identify_one — Mathlib

English

The injection of the unit from any A_i into the free product sends the unit 1 to the unit 1 in the free product.

Русский

Инъекция единицы из A_i в свободное произведение отправляет единицу в единицу во free product.

LaTeX

$$$\iota'_R A (DirectSum.lof\;R\;I\;A\;i\; 1) = 1$$$

Lean4

/-- The injection into the free product of any `1 : A i` is the 1 of the free product. -/
theorem identify_one (i : I) : ι' R A (DirectSum.lof R I A i 1) = 1 :=
  by
  suffices ι' R A (DirectSum.lof R I A i 1) = mkAlgHom R A 1 by simpa
  exact RingQuot.mkAlgHom_rel R <| rel_id R A (i := i)

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