Documentation

Mathlib.Algebra.Ring.Invertible

Theorems about invertible elements in rings #

source
def invertibleNeg {R : Type u} [Mul R] [One R] [HasDistribNeg R] (a : R) [Invertible a] :
Invertible (-a)

-⅟a is the inverse of -a

Equations
    Instances For
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      @[simp]
      theorem invOf_neg {R : Type u} [Monoid R] [HasDistribNeg R] (a : R) [Invertible a] [Invertible (-a)] :
      ⅟ (-a) = -⅟ a
      source
      @[simp]
      theorem one_sub_invOf_two {R : Type u} [Ring R] [Invertible 2] :
      1 - ⅟ 2 = ⅟ 2
      source
      @[simp]
      theorem invOf_two_add_invOf_two {R : Type u} [NonAssocSemiring R] [Invertible 2] :
      ⅟ 2 + ⅟ 2 = 1
      source
      theorem pos_of_invertible_cast {R : Type u} [NonAssocSemiring R] [Nontrivial R] (n : ℕ) [Invertible ↑n] :
      0 < n
      source
      theorem invOf_add_invOf {R : Type u} [Semiring R] (a b : R) [Invertible a] [Invertible b] :
      ⅟ a + ⅟ b = ⅟ a * (a + b) * ⅟ b
      source
      theorem invOf_sub_invOf {R : Type u} [Ring R] (a b : R) [Invertible a] [Invertible b] :
      ⅟ a - ⅟ b = ⅟ a * (b - a) * ⅟ b

      A version of inv_sub_inv' for invOf.

      source
      theorem Ring.inverse_add_inverse {R : Type u} [Semiring R] {a b : R} (h : IsUnit a ↔ IsUnit b) :
      inverse a + inverse b = inverse a * (a + b) * inverse b

      A version of inv_add_inv' for Ring.inverse.

      source
      theorem Ring.inverse_sub_inverse {R : Type u} [Ring R] {a b : R} (h : IsUnit a ↔ IsUnit b) :
      inverse a - inverse b = inverse a * (b - a) * inverse b

      A version of inv_sub_inv' for Ring.inverse.