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| Mirrors > Home > MPE Home > Th. List > elblps | Structured version Visualization version GIF version | ||
| Description: Membership in a ball. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
| Ref | Expression |
|---|---|
| elblps | β’ ((π· β (PsMetβπ) β§ π β π β§ π β β*) β (π΄ β (π(ballβπ·)π ) β (π΄ β π β§ (ππ·π΄) < π ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | blvalps 24111 | . . 3 β’ ((π· β (PsMetβπ) β§ π β π β§ π β β*) β (π(ballβπ·)π ) = {π₯ β π β£ (ππ·π₯) < π }) | |
| 2 | 1 | eleq2d 2817 | . 2 β’ ((π· β (PsMetβπ) β§ π β π β§ π β β*) β (π΄ β (π(ballβπ·)π ) β π΄ β {π₯ β π β£ (ππ·π₯) < π })) |
| 3 | oveq2 7419 | . . . 4 β’ (π₯ = π΄ β (ππ·π₯) = (ππ·π΄)) | |
| 4 | 3 | breq1d 5157 | . . 3 β’ (π₯ = π΄ β ((ππ·π₯) < π β (ππ·π΄) < π )) |
| 5 | 4 | elrab 3682 | . 2 β’ (π΄ β {π₯ β π β£ (ππ·π₯) < π } β (π΄ β π β§ (ππ·π΄) < π )) |
| 6 | 2, 5 | bitrdi 286 | 1 β’ ((π· β (PsMetβπ) β§ π β π β§ π β β*) β (π΄ β (π(ballβπ·)π ) β (π΄ β π β§ (ππ·π΄) < π ))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 β§ w3a 1085 = wceq 1539 β wcel 2104 {crab 3430 class class class wbr 5147 βcfv 6542 (class class class)co 7411 β*cxr 11251 < clt 11252 PsMetcpsmet 21128 ballcbl 21131 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-map 8824 df-xr 11256 df-psmet 21136 df-bl 21139 |
| This theorem is referenced by: elbl2ps 24115 xblpnfps 24121 xblss2ps 24127 xblcntrps 24136 blssps 24150 ballss3 44083 |
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