Theorems · Theorem · functional analysis
pow_mem_ball
∀ {E : Type u_5} [inst : SeminormedCommGroup E] {a b : E} {r : ℝ} {n : ℕ},
0 < n → a ∈ Metric.ball b r → a ^ n ∈ Metric.ball (b ^ n) (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
- SeminormedCommGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
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
- 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
- SeminormedCommGroupstatement and proof · cited by 191
- dist_eq_norm_inv_mulproof · cited by 41
- mul_pos_of_neg_of_negproof · cited by 40
Cited by2
Results whose statement or proof uses this declaration.
- approxOrderOf.image_pow_subsetproof · cited by 0
- approxOrderOf.image_pow_subset_of_coprimeproof · cited by 0