Theorems · Definition · commutative algebra

Function.Surjective.addCommGroupWithOne

{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 : Neg S] →
                [inst_4 : Sub S] →
                  [inst_5 : SMul ℕ S] →
                    [inst_6 : SMul ℤ S] →
                      [inst_7 : NatCast S] →
                        [inst_8 : IntCast S] →
                          [inst_9 : AddCommGroupWithOne R] →
                            f 0 = 0 →
                              f 1 = 1 →
                                (∀ (x y : R), f (x + y) = f x + f y) →
                                  (∀ (x : R), f (-x) = -f x) →
                                    (∀ (x y : R), f (x - y) = f x - f y) →
                                      (∀ (n : ℕ) (x : R), f (n • x) = n • f x) →
                                        (∀ (n : ℤ) (x : R), f (n • x) = n • f x) →
                                          (∀ (n : ℕ), f ↑n = ↑n) → (∀ (n : ℤ), f ↑n = ↑n) → AddCommGroupWithOne S

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

Defined in

Mathlib.Algebra.Ring.InjSurj

Cited by

0 results in Mathlib

Foundations

Depth 22 from the axioms · uses propext

Assumes

AddZeroOneNegSubSMulSMulNatCastIntCastAddCommGroupWithOne