Theorems · Definition · commutative algebra

Function.Surjective.addCommMonoidWithOne

{R : Type u_1} →
  {S : Type u_2} →
    (f : R → S) →
      Function.Surjective f →
        [inst : Add S] →
          [inst_1 : Zero S] →
            [inst_2 : One S] →
              [inst_3 : SMul ℕ S] →
                [inst_4 : NatCast S] →
                  [inst_5 : AddCommMonoidWithOne R] →
                    f 0 = 0 →
                      f 1 = 1 →
                        (∀ (x y : R), f (x + y) = f x + f y) →
                          (∀ (n : ℕ) (x : R), f (n • x) = n • f x) → (∀ (n : ℕ), f ↑n = ↑n) → AddCommMonoidWithOne S

A type endowed with 0, 1 and + is an additive monoid with one, if it admits a surjective map that preserves 0, 1 and * from an additive monoid with one. See note [reducible non-instances].

Defined in

Mathlib.Algebra.Ring.InjSurj

Cited by

0 results in Mathlib

Foundations

Depth 15 from the axioms · uses propext

Assumes

AddZeroOneSMulNatCastAddCommMonoidWithOne