Theorems · Theorem · functional analysis
balanced_iff_neg_mem
∀ {E : Type u_3} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {s : Set E},
Convex ℝ s → (Balanced ℝ s ↔ ∀ ⦃x : E⦄, x ∈ s → -x ∈ s)- Defined in
- Mathlib.Analysis.LocallyConvex.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 157 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddCommGroupModule
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Cites18
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
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- Norm.normproof · cited by 5,413
- Nat.cast_oneproof · cited by 2,501
- Convexstatement and proof · cited by 551
- neg_smulproof · cited by 306
- add_smulproof · cited by 204
- smul_negproof · cited by 181
- div_nonnegproof · cited by 103
- Real.norm_eq_absproof · cited by 88
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