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Theorems · Theorem · order theory

le_mul_of_le_one_left

∀ {R : Type u} [inst : Semiring R] [inst_1 : Preorder R] {a b : R} [ExistsAddOfLE R] [MulPosMono R] [AddRightMono R]
  [AddRightReflectLE R], b ≤ 0 → a ≤ 1 → b ≤ a * b

Variant of mul_le_of_le_one_left for b non-positive instead of non-negative.

Defined in
Mathlib.Algebra.Order.Ring.Unbundled.Basic
Cited by
1 results in Mathlib
Foundations
Depth 13 from the axioms · uses no axioms
Assumes
SemiringPreorderExistsAddOfLEMulPosMonoAddRightMonoAddRightReflectLE
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

parallelepiped_single · cited by 1parallelepiped_singleSemiring · cited by 13802SemiringPreorder · cited by 7952Preorderone_mul · cited by 2841one_mulAddRightMono · cited by 367AddRightMonoExistsAddOfLE · cited by 330ExistsAddOfLEMulPosMono · cited by 128MulPosMonoAddRightReflectLE · cited by 53AddRightReflectLEmul_le_mul_of_nonpos_right · cited by 13mul_le_mul_of_nonpos_rightle_mul_of_le_one_leftCITED BYCITES

Cites8

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Cited by1

Results whose statement or proof uses this declaration.