fiberPullbackEquiv_symm_fst_apply

English

The fiberPullbackEquiv equals a composite of canonical pullback isomorphisms.

Русский

Эквив персона ФиберPullback равен композиции канонических изоморфизмов pullback.

LaTeX

$$$\\mathrm{fiberPullbackEquiv}(F,f,g) = (\\mathrm{PreservesPullback.iso}(F\\circ \\mathrm{incl})\,f\,g) \\;\\circ\\; (\\mathrm{Types.pullbackIsoPullback}(F.mapf)(F.mapg)).toEquiv$$$

Lean4

@[simp]
theorem fiberPullbackEquiv_symm_fst_apply {X A B : C} {f : A ⟶ X} {g : B ⟶ X} (a : F.obj A) (b : F.obj B)
    (h : F.map f a = F.map g b) : F.map (pullback.fst f g) ((fiberPullbackEquiv F f g).symm ⟨(a, b), h⟩) = a :=
  by
  simp [fiberPullbackEquiv]
  change ((Types.pullbackIsoPullback _ _).inv ≫ _ ≫ (F ⋙ FintypeCat.incl).map (pullback.fst f g)) _ = _
  erw [PreservesPullback.iso_inv_fst, Types.pullbackIsoPullback_inv_fst]

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