Theorem: groupCohomology.isOneCoboundary_of_oneCoboundaries
File: Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean

tactic: rcases mem_range_of_mem_oneCoboundaries f.2 with ⟨x, hx⟩
state_before: k✝ G✝ : Type u
inst✝⁷ : CommRing k✝
inst✝⁶ : Group G✝
A✝ : Rep k✝ G✝
k G A : Type u
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup A
inst✝² : Module k A
inst✝¹ : DistribMulAction G A
inst✝ : SMulCommClass G k A
f : ↥(oneCoboundaries (Rep.ofDistribMulAction k G A))
⊢ IsOneCoboundary ↑↑f
state_after: case intro
k✝ G✝ : Type u
inst✝⁷ : CommRing k✝
inst✝⁶ : Group G✝
A✝ : Rep k✝ G✝
k G A : Type u
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup A
inst✝² : Module k A
inst✝¹ : DistribMulAction G A
inst✝ : SMulCommClass G k A
f : ↥(oneCoboundaries (Rep.ofDistribMulAction k G A))
x : CoeSort.coe (Rep.ofDistribMulAction k G A)
hx : (dZero (Rep.ofDistribMulAction k G A)) x = ↑↑f
⊢ IsOneCoboundary ↑↑f

tactic: exact ⟨x, by rw [← hx]; intro g; rfl⟩
state_before: case intro
k✝ G✝ : Type u
inst✝⁷ : CommRing k✝
inst✝⁶ : Group G✝
A✝ : Rep k✝ G✝
k G A : Type u
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup A
inst✝² : Module k A
inst✝¹ : DistribMulAction G A
inst✝ : SMulCommClass G k A
f : ↥(oneCoboundaries (Rep.ofDistribMulAction k G A))
x : CoeSort.coe (Rep.ofDistribMulAction k G A)
hx : (dZero (Rep.ofDistribMulAction k G A)) x = ↑↑f
⊢ IsOneCoboundary ↑↑f
state_after: no goals

tactic: rw [← hx]
state_before: k✝ G✝ : Type u
inst✝⁷ : CommRing k✝
inst✝⁶ : Group G✝
A✝ : Rep k✝ G✝
k G A : Type u
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup A
inst✝² : Module k A
inst✝¹ : DistribMulAction G A
inst✝ : SMulCommClass G k A
f : ↥(oneCoboundaries (Rep.ofDistribMulAction k G A))
x : CoeSort.coe (Rep.ofDistribMulAction k G A)
hx : (dZero (Rep.ofDistribMulAction k G A)) x = ↑↑f
⊢ ∀ (g : G), g • x - x = ↑↑f g
state_after: k✝ G✝ : Type u
inst✝⁷ : CommRing k✝
inst✝⁶ : Group G✝
A✝ : Rep k✝ G✝
k G A : Type u
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup A
inst✝² : Module k A
inst✝¹ : DistribMulAction G A
inst✝ : SMulCommClass G k A
f : ↥(oneCoboundaries (Rep.ofDistribMulAction k G A))
x : CoeSort.coe (Rep.ofDistribMulAction k G A)
hx : (dZero (Rep.ofDistribMulAction k G A)) x = ↑↑f
⊢ ∀ (g : G), g • x - x = (dZero (Rep.ofDistribMulAction k G A)) x g

tactic: intro g
state_before: k✝ G✝ : Type u
inst✝⁷ : CommRing k✝
inst✝⁶ : Group G✝
A✝ : Rep k✝ G✝
k G A : Type u
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup A
inst✝² : Module k A
inst✝¹ : DistribMulAction G A
inst✝ : SMulCommClass G k A
f : ↥(oneCoboundaries (Rep.ofDistribMulAction k G A))
x : CoeSort.coe (Rep.ofDistribMulAction k G A)
hx : (dZero (Rep.ofDistribMulAction k G A)) x = ↑↑f
⊢ ∀ (g : G), g • x - x = (dZero (Rep.ofDistribMulAction k G A)) x g
state_after: k✝ G✝ : Type u
inst✝⁷ : CommRing k✝
inst✝⁶ : Group G✝
A✝ : Rep k✝ G✝
k G A : Type u
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup A
inst✝² : Module k A
inst✝¹ : DistribMulAction G A
inst✝ : SMulCommClass G k A
f : ↥(oneCoboundaries (Rep.ofDistribMulAction k G A))
x : CoeSort.coe (Rep.ofDistribMulAction k G A)
hx : (dZero (Rep.ofDistribMulAction k G A)) x = ↑↑f
g : G
⊢ g • x - x = (dZero (Rep.ofDistribMulAction k G A)) x g

tactic: rfl
state_before: k✝ G✝ : Type u
inst✝⁷ : CommRing k✝
inst✝⁶ : Group G✝
A✝ : Rep k✝ G✝
k G A : Type u
inst✝⁵ : CommRing k
inst✝⁴ : Group G
inst✝³ : AddCommGroup A
inst✝² : Module k A
inst✝¹ : DistribMulAction G A
inst✝ : SMulCommClass G k A
f : ↥(oneCoboundaries (Rep.ofDistribMulAction k G A))
x : CoeSort.coe (Rep.ofDistribMulAction k G A)
hx : (dZero (Rep.ofDistribMulAction k G A)) x = ↑↑f
g : G
⊢ g • x - x = (dZero (Rep.ofDistribMulAction k G A)) x g
state_after: no goals