Theorems · Theorem · order theory

smul_nonneg_iff_neg_imp_nonpos

∀ {α : Type u_1} {β : Type u_2} [inst : Ring α] [inst_1 : LinearOrder α] [IsStrictOrderedRing α]
  [inst_3 : AddCommGroup β] [inst_4 : LinearOrder β] [IsOrderedAddMonoid β] [inst_6 : Module α β]
  [PosSMulStrictMono α β] {a : α} {b : β}, 0 ≤ a • b ↔ (a < 0 → b ≤ 0) ∧ (b < 0 → a ≤ 0)

Defined in

Mathlib.Algebra.Order.Module.Defs

Cited by

0 results in Mathlib

Foundations

Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingLinearOrderIsStrictOrderedRingAddCommGroupLinearOrderIsOrderedAddMonoidModulePosSMulStrictMono