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| Mirrors > Home > MPE Home > Th. List > elbl2ps | Structured version Visualization version GIF version | ||
| Description: Membership in a ball. (Contributed by NM, 9-Mar-2007.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
| Ref | Expression |
|---|---|
| elbl2ps | β’ (((π· β (PsMetβπ) 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 770 | . 2 β’ (((π· β (PsMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β π΄ β π) | |
| 2 | elblps 24203 | . . . . 5 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 ))) | |
| 3 | 2 | 3expa 1115 | . . . 4 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 ))) |
| 4 | 3 | an32s 649 | . . 3 β’ (((π· β (PsMetβπ) 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 714 | . 2 β’ (((π· β (PsMetβπ) 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 704 | 1 β’ (((π· β (PsMetβπ) 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 395 β wcel 2098 class class class wbr 5138 βcfv 6533 (class class class)co 7401 β*cxr 11243 < clt 11244 PsMetcpsmet 21207 ballcbl 21210 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-map 8817 df-xr 11248 df-psmet 21215 df-bl 21218 |
| This theorem is referenced by: elbl3ps 24207 blcomps 24209 |
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