Theorems · Theorem · order theory
smul_le_smul_iff_of_pos_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] [PosSMulMono α β] [PosSMulReflectLE α β], 0 < a → (a • b₁ ≤ a • b₂ ↔ b₁ ≤ b₂)- Defined in
- Mathlib.Algebra.Order.Module.Defs
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 7 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
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
- LT.lt.leproof · cited by 2,189
- PosSMulMonostatement and proof · cited by 188
- PosSMulReflectLEstatement and proof · cited by 51
- smul_le_smul_of_nonneg_leftproof · cited by 47
- le_of_smul_le_smul_leftproof · cited by 6
Cited by7
Results whose statement or proof uses this declaration.
- le_inv_smul_iff_of_posproof · cited by 5
- inv_smul_le_iff_of_posproof · cited by 3
- ConvexOn.exists_ge_of_centerMassproof · cited by 2
- smul_le_smul_iff_of_neg_leftproof · cited by 2
- lineMap_le_lineMap_iff_of_ltproof · cited by 2
- IsMinOn.of_isLocalMinOn_of_convexOn_Iccproof · cited by 1
- CStarModule.inner_mul_inner_swap_leproof · cited by 1