Theorem: Monoid.PushoutI.Reduced.eq_empty_of_mem_range
File: Mathlib/GroupTheory/PushoutI.lean

Proof steps:
state_before: ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
w : Word G
hw : Reduced φ w
h : ofCoprodI w.prod ∈ (base φ).range
⊢ w = Word.empty
state_after: case intro
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
w : Word G
hw : Reduced φ w
h : ofCoprodI w.prod ∈ (base φ).range
d : Transversal φ
⊢ w = Word.empty
tactic: rcases transversal_nonempty φ hφ with ⟨d⟩
state_before: case intro
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
w : Word G
hw : Reduced φ w
h : ofCoprodI w.prod ∈ (base φ).range
d : Transversal φ
⊢ w = Word.empty
state_after: case intro.intro.intro
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
w : Word G
hw : Reduced φ w
h : ofCoprodI w.prod ∈ (base φ).range
d : Transversal φ
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
⊢ w = Word.empty
tactic: rcases hw.exists_normalWord_prod_eq d with ⟨w', hw'prod, hw'map⟩
state_before: case intro.intro.intro
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
w : Word G
hw : Reduced φ w
h : ofCoprodI w.prod ∈ (base φ).range
d : Transversal φ
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
⊢ w = Word.empty
state_after: case intro.intro.intro.intro
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
w : Word G
hw : Reduced φ w
d : Transversal φ
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
h : H
heq : (base φ) h = ofCoprodI w.prod
⊢ w = Word.empty
tactic: rcases h with ⟨h, heq⟩
state_before: case intro.intro.intro.intro
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
w : Word G
hw : Reduced φ w
d : Transversal φ
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
h : H
heq : (base φ) h = ofCoprodI w.prod
⊢ w = Word.empty
state_after: case intro.intro.intro.intro
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
w : Word G
hw : Reduced φ w
d : Transversal φ
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
h : H
heq : (base φ) h = ofCoprodI w.prod
this : { toWord := Word.empty, head := h, normalized := ⋯ }.prod = (base φ) h
⊢ w = Word.empty
tactic: have : (NormalWord.prod (d := d) ⟨.empty, h, by simp⟩) = base φ h := by
    simp [NormalWord.prod]
state_before: case intro.intro.intro.intro
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
w : Word G
hw : Reduced φ w
d : Transversal φ
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
h : H
heq : (base φ) h = ofCoprodI w.prod
this : { toWord := Word.empty, head := h, normalized := ⋯ }.prod = (base φ) h
⊢ w = Word.empty
state_after: case intro.intro.intro.intro
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
w : Word G
hw : Reduced φ w
d : Transversal φ
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
h : H
heq : { toWord := Word.empty, head := h, normalized := ⋯ }.prod = w'.prod
this : { toWord := Word.empty, head := h, normalized := ⋯ }.prod = (base φ) h
⊢ w = Word.empty
tactic: rw [← hw'prod, ← this] at heq
state_before: case intro.intro.intro.intro
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
w : Word G
hw : Reduced φ w
d : Transversal φ
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
h : H
heq : { toWord := Word.empty, head := h, normalized := ⋯ }.prod = w'.prod
this : { toWord := Word.empty, head := h, normalized := ⋯ }.prod = (base φ) h
⊢ w = Word.empty
state_after: case intro.intro.intro.intro
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
w : Word G
hw : Reduced φ w
d : Transversal φ
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
h : H
heq : { toWord := Word.empty, head := h, normalized := ⋯ }.prod = w'.prod
this : { toWord := Word.empty, head := h, normalized := ⋯ }.prod = (base φ) h
⊢ w'.toWord = Word.empty
tactic: suffices w'.toWord = .empty by
    simp [this, @eq_comm _ []] at hw'map
    ext
    simp [hw'map]
state_before: case intro.intro.intro.intro
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
hφ : ∀ (i : ι), Injective ⇑(φ i)
w : Word G
hw : Reduced φ w
d : Transversal φ
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
h : H
heq : { toWord := Word.empty, head := h, normalized := ⋯ }.prod = w'.prod
this : { toWord := Word.empty, head := h, normalized := ⋯ }.prod = (base φ) h
⊢ w'.toWord = Word.empty
state_after: no goals
tactic: rw [← prod_injective heq]