Theorem: mderivEquiv_eq (Ray/AnalyticManifold/OneDimension.lean)

State before:
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
f0 : f ≠ 0
⊢ ↑(mderivEquiv f f0) = f
Tactic:
apply ContinuousLinearMap.ext
State after:
case h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
f0 : f ≠ 0
⊢ ∀ (x : TangentSpace I z), ↑(mderivEquiv f f0) x = f x

State before:
case h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
f0 : f ≠ 0
⊢ ∀ (x : TangentSpace I z), ↑(mderivEquiv f f0) x = f x
Tactic:
intro x
State after:
case h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
f0 : f ≠ 0
x : TangentSpace I z
⊢ ↑(mderivEquiv f f0) x = f x

State before:
case h
S : Type
inst✝⁵ : TopologicalSpace S
cs : ChartedSpace ℂ S
inst✝⁴ : AnalyticManifold I S
T : Type
inst✝³ : TopologicalSpace T
ct : ChartedSpace ℂ T
inst✝² : AnalyticManifold I T
U : Type
inst✝¹ : TopologicalSpace U
cu : ChartedSpace ℂ U
inst✝ : AnalyticManifold I U
z : S
w : T
f : TangentSpace I z →L[ℂ] TangentSpace I w
f0 : f ≠ 0
x : TangentSpace I z
⊢ ↑(mderivEquiv f f0) x = f x
Tactic:
rfl
State after:
no goals