English
Let f: R → S be surjective and suppose R has a NonUnitalCommRing structure. Then S inherits a NonUnitalCommRing structure transported along f.
Русский
Пусть f: R → S — суръекция и R имеет структуру NonUnitalCommRing. Тогда S наследует структуру NonUnitalCommRing, перенесённую по f.
LaTeX
$$$\text{Pushforward}_{f}({\text{NonUnitalCommRing}}\,R) = {\text{NonUnitalCommRing}}\,S.$$$
Lean4
/-- Pushforward a `NonUnitalCommRing` instance along a surjective function. -/
-- See note [reducible non-instances]
protected abbrev nonUnitalCommRing [NonUnitalCommRing 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) : NonUnitalCommRing S
where
toNonUnitalRing := hf.nonUnitalRing f zero add mul neg sub nsmul zsmul
__ := hf.nonUnitalNonAssocCommRing f zero add mul neg sub nsmul zsmul
