Theorems · Theorem · ring theory
RingHom.map_mul
∀ {α : Type u_2} {β : Type u_3} {x : NonAssocSemiring α} {x_1 : NonAssocSemiring β} (f : α →+* β) (a b : α),
f (a * b) = f a * f bRing homomorphisms preserve multiplication.
- Defined in
- Mathlib.Algebra.Ring.Hom.Defs
- Cited by
- 45 results in Mathlib
- Foundations
- Depth 17 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- RingHomstatement and proof · cited by 10,189
- map_mulproof · cited by 1,137
- NonAssocSemiringstatement and proof · cited by 805
Cited by45
Results whose statement or proof uses this declaration.
- Polynomial.C_mulproof · cited by 35
- MvPolynomial.coeff_mapproof · cited by 13
- Algebra.adjoin_eq_spanproof · cited by 13
- Polynomial.eval₂_at_applyproof · cited by 13
- WittVector.frobenius_verschiebungproof · cited by 7
- Polynomial.C_mul_dvdproof · cited by 4
- Valuation.Integers.dvd_of_leproof · cited by 4
- IsLocalization.exist_integer_multiplesproof · cited by 4
- MvPolynomial.eval₂_comp_rightproof · cited by 3
- Matrix.adjugate_mul_distribproof · cited by 3
- Valuation.Integers.one_of_isUnit'proof · cited by 3
- IsLocalization.isInteger_smulproof · cited by 2