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feat(Topology/FiberBundle): continuousAt_symm_prodMk_left #38020

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feat(Topology/FiberBundle): continuousAt_symm_prodMk_left #38020

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6d439df
Added continuousAt_symm_prodMk_left.
Deicyde Apr 13, 2026
1b08856
Chaned theorem statements to make prodMk_left more explicit.
Deicyde Apr 13, 2026
c3aef65
Made theorem statement simpler by removing explicit .toOpenPartialHom…
Deicyde Apr 13, 2026
a7cc26b
Shorted docstrings
Deicyde Apr 13, 2026
b1250be
golf
Deicyde Apr 13, 2026
cfd5320
Fixed grunweg comments
Deicyde Apr 16, 2026
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13 changes: 13 additions & 0 deletions Mathlib/Topology/FiberBundle/Trivialization.lean
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Expand Up @@ -592,6 +592,19 @@ theorem continuousAt_proj (ex : x ∈ e.source) : ContinuousAt proj x :=
theorem continuousOn_proj : ContinuousOn proj e.source :=
continuousOn_of_forall_continuousAt fun _ ↦ e.continuousAt_proj

/-- The inverse of a trivialization is continuous at any point in the target. -/
theorem continuousAt_symm {p : B × F} (hp : p ∈ e.target) : ContinuousAt e.symm p :=
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@grunweg grunweg Apr 16, 2026

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I was thinking of OpenPartialHomeomorph.continuousAt_symm instead --- that will also apply inside Homotopy/Lifting. (And doing that within this PR is fine with me.)

e.continuousOn_symm.continuousAt (e.open_target.mem_nhds hp)

/-- For fixed `v ∈ F`, `x ↦ e.symm (x,v)` is continuous at any point in the base set. -/
theorem continuousAt_symm_prodMk_left {b : B} {v : F} (hb : b ∈ e.baseSet) :
ContinuousAt (fun x ↦ e.symm (x, v)) b :=
(e.continuousAt_symm (e.mem_target.mpr hb)).comp (by fun_prop)

/-- For fixed `v ∈ F`, `x ↦ e.symm (x,v)` is continuous on `e.baseSet`. -/
theorem continuousOn_symm_prodMk_left {v : F} : ContinuousOn (fun x ↦ e.symm (x, v)) e.baseSet :=
fun _ hb ↦ (e.continuousAt_symm_prodMk_left hb).continuousWithinAt

/-- Pre-composition of a `Bundle.Trivialization` and a `Homeomorph`. -/
protected def compHomeomorph {Z' : Type*} [TopologicalSpace Z'] (h : Z' ≃ₜ Z) :
Trivialization F (proj ∘ h) where
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