Theorems · Theorem · ring theory
AddMonoidHom.map_mul_iff
∀ {R : Type u_1} {S : Type u_2} [inst : NonUnitalNonAssocSemiring R] [inst_1 : NonUnitalNonAssocSemiring S]
(f : R →+ S), (∀ (x y : R), f (x * y) = f x * f y) ↔ AddMonoidHom.mul.compr₂ f = (AddMonoidHom.mul.comp f).compl₂ fAn AddMonoidHom preserves multiplication if pre- and post- composition with
mul are equivalent. By converting the statement into an equality of
AddMonoidHoms, this lemma allows various specialized ext lemmas about →+ to then be applied.
- Defined in
- Mathlib.Algebra.Ring.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 27 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- AddMonoidHomstatement and proof · cited by 3,230
- NonUnitalNonAssocSemiringstatement and proof · cited by 1,081
- AddMonoidHom.compstatement · cited by 339
- AddMonoidHom.mulstatement · cited by 7
- AddMonoidHom.compl₂statement · cited by 4
- AddMonoidHom.compr₂statement · cited by 4
- AddMonoidHom.ext_iff₂proof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- AddMonoidAlgebra.toDirectSum_mulproof · cited by 1