mkInv_comp_mk — Mathlib

English

The inverse of f composed with mk_W f equals the identity on X: (mkInv f) ∘ (mk_W f) = mk_W(id_X).

Русский

Обратный к f, композиция с mk_W f даёт тождественный на X: (mkInv f) ∘ (mk_W f) = mk_W(id_X).

LaTeX

$$$(\mathrm{mkInv}\,f\,hf) \cdot (\mathrm{mk}\,W\,f) = \mathrm{mk}\,W\,(\mathrm{id}_X)$$$

Lean4

@[simp]
theorem mkInv_comp_mk [HasSmallLocalizedHom.{w} W X X] [HasSmallLocalizedHom.{w} W X Y] [HasSmallLocalizedHom.{w} W Y X]
    (f : Y ⟶ X) (hf : W f) : (mkInv f hf).comp (mk W f) = mk W (𝟙 X) :=
  (equiv W W.Q).injective (by simp [equiv_comp])

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