Theorems · Theorem · functional analysis
affinity_unitBall
∀ {E : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {r : ℝ},
0 < r → ∀ (x : E), x +ᵥ r • Metric.ball 0 1 = Metric.ball x rAny ball Metric.ball x r, 0 < r is the image of the unit ball under fun y ↦ x + r • y.
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- Foundations
- Depth 163 from the axioms · uses propext, Classical.choice, Quot.sound
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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
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- HVAdd.hVAddstatement and proof · cited by 1,820
- Metric.ballstatement and proof · cited by 735
- Set.smulSetstatement · cited by 608
- Set.vaddSetstatement · cited by 403
- smul_unitBall_of_posproof · cited by 2
- vadd_ball_zeroproof · cited by 2
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