Theorems · Theorem · order theory
le_of_smul_le_smul_left
∀ {α : Type u_1} {β : Type u_2} {a : α} {b₁ b₂ : β} [inst : SMul α β] [inst_1 : Preorder α] [inst_2 : Preorder β]
[inst_3 : Zero α] [PosSMulReflectLE α β], a • b₁ ≤ a • b₂ → 0 < a → b₁ ≤ b₂- Defined in
- Mathlib.Algebra.Order.Module.Defs
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 5 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites3
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Preorderstatement and proof · cited by 7,952
- PosSMulReflectLEstatement and proof · cited by 51
- PosSMulReflectLE.le_of_smul_le_smul_leftproof · cited by 1
Cited by6
Results whose statement or proof uses this declaration.
- smul_le_smul_iff_of_pos_leftproof · cited by 7
- le_of_smul_le_smul_of_pos_leftproof · cited by 3
- SpectrumRestricts.smul_of_nonnegproof · cited by 1
- PosSMulReflectLE.toPosSMulStrictMonoproof · cited by 1
- PosSMulReflectLE.liftproof · cited by 0
- floorDiv_le_ceilDivproof · cited by 0