = z
⊢ ∃ x, g x = z | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.map_one | [45, 1] | [51, 28] | have h : f 1 * f 1 = f 1 * 1 := by
rw [← map_mul, mul_one, mul_one] | G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
⊢ f 1 = 1 | G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
h : f 1 * f 1 = f 1 * 1
⊢ f 1 = 1 |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.map_one | [45, 1] | [51, 28] | exact mul_left_cancel _ h | G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
h : f 1 * f 1 = f 1 * 1
⊢ f 1 = 1 | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.map_one | [45, 1] | [51, 28] | rw [← map_mul, mul_one, mul_one] | G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
⊢ f 1 * f 1 = f 1 * 1 | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.map_inv | [57, 1] | [60, 40] | apply eq_inv_of_mul_eq_one_left | G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
a : G₁
⊢ f a⁻¹ = (f a)⁻¹ | case a
G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
a : G₁
⊢ f a⁻¹ * f a = 1 |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.map_inv | [57, 1] | [60, 40] | rw [← map_mul, inv_mul_self, map_one] | case a
G₁ G₂ : Type
inst✝¹ : Group G₁
inst✝ : Group G₂
f : G₁ →* G₂
a : G₁
⊢ f a⁻¹ * f a = 1 | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.injective_iff_map_eq_one | [219, 1] | [230, 31] | constructor | G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ Function.Injective ⇑f ↔ ∀ (a : G₁), f a = 1 → a = 1 | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ Function.Injective ⇑f → ∀ (a : G₁), f a = 1 → a = 1
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ (∀ (a : G₁), f a = 1 → a = 1) → Function.Injective ⇑f |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.injective_iff_map_eq_one | [219, 1] | [230, 31] | intro hf a ha | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ Function.Injective ⇑f → ∀ (a : G₁), f a = 1 → a = 1 | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hf : Function.Injective ⇑f
a : G₁
ha : f a = 1
⊢ a = 1 |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.injective_iff_map_eq_one | [219, 1] | [230, 31] | apply hf | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hf : Function.Injective ⇑f
a : G₁
ha : f a = 1
⊢ a = 1 | case mp.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hf : Function.Injective ⇑f
a : G₁
ha : f a = 1
⊢ f a = f 1 |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.injective_iff_map_eq_one | [219, 1] | [230, 31] | rw [ha, map_one] | case mp.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hf : Function.Injective ⇑f
a : G₁
ha : f a = 1
⊢ f a = f 1 | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.injective_iff_map_eq_one | [219, 1] | [230, 31] | intro h a b hab | case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ (∀ (a : G₁), f a = 1 → a = 1) → Function.Injective ⇑f | case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a b : G₁
hab : f a = f b
⊢ a = b |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.injective_iff_map_eq_one | [219, 1] | [230, 31] | rw [← mul_inv_eq_one] at * | case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a b : G₁
hab : f a = f b
⊢ a = b | case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a b : G₁
hab : f a * (f b)⁻¹ = 1
⊢ a * b⁻¹ = 1 |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.injective_iff_map_eq_one | [219, 1] | [230, 31] | apply h | case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a b : G₁
hab : f a * (f b)⁻¹ = 1
⊢ a * b⁻¹ = 1 | case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a b : G₁
hab : f a * (f b)⁻¹ = 1
⊢ f (a * b⁻¹) = 1 |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.injective_iff_map_eq_one | [219, 1] | [230, 31] | rw [map_mul, map_inv, hab] | case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a b : G₁
hab : f a * (f b)⁻¹ = 1
⊢ f (a * b⁻¹) = 1 | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.ker_eq_bot | [236, 1] | [253, 23] | rw [injective_iff_map_eq_one] | G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ ker f = ⊥ ↔ Function.Injective ⇑f | G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ ker f = ⊥ ↔ ∀ (a : G₁), f a = 1 → a = 1 |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.ker_eq_bot | [236, 1] | [253, 23] | constructor | G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ ker f = ⊥ ↔ ∀ (a : G₁), f a = 1 → a = 1 | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ ker f = ⊥ → ∀ (a : G₁), f a = 1 → a = 1
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ (∀ (a : G₁), f a = 1 → a = 1) → ker f = ⊥ |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.ker_eq_bot | [236, 1] | [253, 23] | intro h a (hf : a ∈ f.ker) | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ ker f = ⊥ → ∀ (a : G₁), f a = 1 → a = 1 | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ker f = ⊥
a : G₁
hf : a ∈ ker f
⊢ a = 1 |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.ker_eq_bot | [236, 1] | [253, 23] | simpa [h] using hf | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ker f = ⊥
a : G₁
hf : a ∈ ker f
⊢ a = 1 | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.ker_eq_bot | [236, 1] | [253, 23] | intro h | case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ (∀ (a : G₁), f a = 1 → a = 1) → ker f = ⊥ | case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
⊢ ker f = ⊥ |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.ker_eq_bot | [236, 1] | [253, 23] | ext a | case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
⊢ ker f = ⊥ | case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
⊢ a ∈ ker f ↔ a ∈ ⊥ |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.ker_eq_bot | [236, 1] | [253, 23] | simp only [mem_ker, mem_bot] | case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
⊢ a ∈ ker f ↔ a ∈ ⊥ | case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
⊢ f a = 1 ↔ a = 1 |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.ker_eq_bot | [236, 1] | [253, 23] | constructor | case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
⊢ f a = 1 ↔ a = 1 | case mpr.a.mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
⊢ f a = 1 → a = 1
case mpr.a.mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
⊢ a = 1 → f a = 1 |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.ker_eq_bot | [236, 1] | [253, 23] | intro ha | case mpr.a.mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
⊢ f a = 1 → a = 1 | case mpr.a.mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
ha : f a = 1
⊢ a = 1 |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.ker_eq_bot | [236, 1] | [253, 23] | apply h | case mpr.a.mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
ha : f a = 1
⊢ a = 1 | case mpr.a.mp.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
ha : f a = 1
⊢ f a = 1 |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.ker_eq_bot | [236, 1] | [253, 23] | apply ha | case mpr.a.mp.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
ha : f a = 1
⊢ f a = 1 | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.ker_eq_bot | [236, 1] | [253, 23] | intro ha | case mpr.a.mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
⊢ a = 1 → f a = 1 | case mpr.a.mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
ha : a = 1
⊢ f a = 1 |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.ker_eq_bot | [236, 1] | [253, 23] | rw [ha, map_one] | case mpr.a.mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
h : ∀ (a : G₁), f a = 1 → a = 1
a : G₁
ha : a = 1
⊢ f a = 1 | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.range_eq_top | [257, 1] | [273, 18] | constructor | G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ range f = ⊤ ↔ Function.Surjective ⇑f | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ range f = ⊤ → Function.Surjective ⇑f
case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ Function.Surjective ⇑f → range f = ⊤ |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.range_eq_top | [257, 1] | [273, 18] | intro hrange y | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ range f = ⊤ → Function.Surjective ⇑f | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
⊢ ∃ a, f a = y |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.range_eq_top | [257, 1] | [273, 18] | have hy : y ∈ (⊤ : Subgroup G₂) := by
simp | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
⊢ ∃ a, f a = y | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
hy : y ∈ ⊤
⊢ ∃ a, f a = y |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.range_eq_top | [257, 1] | [273, 18] | rw [← hrange] at hy | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
hy : y ∈ ⊤
⊢ ∃ a, f a = y | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
hy : y ∈ range f
⊢ ∃ a, f a = y |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.range_eq_top | [257, 1] | [273, 18] | have ⟨x, hx⟩ := hy | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
hy : y ∈ range f
⊢ ∃ a, f a = y | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
hy : y ∈ range f
x : G₁
hx : f x = y
⊢ ∃ a, f a = y |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.range_eq_top | [257, 1] | [273, 18] | exists x | case mp
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
hy : y ∈ range f
x : G₁
hx : f x = y
⊢ ∃ a, f a = y | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.range_eq_top | [257, 1] | [273, 18] | simp | G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hrange : range f = ⊤
y : G₂
⊢ y ∈ ⊤ | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.range_eq_top | [257, 1] | [273, 18] | intro hsurj | case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
⊢ Function.Surjective ⇑f → range f = ⊤ | case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hsurj : Function.Surjective ⇑f
⊢ range f = ⊤ |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.range_eq_top | [257, 1] | [273, 18] | ext y | case mpr
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hsurj : Function.Surjective ⇑f
⊢ range f = ⊤ | case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hsurj : Function.Surjective ⇑f
y : G₂
⊢ y ∈ range f ↔ y ∈ ⊤ |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.range_eq_top | [257, 1] | [273, 18] | simp only [mem_range, mem_top, iff_true] | case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hsurj : Function.Surjective ⇑f
y : G₂
⊢ y ∈ range f ↔ y ∈ ⊤ | case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hsurj : Function.Surjective ⇑f
y : G₂
⊢ ∃ a, f a = y |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.GroupHom.range_eq_top | [257, 1] | [273, 18] | apply hsurj y | case mpr.a
G₁ G₂ G : Type
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : Group G
f : G₁ →* G₂
hsurj : Function.Surjective ⇑f
y : G₂
⊢ ∃ a, f a = y | no goals |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.homToPerm_injective | [346, 1] | [360, 15] | rw [injective_iff_map_eq_one] | G : Type
inst✝ : Group G
⊢ Function.Injective ⇑(homToPerm G) | G : Type
inst✝ : Group G
⊢ ∀ (a : G), (homToPerm G) a = 1 → a = 1 |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.homToPerm_injective | [346, 1] | [360, 15] | intro a h | G : Type
inst✝ : Group G
⊢ ∀ (a : G), (homToPerm G) a = 1 → a = 1 | G : Type
inst✝ : Group G
a : G
h : (homToPerm G) a = 1
⊢ a = 1 |
https://github.com/yuma-mizuno/lean-math-workshop.git | 4a69b0130b276b45212e2b12b90032b146b56d67 | Solution/Advanced/Algebra/Lecture2.lean | Tutorial.homToPerm_injective | [346, 1] | [360, 15] | calc
a = a * 1 := by simp
_ = (homToPerm G a) 1 := by
rfl
_ = 1 := by
simp [h] | G : Type
inst✝ : Group G
a : G
h : (homToPerm G) a = 1
⊢ a = 1 | no goals |
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