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Theorems · Theorem · order theory

nsmul_nonneg_iff

∀ {M : Type u_3} [inst : AddMonoid M] [inst_1 : LinearOrder M] [AddLeftMono M] {x : M} {n : ℕ},
  n ≠ 0 → (0 ≤ n • x ↔ 0 ≤ x)
Defined in
Mathlib.Algebra.Order.Monoid.Unbundled.Pow
Cited by
2 results in Mathlib
Foundations
Depth 17 from the axioms · uses propext
Assumes
AddMonoidLinearOrderAddLeftMono
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Around this declaration

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

nsmul_nonpos_iff · cited by 1nsmul_nonpos_iffnsmul_neg_iff · cited by 1nsmul_neg_iffLinearOrder · cited by 8572LinearOrderAddMonoid · cited by 2864AddMonoidAddLeftMono · cited by 687AddLeftMononsmul_nonneg · cited by 9nsmul_nonnegle_imp_le_of_lt_imp_lt · cited by 3le_imp_le_of_lt_imp_ltnsmul_neg · cited by 2nsmul_negnsmul_nonneg_iffCITED BYCITES

Cites6

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by2

Results whose statement or proof uses this declaration.