Theorems · Theorem · field theory
Real.sSup_nonneg
∀ {s : Set ℝ}, (∀ x ∈ s, 0 ≤ x) → 0 ≤ sSup sAs sSup s = 0 when s is a set of reals that's either empty or unbounded above,
it suffices to show that all elements of s are nonnegative to show that 0 ≤ sSup s.
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 118 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Realstatement and proof · cited by 25,697
- Set.Nonemptyproof · cited by 2,627
- SupSet.sSupstatement · cited by 954
- Eq.geproof · cited by 375
- Set.eq_empty_or_nonemptyproof · cited by 248
- Real.sSup_emptyproof · cited by 17
- Real.sSup_nonneg'proof · cited by 2
Cited by4
Results whose statement or proof uses this declaration.
- Real.iSup_nonnegproof · cited by 6
- NNReal.coe_sSupproof · cited by 5
- Complex.HadamardThreeLines.sSupNormIm_nonnegproof · cited by 5
- Real.tendstoLocallyUniformlyOn_rpow_sub_one_logproof · cited by 2