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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
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golf
Deicyde Apr 13, 2026
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11 changes: 11 additions & 0 deletions Mathlib/Topology/FiberBundle/Trivialization.lean
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Expand Up @@ -592,6 +592,17 @@ 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

/-- For fixed `v ∈ F`, `x ↦ e.symm (x,v)` is continuous at any point in the base set -/
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@grunweg grunweg Apr 15, 2026

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Suggested change
/-- For fixed `v ∈ F`, `x ↦ e.symm (x,v)` is continuous at any point in the base set -/
/-- 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 (e.symm ∘ (·, v)) b :=
ContinuousAt.comp (e.continuousOn_symm.continuousAt
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@grunweg grunweg Apr 15, 2026

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Can you extract e.continuousAt_symm as a separate lemma? (Topology/Homotopy/Lifting also would use that lemma)

(e.open_target.mem_nhds (e.mem_target.mpr hb))) (continuousAt_id.prodMk continuousAt_const)
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@grunweg grunweg Apr 15, 2026

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Two more small golfs:

  • the last argument can be by fun_prop
  • you can use dot notation: (....).comp (by fun_prop)

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@grunweg grunweg Apr 15, 2026

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Actually, I wonder if (fun x ↦ e.symm (x, v)) is a better statement here. (That matches how the other statements are written.)


/-- For fixed `v ∈ F`, `x ↦ e.symm (x,v)` is continuous on `e.baseSet` -/
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@grunweg grunweg Apr 15, 2026

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Suggested change
/-- For fixed `v ∈ F`, `x ↦ e.symm (x,v)` is continuous on `e.baseSet` -/
/-- For fixed `v ∈ F`, `x ↦ e.symm (x,v)` is continuous on `e.baseSet`. -/

theorem continuousOn_symm_prodMk_left {v : F} :
ContinuousOn (e.symm ∘ (·, v)) e.baseSet :=
fun _ hb => (e.continuousAt_symm_prodMk_left hb).continuousWithinAt
Comment on lines +602 to +604
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@grunweg grunweg Apr 15, 2026

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Minor reformatting, might be a matter of taste:

Suggested change
theorem continuousOn_symm_prodMk_left {v : F} :
ContinuousOn (e.symm ∘ (·, v)) e.baseSet :=
fun _ hb => (e.continuousAt_symm_prodMk_left hb).continuousWithinAt
theorem continuousOn_symm_prodMk_left {v : F} : ContinuousOn (e.symm ∘ (·, 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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