Theorem: Subgroup.IsComplement.equiv_fst_eq_iff_leftCosetEquivalence
File: Mathlib/GroupTheory/Complement.lean

tactic: rw [LeftCosetEquivalence, leftCoset_eq_iff]
state_before: G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
⊢ (hSK.equiv g₁).1 = (hSK.equiv g₂).1 ↔ LeftCosetEquivalence (↑K) g₁ g₂
state_after: G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
⊢ (hSK.equiv g₁).1 = (hSK.equiv g₂).1 ↔ g₁⁻¹ * g₂ ∈ K

tactic: constructor
state_before: G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
⊢ (hSK.equiv g₁).1 = (hSK.equiv g₂).1 ↔ g₁⁻¹ * g₂ ∈ K
state_after: case mp
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
⊢ (hSK.equiv g₁).1 = (hSK.equiv g₂).1 → g₁⁻¹ * g₂ ∈ K

case mpr
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
⊢ g₁⁻¹ * g₂ ∈ K → (hSK.equiv g₁).1 = (hSK.equiv g₂).1

tactic: intro h
state_before: case mp
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
⊢ (hSK.equiv g₁).1 = (hSK.equiv g₂).1 → g₁⁻¹ * g₂ ∈ K
state_after: case mp
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
h : (hSK.equiv g₁).1 = (hSK.equiv g₂).1
⊢ g₁⁻¹ * g₂ ∈ K

tactic: rw [← hSK.equiv_fst_mul_equiv_snd g₂, ← hSK.equiv_fst_mul_equiv_snd g₁, ← h,
  mul_inv_rev, ← mul_assoc, inv_mul_cancel_right, ← coe_inv, ← coe_mul]
state_before: case mp
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
h : (hSK.equiv g₁).1 = (hSK.equiv g₂).1
⊢ g₁⁻¹ * g₂ ∈ K
state_after: case mp
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
h : (hSK.equiv g₁).1 = (hSK.equiv g₂).1
⊢ ↑((hSK.equiv g₁).2⁻¹ * (hSK.equiv g₂).2) ∈ K

tactic: exact Subtype.property _
state_before: case mp
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
h : (hSK.equiv g₁).1 = (hSK.equiv g₂).1
⊢ ↑((hSK.equiv g₁).2⁻¹ * (hSK.equiv g₂).2) ∈ K
state_after: no goals

tactic: intro h
state_before: case mpr
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
⊢ g₁⁻¹ * g₂ ∈ K → (hSK.equiv g₁).1 = (hSK.equiv g₂).1
state_after: case mpr
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (hSK.equiv g₁).1 = (hSK.equiv g₂).1

tactic: apply (mem_leftTransversals_iff_existsUnique_inv_mul_mem.1 hSK g₁).unique
state_before: case mpr
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (hSK.equiv g₁).1 = (hSK.equiv g₂).1
state_after: case mpr.py₁
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (↑(hSK.equiv g₁).1)⁻¹ * g₁ ∈ ↑K

case mpr.py₂
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (↑(hSK.equiv g₂).1)⁻¹ * g₁ ∈ ↑K

tactic: rw [equiv_fst_eq_mul_inv]
state_before: case mpr.py₁
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (↑(hSK.equiv g₁).1)⁻¹ * g₁ ∈ ↑K
state_after: case mpr.py₁
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (g₁ * (↑(hSK.equiv g₁).2)⁻¹)⁻¹ * g₁ ∈ ↑K

tactic: simp
state_before: case mpr.py₁
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (g₁ * (↑(hSK.equiv g₁).2)⁻¹)⁻¹ * g₁ ∈ ↑K
state_after: no goals

tactic: rw [SetLike.mem_coe, ← mul_mem_cancel_right h]
state_before: case mpr.py₂
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (↑(hSK.equiv g₂).1)⁻¹ * g₁ ∈ ↑K
state_after: case mpr.py₂
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (↑(hSK.equiv g₂).1)⁻¹ * g₁ * (g₁⁻¹ * g₂) ∈ K

tactic: rw [equiv_fst_eq_mul_inv]
state_before: case mpr.py₂
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (↑(hSK.equiv g₂).1)⁻¹ * g₁ * (g₁⁻¹ * g₂) ∈ K
state_after: case mpr.py₂
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (g₂ * (↑(hSK.equiv g₂).2)⁻¹)⁻¹ * g₁ * (g₁⁻¹ * g₂) ∈ K

tactic: simp [equiv_fst_eq_mul_inv, ← mul_assoc]
state_before: case mpr.py₂
G : Type u_1
inst✝ : Group G
H K : Subgroup G
S T : Set G
hST : IsComplement S T
hHT : IsComplement (↑H) T
hSK : IsComplement S ↑K
g₁ g₂ : G
h : g₁⁻¹ * g₂ ∈ K
⊢ (g₂ * (↑(hSK.equiv g₂).2)⁻¹)⁻¹ * g₁ * (g₁⁻¹ * g₂) ∈ K
state_after: no goals