Theorems · Theorem · functional analysis
nsmul_mem_ball
∀ {E : Type u_5} [inst : SeminormedAddCommGroup E] {a b : E} {r : ℝ} {n : ℕ},
0 < n → a ∈ Metric.ball b r → n • a ∈ Metric.ball (n • b) (n • r)- Defined in
- Mathlib.Analysis.Normed.Group.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 110 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.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- Norm.normproof · cited by 5,413
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Nat.cast_oneproof · cited by 2,501
- Nat.cast_zeroproof · cited by 1,870
- Metric.ballstatement and proof · cited by 735
- lt_of_le_of_ltproof · cited by 432
- lt_of_not_geproof · cited by 374
- nsmul_eq_mulproof · cited by 369
- Nat.cast_ltproof · cited by 80
- Metric.mem_ballproof · cited by 47
Cited by2
Results whose statement or proof uses this declaration.
- approxAddOrderOf.image_nsmul_subsetproof · cited by 1
- approxAddOrderOf.image_nsmul_subset_of_coprimeproof · cited by 1