blssp - Mathbox for Jeff Madsen

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Theorem blssp 37716

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 24365	. . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
2	1	ad2antrr 725	. 2 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → 𝑀 ∈ (∞Met‘𝑋))
3		simprl 770	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → 𝑌 ∈ 𝑆)
4		simplr 768	. . . 4 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → 𝑆 ⊆ 𝑋)
5		sseqin2 4244	. . . 4 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑋 ∩ 𝑆) = 𝑆)
6	4, 5	sylib 218	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → (𝑋 ∩ 𝑆) = 𝑆)
7	3, 6	eleqtrrd 2847	. 2 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → 𝑌 ∈ (𝑋 ∩ 𝑆))
8		rpxr 13066	. . 3 ⊢ (𝑅 ∈ ℝ⁺ → 𝑅 ∈ ℝ^*)
9	8	ad2antll 728	. 2 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → 𝑅 ∈ ℝ^*)
10		blssp.2	. . 3 ⊢ 𝑁 = (𝑀 ↾ (𝑆 × 𝑆))
11	10	blres 24462	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑌 ∈ (𝑋 ∩ 𝑆) ∧ 𝑅 ∈ ℝ^*) → (𝑌(ball‘𝑁)𝑅) = ((𝑌(ball‘𝑀)𝑅) ∩ 𝑆))
12	2, 7, 9, 11	syl3anc 1371	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 395 = wceq 1537 ∈ wcel 2108 ∩ cin 3975 ⊆ wss 3976 × cxp 5698 ↾ cres 5702 ‘cfv 6573 (class class class)co 7448 ℝ^*cxr 11323 ℝ⁺crp 13057 ∞Metcxmet 21372 Metcmet 21373 ballcbl 21374

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-mulcl 11246 ax-i2m1 11252

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-rp 13058 df-xadd 13176 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382

This theorem is referenced by: bndss 37746

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