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| Mirrors > Home > MPE Home > Th. List > elbl2 | Structured version Visualization version GIF version | ||
| Description: Membership in a ball. (Contributed by NM, 9-Mar-2007.) |
| Ref | Expression |
|---|---|
| elbl2 | β’ (((π· β (βMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β (π΄ β (π(ballβπ·)π ) β (ππ·π΄) < π )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simprr 769 | . 2 β’ (((π· β (βMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β π΄ β π) | |
| 2 | elbl 24114 | . . . . 5 β’ ((π· β (βMetβπ) β§ π β π β§ π β β*) β (π΄ β (π(ballβπ·)π ) β (π΄ β π β§ (ππ·π΄) < π ))) | |
| 3 | 2 | 3expa 1116 | . . . 4 β’ (((π· β (βMetβπ) β§ π β π) β§ π β β*) β (π΄ β (π(ballβπ·)π ) β (π΄ β π β§ (ππ·π΄) < π ))) |
| 4 | 3 | an32s 648 | . . 3 β’ (((π· β (βMetβπ) β§ π β β*) β§ π β π) β (π΄ β (π(ballβπ·)π ) β (π΄ β π β§ (ππ·π΄) < π ))) |
| 5 | 4 | adantrr 713 | . 2 β’ (((π· β (βMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β (π΄ β (π(ballβπ·)π ) β (π΄ β π β§ (ππ·π΄) < π ))) |
| 6 | 1, 5 | mpbirand 703 | 1 β’ (((π· β (βMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β (π΄ β (π(ballβπ·)π ) β (ππ·π΄) < π )) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 394 β wcel 2104 class class class wbr 5147 βcfv 6542 (class class class)co 7411 β*cxr 11251 < clt 11252 βMetcxmet 21129 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-xmet 21137 df-bl 21139 |
| This theorem is referenced by: elbl3 24118 blcom 24120 imasf1obl 24217 prdsbl 24220 blsscls2 24233 metcnp 24270 ngpocelbl 24441 zdis 24552 metdsge 24585 cfil3i 25017 iscfil3 25021 iscmet3lem2 25040 caubl 25056 dvlog2lem 26396 lgamucov 26778 isbnd3 36955 cntotbnd 36967 ismtyima 36974 stirlinglem5 45092 hoiqssbllem2 45637 |
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