Theorem: Trivialization.symm_apply_eq_mk_continuousLinearEquivAt_symm
File: Mathlib/Topology/VectorBundle/Basic.lean

tactic: have h : (b, z) ∈ e.target
state_before: R : Type u_4
B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝⁹ : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace E)
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
z : F
⊢ ↑(LocalHomeomorph.symm e.toLocalHomeomorph) (b, z) =
    totalSpaceMk b (↑(ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b hb)) z)

tactic: apply e.injOn (e.map_target h)
state_before: R : Type u_4
B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝⁹ : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace E)
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
z : F
h : (b, z) ∈ e.target
⊢ ↑(LocalHomeomorph.symm e.toLocalHomeomorph) (b, z) =
    totalSpaceMk b (↑(ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b hb)) z)

tactic: rw [e.target_eq]
state_before: case h
R : Type u_4
B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝⁹ : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace E)
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
z : F
⊢ (b, z) ∈ e.target

tactic: exact ⟨hb, mem_univ _⟩
state_before: case h
R : Type u_4
B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝⁹ : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace E)
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
z : F
⊢ (b, z) ∈ e.baseSet ×ˢ univ

tactic: simpa only [e.source_eq, mem_preimage]
state_before: case a
R : Type u_4
B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝⁹ : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace E)
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
z : F
h : (b, z) ∈ e.target
⊢ totalSpaceMk b (↑(ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b hb)) z) ∈ e.source

tactic: simp_rw [e.right_inv h, coe_coe, e.apply_eq_prod_continuousLinearEquivAt R b hb,
  ContinuousLinearEquiv.apply_symm_apply]
state_before: case a
R : Type u_4
B : Type u_1
F : Type u_2
E : B → Type u_3
inst✝⁹ : NontriviallyNormedField R
inst✝⁸ : (x : B) → AddCommMonoid (E x)
inst✝⁷ : (x : B) → Module R (E x)
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace R F
inst✝⁴ : TopologicalSpace B
inst✝³ : TopologicalSpace (TotalSpace E)
inst✝² : (x : B) → TopologicalSpace (E x)
inst✝¹ : FiberBundle F E
e : Trivialization F TotalSpace.proj
inst✝ : Trivialization.IsLinear R e
b : B
hb : b ∈ e.baseSet
z : F
h : (b, z) ∈ e.target
⊢ ↑e.toLocalHomeomorph (↑(LocalHomeomorph.symm e.toLocalHomeomorph) (b, z)) =
    ↑e.toLocalHomeomorph (totalSpaceMk b (↑(ContinuousLinearEquiv.symm (continuousLinearEquivAt R e b hb)) z))