Theorems · Theorem · functional analysis
exists_lt_of_gauge_lt
∀ {E : Type u_2} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {s : Set E} {x : E} {a : ℝ},
Absorbent ℝ s → gauge s x < a → ∃ b, 0 < b ∧ b < a ∧ x ∈ b • s- Defined in
- Mathlib.Analysis.Convex.Gauge
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 159 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.
Cites10
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
- Set.ofPredproof · cited by 6,101
- Set.smulSetstatement · cited by 608
- gaugestatement and proof · cited by 85
- Absorbentstatement and proof · cited by 60
- exists_lt_of_csInf_ltproof · cited by 10
- Absorbent.gauge_set_nonemptyproof · cited by 2
Cited by6
Results whose statement or proof uses this declaration.
- gauge_add_leproof · cited by 4
- setOfPred_gauge_le_eqproof · cited by 3
- comap_gauge_nhds_zero_leproof · cited by 2
- mem_openSegment_of_gauge_lt_oneproof · cited by 2
- setOfPred_gauge_lt_eqproof · cited by 2
- setOfPred_gauge_lt_eq'proof · cited by 2