Theorems · Theorem · functional analysis
mem_openSegment_of_gauge_lt_one
∀ {E : Type u_2} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {s : Set E} {x : E},
Absorbent ℝ s → gauge s x < 1 → ∃ y ∈ s, x ∈ openSegment ℝ 0 y- Defined in
- Mathlib.Analysis.Convex.Gauge
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 160 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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
- zero_addproof · cited by 2,366
- smul_zeroproof · cited by 665
- sub_add_cancelproof · cited by 344
- openSegmentstatement · cited by 102
- gaugestatement and proof · cited by 85
- Absorbentstatement and proof · cited by 60
- exists_lt_of_gauge_ltproof · cited by 6
Cited by2
Results whose statement or proof uses this declaration.
- setOfPred_gauge_lt_one_subset_selfproof · cited by 5
- setOfPred_gauge_lt_one_eq_interiorproof · cited by 3