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Theorem blssp 38130

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 24324	. . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
2	1	ad2antrr 732	. 2 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → 𝑀 ∈ (∞Met‘𝑋))
3		simprl 776	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → 𝑌 ∈ 𝑆)
4		simplr 774	. . . 4 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → 𝑆 ⊆ 𝑋)
5		sseqin2 4159	. . . 4 ⊢ (𝑆 ⊆ 𝑋 ↔ (𝑋 ∩ 𝑆) = 𝑆)
6	4, 5	sylib 219	. . 3 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → (𝑋 ∩ 𝑆) = 𝑆)
7	3, 6	eleqtrrd 2843	. 2 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → 𝑌 ∈ (𝑋 ∩ 𝑆))
8		rpxr 12950	. . 3 ⊢ (𝑅 ∈ ℝ⁺ → 𝑅 ∈ ℝ^*)
9	8	ad2antll 735	. 2 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ⁺)) → 𝑅 ∈ ℝ^*)
10		blssp.2	. . 3 ⊢ 𝑁 = (𝑀 ↾ (𝑆 × 𝑆))
11	10	blres 24421	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑌 ∈ (𝑋 ∩ 𝑆) ∧ 𝑅 ∈ ℝ^*) → (𝑌(ball‘𝑁)𝑅) = ((𝑌(ball‘𝑀)𝑅) ∩ 𝑆))
12	2, 7, 9, 11	syl3anc 1379	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 396 = wceq 1547 ∈ wcel 2119 ∩ cin 3889 ⊆ wss 3890 × cxp 5623 ↾ cres 5627 ‘cfv 6492 (class class class)co 7363 ℝ^*cxr 11176 ℝ⁺crp 12940 ∞Metcxmet 21339 Metcmet 21340 ballcbl 21341

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-mulcl 11098 ax-i2m1 11104

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7938 df-2nd 7939 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11179 df-mnf 11180 df-xr 11181 df-rp 12941 df-xadd 13062 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349

This theorem is referenced by: bndss 38160

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