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| Mirrors > Home > MPE Home > Th. List > Mathboxes > blssp | Structured version Visualization version GIF version | ||
| Description: A ball in the subspace metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jan-2014.) | 
| Ref | Expression | 
|---|---|
| blssp.2 | ⊢ 𝑁 = (𝑀 ↾ (𝑆 × 𝑆)) | 
| Ref | Expression | 
|---|---|
| blssp | ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+)) → (𝑌(ball‘𝑁)𝑅) = ((𝑌(ball‘𝑀)𝑅) ∩ 𝑆)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | metxmet 24345 | . . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋)) | |
| 2 | 1 | ad2antrr 726 | . 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 4222 | . . . 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 2843 | . 2 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+)) → 𝑌 ∈ (𝑋 ∩ 𝑆)) | 
| 8 | rpxr 13045 | . . 3 ⊢ (𝑅 ∈ ℝ+ → 𝑅 ∈ ℝ*) | |
| 9 | 8 | ad2antll 729 | . 2 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+)) → 𝑅 ∈ ℝ*) | 
| 10 | blssp.2 | . . 3 ⊢ 𝑁 = (𝑀 ↾ (𝑆 × 𝑆)) | |
| 11 | 10 | blres 24442 | . 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑌 ∈ (𝑋 ∩ 𝑆) ∧ 𝑅 ∈ ℝ*) → (𝑌(ball‘𝑁)𝑅) = ((𝑌(ball‘𝑀)𝑅) ∩ 𝑆)) | 
| 12 | 2, 7, 9, 11 | syl3anc 1372 | 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 1539 ∈ wcel 2107 ∩ cin 3949 ⊆ wss 3950 × cxp 5682 ↾ cres 5686 ‘cfv 6560 (class class class)co 7432 ℝ*cxr 11295 ℝ+crp 13035 ∞Metcxmet 21350 Metcmet 21351 ballcbl 21352 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-mulcl 11218 ax-i2m1 11224 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-rp 13036 df-xadd 13156 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 | 
| This theorem is referenced by: bndss 37794 | 
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