Documentation

Mathlib.Algebra.Order.Invertible

Lemmas about invOf in ordered (semi)rings. #

source
@[simp]
theorem invOf_pos {α : Type u_1} [Semiring α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} [Invertible a] :
0 < ⅟ a ↔ 0 < a
source
@[simp]
theorem invOf_nonpos {α : Type u_1} [Semiring α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} [Invertible a] :
⅟ a ≤ 0 ↔ a ≤ 0
source
@[simp]
theorem invOf_nonneg {α : Type u_1} [Semiring α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} [Invertible a] :
0 ≤ ⅟ a ↔ 0 ≤ a
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@[simp]
theorem invOf_lt_zero {α : Type u_1} [Semiring α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} [Invertible a] :
⅟ a < 0 ↔ a < 0
source
@[simp]
theorem invOf_le_one {α : Type u_1} [Semiring α] [LinearOrder α] [IsStrictOrderedRing α] {a : α} [Invertible a] (h : 1 ≤ a) :
⅟ a ≤ 1
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theorem pos_invOf_of_invertible_cast {α : Type u_1} [Semiring α] [LinearOrder α] [IsStrictOrderedRing α] [Nontrivial α] (n : ℕ) [Invertible ↑n] :
0 < ⅟ ↑n