Theorems · Theorem · functional analysis
pow_mem_closedBall
∀ {E : Type u_5} [inst : SeminormedCommGroup E] {a b : E} {r : ℝ} {n : ℕ},
a ∈ Metric.closedBall b r → a ^ n ∈ Metric.closedBall (b ^ n) (n • r)- Defined in
- Mathlib.Analysis.Normed.Group.Basic
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SeminormedCommGroup
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Cites11
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
- LE.le.transproof · cited by 3,151
- Metric.closedBallstatement and proof · cited by 704
- nsmul_eq_mulproof · cited by 369
- mul_le_mul_of_nonneg_leftproof · cited by 361
- SeminormedCommGroupstatement and proof · cited by 191
- Nat.cast_nonnegproof · cited by 109
- dist_eq_norm_inv_mulproof · cited by 41
- norm_pow_le_mul_normproof · cited by 4
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