eqToHom_comp_ι — Mathlib

English

If j = j', then the projection composed with the equality-induced isomorphism equals the projection on the same index.

Русский

Если j = j', то проекция, композиция с единичной изоморфностью по равенству, равна проекции на тот же индекс.

LaTeX

$$$\forall J\, f\; [HasBiproduct f] \; (w: j=j') \; biproduct.π\, f\, j \; ≫ \; eqToHom (by simp [w]) = biproduct.π\, f\, j'$$$

Lean4

@[reassoc (attr := simp)]
theorem eqToHom_comp_ι (f : J → C) [HasBiproduct f] {j j' : J} (w : j = j') :
    eqToHom (by simp [w]) ≫ biproduct.ι f j' = biproduct.ι f j :=
  by
  cases w
  simp
    -- TODO?: simp can prove this using `eqToHom_naturality`
    -- but `eqToHom_naturality` applies less easily than this lemma

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