Theorem: Graph.SubgroupCover.Quot.toFuncE_bar (GroupComp/Graph/SubgroupCover.lean)

State before:
V : Type u
E : Type v
inst✝¹ : DecidableEq V
inst✝ : DecidableEq E
G : Graph V E
x₀ : V
H : Subgroup (π₁ G x₀)
⊢ ∀ (ebar : Quotient (edgeSetoid H)), toFuncE H ((groupCover H).bar ebar) = G.bar (toFuncE H ebar)
Tactic:
apply Quotient.ind
State after:
case a
V : Type u
E : Type v
inst✝¹ : DecidableEq V
inst✝ : DecidableEq E
G : Graph V E
x₀ : V
H : Subgroup (π₁ G x₀)
⊢ ∀ (a : Edge G x₀), toFuncE H ((groupCover H).bar ⟦a⟧) = G.bar (toFuncE H ⟦a⟧)

State before:
case a
V : Type u
E : Type v
inst✝¹ : DecidableEq V
inst✝ : DecidableEq E
G : Graph V E
x₀ : V
H : Subgroup (π₁ G x₀)
⊢ ∀ (a : Edge G x₀), toFuncE H ((groupCover H).bar ⟦a⟧) = G.bar (toFuncE H ⟦a⟧)
Tactic:
intro e
State after:
case a
V : Type u
E : Type v
inst✝¹ : DecidableEq V
inst✝ : DecidableEq E
G : Graph V E
x₀ : V
H : Subgroup (π₁ G x₀)
e : Edge G x₀
⊢ toFuncE H ((groupCover H).bar ⟦e⟧) = G.bar (toFuncE H ⟦e⟧)

State before:
case a
V : Type u
E : Type v
inst✝¹ : DecidableEq V
inst✝ : DecidableEq E
G : Graph V E
x₀ : V
H : Subgroup (π₁ G x₀)
e : Edge G x₀
⊢ toFuncE H ((groupCover H).bar ⟦e⟧) = G.bar (toFuncE H ⟦e⟧)
Tactic:
rw [toFuncE_defn, bar_eq_barn']
State after:
case a
V : Type u
E : Type v
inst✝¹ : DecidableEq V
inst✝ : DecidableEq E
G : Graph V E
x₀ : V
H : Subgroup (π₁ G x₀)
e : Edge G x₀
⊢ toFuncE H ⟦(univ G x₀).bar e⟧ = G.bar e.nxt.edge

State before:
case a
V : Type u
E : Type v
inst✝¹ : DecidableEq V
inst✝ : DecidableEq E
G : Graph V E
x₀ : V
H : Subgroup (π₁ G x₀)
e : Edge G x₀
⊢ toFuncE H ⟦(univ G x₀).bar e⟧ = G.bar e.nxt.edge
Tactic:
apply (proj G x₀).toFuncE_bar
State after:
no goals