prod — Mathlib

English

The cartesian product of two homomorphisms f₁ and f₂ induces a homomorphism f₁.prodMap f₂ from A×C to B×D, by composing with projections fst and snd.

Русский

Картезианское произведение гомоморфизмов порождает гомоморфизм f₁.prodMap f₂: A×C → B×D, заданный композициями с проекциями fst и snd.

LaTeX

$$def prodMap {D} (f₁ : A →A[R] B) (f₂ : C →A[R] D) : A × C →A[R] B × D := (f₁.comp (fst R A C)).prod (f₂.comp (snd R A C))$$

Lean4

/-- The Cartesian product of two continuous algebra morphisms as a continuous algebra morphism. -/
protected def prod (f₁ : A →A[R] B) (f₂ : A →A[R] C) : A →A[R] B × C :=
  ⟨(f₁ : A →ₐ[R] B).prod f₂, f₁.2.prodMk f₂.2⟩

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