blssp - Mathbox for Jeff Madsen

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Theorem blssp 38215

Description: A ball in the subspace metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jan-2014.)

**Hypothesis**
Ref	Expression
blssp.2	⊢ 𝑁 = (𝑀 ↾ (𝑆 × 𝑆))

**Assertion**
Ref	Expression
blssp	⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → (𝑌(ball‘𝑁)𝑅) = ((𝑌(ball‘𝑀)𝑅) ∩ 𝑆))

**Proof of Theorem blssp**
Step	Hyp	Ref	Expression
1		metxmet 24381	. . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
2	1	ad2antrr 736	. 2 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → 𝑀 ∈ (∞Met‘𝑋))
3		simprl 780	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → 𝑌 ∈ 𝑆)
4		simplr 778	. . . 4 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → 𝑆 ⊆ 𝑋)
5		sseqin2 4173	. . . 4 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑋 ∩ 𝑆) = 𝑆)
6	4, 5	sylib 220	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → (𝑋 ∩ 𝑆) = 𝑆)
7	3, 6	eleqtrrd 2864	. 2 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → 𝑌 ∈ (𝑋 ∩ 𝑆))
8		rpxr 12996	. . 3 ⊢ (𝑅 ∈ ℝ⁺ → 𝑅 ∈ ℝ^*)
9	8	ad2antll 739	. 2 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → 𝑅 ∈ ℝ^*)
10		blssp.2	. . 3 ⊢ 𝑁 = (𝑀 ↾ (𝑆 × 𝑆))
11	10	blres 24478	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑌 ∈ (𝑋 ∩ 𝑆) ∧ 𝑅 ∈ ℝ^*) → (𝑌(ball‘𝑁)𝑅) = ((𝑌(ball‘𝑀)𝑅) ∩ 𝑆))
12	2, 7, 9, 11	syl3anc 1389	1 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → (𝑌(ball‘𝑁)𝑅) = ((𝑌(ball‘𝑀)𝑅) ∩ 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ∩ cin 3901 ⊆ wss 3902 × cxp 5641 ↾ cres 5645 ‘cfv 6515 (class class class)co 7390 ℝ^*cxr 11208 ℝ⁺crp 12986 ∞Metcxmet 21396 Metcmet 21397 ballcbl 21398

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-mulcl 11128 ax-i2m1 11134

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5538 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7964 df-2nd 7965 df-er 8671 df-map 8803 df-en 8921 df-dom 8922 df-sdom 8923 df-pnf 11211 df-mnf 11212 df-xr 11213 df-rp 12987 df-xadd 13108 df-psmet 21403 df-xmet 21404 df-met 21405 df-bl 21406

This theorem is referenced by: bndss 38245

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