English
Under a surjective map and a NonUnitalRing structure on the domain, the codomain inherits a NonUnitalRing structure.
Русский
При сюръективном отображении и структуре NonUnitalRing в области определения, кодомоноид получает структуру.
LaTeX
$$$\\exists\\; \\mathcal{S}: \\text{NonUnitalRing}(S)$$$
Lean4
/-- Pushforward a `NonUnitalRing` instance along a surjective function. -/
-- See note [reducible non-instances]
protected abbrev nonUnitalRing [NonUnitalRing R] (zero : f 0 = 0) (add : ∀ x y, f (x + y) = f x + f y)
(mul : ∀ x y, f (x * y) = f x * f y) (neg : ∀ x, f (-x) = -f x) (sub : ∀ x y, f (x - y) = f x - f y)
(nsmul : ∀ (n : ℕ) (x), f (n • x) = n • f x) (zsmul : ∀ (n : ℤ) (x), f (n • x) = n • f x) : NonUnitalRing S
where
toNonUnitalNonAssocRing := hf.nonUnitalNonAssocRing f zero add mul neg sub nsmul zsmul
__ := hf.nonUnitalSemiring f zero add mul nsmul
