lift_inv_comp_left — Mathlib

English

A variant of the left-inverse lifting identity: lift (f ≫ ι) f ≫ μ equals toUnit ≫ η.

Русский

Вариант левой инверсной подстановки: lift (f ≫ ι) f ≫ μ = toUnit ≫ η.

LaTeX

$$$\mathrm{lift}(f \circ \iota) f \circ \mu = \mathrm{toUnit} \circ \eta$$$

Lean4

@[reassoc]
theorem lift_inv_comp_left [GrpObj A] [GrpObj B] (f : A ⟶ B) [IsMonHom f] : lift (ι ≫ f) f ≫ μ = toUnit _ ≫ η :=
  by
  have := left_inv A =≫ f
  rwa [assoc, IsMonHom.mul_hom, assoc, IsMonHom.one_hom, lift_map_assoc, id_comp] at this

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