Theorems · Definition · commutative algebra

Function.Injective.ring

{R : Type u_1} →
  {S : Type u_2} →
    (f : S → R) →
      Function.Injective f →
        [inst : Add S] →
          [inst_1 : Mul S] →
            [inst_2 : Zero S] →
              [inst_3 : One S] →
                [inst_4 : Neg S] →
                  [inst_5 : Sub S] →
                    [inst_6 : SMul ℕ S] →
                      [inst_7 : SMul ℤ S] →
                        [inst_8 : Pow S ℕ] →
                          [inst_9 : NatCast S] →
                            [inst_10 : IntCast S] →
                              [inst_11 : Ring R] →
                                f 0 = 0 →
                                  f 1 = 1 →
                                    (∀ (x y : S), f (x + y) = f x + f y) →
                                      (∀ (x y : S), f (x * y) = f x * f y) →
                                        (∀ (x : S), f (-x) = -f x) →
                                          (∀ (x y : S), f (x - y) = f x - f y) →
                                            (∀ (n : ℕ) (x : S), f (n • x) = n • f x) →
                                              (∀ (n : ℤ) (x : S), f (n • x) = n • f x) →
                                                (∀ (x : S) (n : ℕ), f (x ^ n) = f x ^ n) →
                                                  (∀ (n : ℕ), f ↑n = ↑n) → (∀ (n : ℤ), f ↑n = ↑n) → Ring S

Pullback a Ring instance along an injective function.

Defined in

Mathlib.Algebra.Ring.InjSurj

Cited by

0 results in Mathlib

Foundations

Depth 22 from the axioms · uses propext

Assumes

AddMulZeroOneNegSubSMulSMulPowNatCastIntCastRing