Theorems · Definition · ring theory

Function.Injective.idemSemiring

{α : Type u_1} →
  {β : Type u_2} →
    [inst : IdemSemiring α] →
      [inst_1 : LE β] →
        [inst_2 : LT β] →
          [inst_3 : Zero β] →
            [inst_4 : One β] →
              [inst_5 : Add β] →
                [inst_6 : Mul β] →
                  [inst_7 : Pow β ℕ] →
                    [inst_8 : SMul ℕ β] →
                      [inst_9 : NatCast β] →
                        [inst_10 : Max β] →
                          [inst_11 : Bot β] →
                            (f : β → α) →
                              Function.Injective f →
                                (∀ {x y : β}, f x ≤ f y ↔ x ≤ y) →
                                  (∀ {x y : β}, f x < f y ↔ x < y) →
                                    f 0 = 0 →
                                      f 1 = 1 →
                                        (∀ (x y : β), f (x + y) = f x + f y) →
                                          (∀ (x y : β), f (x * y) = f x * f y) →
                                            (∀ (n : ℕ) (x : β), f (n • x) = n • f x) →
                                              (∀ (x : β) (n : ℕ), f (x ^ n) = f x ^ n) →
                                                (∀ (n : ℕ), f ↑n = ↑n) →
                                                  (∀ (a b : β), f (a ⊔ b) = f a ⊔ f b) → f ⊥ = ⊥ → IdemSemiring β

Pullback an IdemSemiring instance along an injective function.

Defined in

Mathlib.Algebra.Order.Kleene

Cited by

0 results in Mathlib

Foundations

Depth 20 from the axioms · uses propext

Assumes

IdemSemiringLELTZeroOneAddMulPowSMulNatCastMaxBot