map_dotProduct — Mathlib

English

Let R,S be non-associative semirings and f:R→+*S a ring homomorphism. For vectors v,w: n→R, f(v ⬝ᵥ w) = (f∘v) ⬝ᵥ (f∘w).

Русский

Пусть R,S — полусSemiкола; f: R → S гомоморф. Для векторов v,w: n→R верно: f(v ⬝ᵥ w) = (f∘v) ⬝ᵥ (f∘w).

LaTeX

$$$\\text{Let } f: R \\to S \\text{ be a ring homomorphism and } v,w: n \\to R.\\quad f(v \\ ⬝ᵥ w) = (f\\circ v) \\ ⬝ᵥ (f\\circ w).$$$

Lean4

theorem map_dotProduct [NonAssocSemiring R] [NonAssocSemiring S] (f : R →+* S) (v w : n → R) :
    f (v ⬝ᵥ w) = f ∘ v ⬝ᵥ f ∘ w := by simp only [dotProduct, map_sum f, f.map_mul, Function.comp]

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