Theorems · Theorem · order theory
mul_nonpos_iff_pos_imp_nonpos
∀ {R : Type u} [inst : Ring R] [inst_1 : LinearOrder R] {a b : R} [PosMulStrictMono R] [MulPosStrictMono R]
[AddLeftMono R] [AddLeftReflectLE R], a * b ≤ 0 ↔ (0 < a → b ≤ 0) ∧ (b < 0 → 0 ≤ a)- Cited by
- 0 results in Mathlib
- Foundations
- Depth 20 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- LinearOrderstatement and proof · cited by 8,572
- Ringstatement and proof · cited by 7,463
- AddLeftMonostatement and proof · cited by 687
- mul_negproof · cited by 590
- PosMulStrictMonostatement and proof · cited by 151
- AddLeftReflectLEstatement and proof · cited by 119
- MulPosStrictMonostatement and proof · cited by 94
- neg_nonnegproof · cited by 60
- mul_nonneg_iff_pos_imp_nonnegproof · cited by 3
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