Theorems · Theorem · functional analysis
QuotientAddGroup.norm_lt_iff
∀ {M : Type u_1} [inst : SeminormedAddCommGroup M] {S : AddSubgroup M} {x : M ⧸ S} {r : ℝ},
‖x‖ < r ↔ ∃ m, ↑m = x ∧ ‖m‖ < r- Defined in
- Mathlib.Analysis.Normed.Group.Quotient
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 155 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SeminormedAddCommGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- Norm.normstatement and proof · cited by 5,413
- AddSubgroupstatement and proof · cited by 3,232
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- HasQuotient.Quotientstatement and proof · cited by 2,301
- PseudoMetricSpaceproof · cited by 1,550
- QuotientAddGroup.mkstatement and proof · cited by 348
- dist_zeroproof · cited by 16
- Metric.infDist_lt_iffproof · cited by 4
Cited by3
Results whose statement or proof uses this declaration.
- QuotientAddGroup.exists_norm_mk_ltproof · cited by 3
- QuotientAddGroup.norm_lift_apply_leproof · cited by 2
- QuotientAddGroup.nhds_zero_hasBasisproof · cited by 0