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Mirrors > Home > MPE Home > Th. List > blssm | Structured version Visualization version GIF version |
Description: A ball is a subset of the base set of a metric space. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
blssm | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blf 24404 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋) | |
2 | fovcdm 7596 | . . 3 ⊢ (((ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋 ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝒫 𝑋) | |
3 | 1, 2 | syl3an1 1160 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝒫 𝑋) |
4 | 3 | elpwid 4616 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 ∈ wcel 2099 ⊆ wss 3947 𝒫 cpw 4607 × cxp 5680 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 ℝ*cxr 11297 ∞Metcxmet 21328 ballcbl 21330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-map 8857 df-xr 11302 df-psmet 21335 df-xmet 21336 df-bl 21338 |
This theorem is referenced by: blpnfctr 24433 xmetresbl 24434 imasf1oxms 24489 prdsbl 24491 blcld 24505 blcls 24506 prdsxmslem2 24529 metcnp 24541 cnllycmp 24973 lebnumlem3 24980 lebnum 24981 cfil3i 25288 iscfil3 25292 cfilfcls 25293 iscmet3lem2 25311 equivcfil 25318 caublcls 25328 relcmpcmet 25337 cmpcmet 25338 cncmet 25341 bcthlem2 25344 bcthlem4 25346 dvlip2 26019 dv11cn 26025 pserdvlem2 26458 pserdv 26459 abelthlem3 26463 abelthlem5 26465 dvlog2lem 26679 dvlog2 26680 efopnlem2 26684 efopn 26685 logtayl 26687 efrlim 26997 efrlimOLD 26998 blsconn 35072 sstotbnd2 37475 equivtotbnd 37479 isbnd2 37484 blbnd 37488 totbndbnd 37490 prdstotbnd 37495 prdsbnd2 37496 ismtyima 37504 heiborlem3 37514 heiborlem8 37519 |
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