Theorems · Definition · commutative algebra

Function.Injective.addCommMonoidWithOne

{R : Type u_1} →
  {S : Type u_3} →
    [inst : Zero S] →
      [inst_1 : One S] →
        [inst_2 : Add S] →
          [inst_3 : SMul ℕ S] →
            [inst_4 : NatCast S] →
              [inst_5 : AddCommMonoidWithOne R] →
                (f : S → R) →
                  Function.Injective f →
                    f 0 = 0 →
                      f 1 = 1 →
                        (∀ (x y : S), f (x + y) = f x + f y) →
                          (∀ (n : ℕ) (x : S), f (n • x) = n • f x) → (∀ (n : ℕ), f ↑n = ↑n) → AddCommMonoidWithOne S

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

Defined in

Mathlib.Algebra.Ring.InjSurj

Cited by

0 results in Mathlib

Foundations

Depth 16 from the axioms · uses no axioms

Assumes

ZeroOneAddSMulNatCastAddCommMonoidWithOne