![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > elbl | Structured version Visualization version GIF version |
Description: Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
elbl | ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | blval 22568 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) = {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅}) | |
2 | 1 | eleq2d 2892 | . 2 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ 𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅})) |
3 | oveq2 6918 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑃𝐷𝑥) = (𝑃𝐷𝐴)) | |
4 | 3 | breq1d 4885 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑃𝐷𝑥) < 𝑅 ↔ (𝑃𝐷𝐴) < 𝑅)) |
5 | 4 | elrab 3585 | . 2 ⊢ (𝐴 ∈ {𝑥 ∈ 𝑋 ∣ (𝑃𝐷𝑥) < 𝑅} ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅)) |
6 | 2, 5 | syl6bb 279 | 1 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐴 ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝐴 ∈ 𝑋 ∧ (𝑃𝐷𝐴) < 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 {crab 3121 class class class wbr 4875 ‘cfv 6127 (class class class)co 6910 ℝ*cxr 10397 < clt 10398 ∞Metcxmet 20098 ballcbl 20100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-map 8129 df-xr 10402 df-psmet 20105 df-xmet 20106 df-bl 20108 |
This theorem is referenced by: elbl2 22572 xblpnf 22578 bldisj 22580 blgt0 22581 xblss2 22584 blhalf 22587 xblcntr 22593 xbln0 22596 blin 22603 blss 22607 blres 22613 imasf1obl 22670 prdsbl 22673 blcls 22688 metcnp 22723 dscopn 22755 cnbl0 22954 bl2ioo 22972 blcvx 22978 xrsmopn 22992 recld2 22994 cnheibor 23131 nmhmcn 23296 lmmbr2 23434 iscau2 23452 dvlip2 24164 psercn 24586 abelth 24601 logtayl 24812 logtayl2 24814 poimirlem29 33977 heicant 33983 iooabslt 40514 limcrecl 40650 islpcn 40660 qndenserrnbllem 41299 |
Copyright terms: Public domain | W3C validator |