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| Mirrors > Home > MPE Home > Th. List > elbl3 | Structured version Visualization version GIF version | ||
| Description: Membership in a ball, with reversed distance function arguments. (Contributed by NM, 10-Nov-2007.) |
| Ref | Expression |
|---|---|
| elbl3 | β’ (((π· β (βMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β (π΄ β (π(ballβπ·)π ) β (π΄π·π) < π )) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elbl2 24216 | . 2 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 )) | |
| 2 | xmetsym 24173 | . . . . 5 β’ ((π· β (βMetβπ) β§ π β π β§ π΄ β π) β (ππ·π΄) = (π΄π·π)) | |
| 3 | 2 | 3expb 1119 | . . . 4 β’ ((π· β (βMetβπ) β§ (π β π β§ π΄ β π)) β (ππ·π΄) = (π΄π·π)) |
| 4 | 3 | adantlr 712 | . . 3 β’ (((π· β (βMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β (ππ·π΄) = (π΄π·π)) |
| 5 | 4 | breq1d 5158 | . 2 β’ (((π· β (βMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β ((ππ·π΄) < π β (π΄π·π) < π )) |
| 6 | 1, 5 | bitrd 279 | 1 β’ (((π· β (βMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β (π΄ β (π(ballβπ·)π ) β (π΄π·π) < π )) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1540 β wcel 2105 class class class wbr 5148 βcfv 6543 (class class class)co 7412 β*cxr 11254 < clt 11255 βMetcxmet 21218 ballcbl 21220 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-xadd 13100 df-psmet 21225 df-xmet 21226 df-bl 21228 |
| This theorem is referenced by: blcom 24220 reperflem 24654 reconnlem2 24663 ellimc3 25728 dvlip2 25848 lhop1lem 25866 ulmdvlem1 26251 pserdvlem2 26280 abelthlem2 26284 abelthlem3 26285 abelthlem5 26287 abelthlem7 26290 efopn 26506 logtayl 26508 xrlimcnp 26814 efrlim 26815 lgamucov 26883 lgamcvg2 26900 tpr2rico 33356 heibor1lem 37141 |
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