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| Mirrors > Home > MPE Home > Th. List > xblpnfps | Structured version Visualization version GIF version | ||
| Description: The infinity ball in an extended metric is the set of all points that are a finite distance from the center. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
| Ref | Expression |
|---|---|
| xblpnfps | β’ ((π· β (PsMetβπ) β§ π β π) β (π΄ β (π(ballβπ·)+β) β (π΄ β π β§ (ππ·π΄) β β))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pnfxr 11269 | . . 3 β’ +β β β* | |
| 2 | elblps 24244 | . . 3 β’ ((π· β (PsMetβπ) β§ π β π β§ +β β β*) β (π΄ β (π(ballβπ·)+β) β (π΄ β π β§ (ππ·π΄) < +β))) | |
| 3 | 1, 2 | mp3an3 1446 | . 2 β’ ((π· β (PsMetβπ) β§ π β π) β (π΄ β (π(ballβπ·)+β) β (π΄ β π β§ (ππ·π΄) < +β))) |
| 4 | psmetcl 24164 | . . . . . . . 8 β’ ((π· β (PsMetβπ) β§ π β π β§ π΄ β π) β (ππ·π΄) β β*) | |
| 5 | psmetge0 24169 | . . . . . . . 8 β’ ((π· β (PsMetβπ) β§ π β π β§ π΄ β π) β 0 β€ (ππ·π΄)) | |
| 6 | ge0nemnf 13155 | . . . . . . . 8 β’ (((ππ·π΄) β β* β§ 0 β€ (ππ·π΄)) β (ππ·π΄) β -β) | |
| 7 | 4, 5, 6 | syl2anc 583 | . . . . . . 7 β’ ((π· β (PsMetβπ) β§ π β π β§ π΄ β π) β (ππ·π΄) β -β) |
| 8 | ngtmnft 13148 | . . . . . . . . 9 β’ ((ππ·π΄) β β* β ((ππ·π΄) = -β β Β¬ -β < (ππ·π΄))) | |
| 9 | 4, 8 | syl 17 | . . . . . . . 8 β’ ((π· β (PsMetβπ) β§ π β π β§ π΄ β π) β ((ππ·π΄) = -β β Β¬ -β < (ππ·π΄))) |
| 10 | 9 | necon2abid 2977 | . . . . . . 7 β’ ((π· β (PsMetβπ) β§ π β π β§ π΄ β π) β (-β < (ππ·π΄) β (ππ·π΄) β -β)) |
| 11 | 7, 10 | mpbird 257 | . . . . . 6 β’ ((π· β (PsMetβπ) β§ π β π β§ π΄ β π) β -β < (ππ·π΄)) |
| 12 | 11 | biantrurd 532 | . . . . 5 β’ ((π· β (PsMetβπ) β§ π β π β§ π΄ β π) β ((ππ·π΄) < +β β (-β < (ππ·π΄) β§ (ππ·π΄) < +β))) |
| 13 | xrrebnd 13150 | . . . . . 6 β’ ((ππ·π΄) β β* β ((ππ·π΄) β β β (-β < (ππ·π΄) β§ (ππ·π΄) < +β))) | |
| 14 | 4, 13 | syl 17 | . . . . 5 β’ ((π· β (PsMetβπ) β§ π β π β§ π΄ β π) β ((ππ·π΄) β β β (-β < (ππ·π΄) β§ (ππ·π΄) < +β))) |
| 15 | 12, 14 | bitr4d 282 | . . . 4 β’ ((π· β (PsMetβπ) β§ π β π β§ π΄ β π) β ((ππ·π΄) < +β β (ππ·π΄) β β)) |
| 16 | 15 | 3expa 1115 | . . 3 β’ (((π· β (PsMetβπ) β§ π β π) β§ π΄ β π) β ((ππ·π΄) < +β β (ππ·π΄) β β)) |
| 17 | 16 | pm5.32da 578 | . 2 β’ ((π· β (PsMetβπ) β§ π β π) β ((π΄ β π β§ (ππ·π΄) < +β) β (π΄ β π β§ (ππ·π΄) β β))) |
| 18 | 3, 17 | bitrd 279 | 1 β’ ((π· β (PsMetβπ) β§ π β π) β (π΄ β (π(ballβπ·)+β) β (π΄ β π β§ (ππ·π΄) β β))) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wne 2934 class class class wbr 5141 βcfv 6536 (class class class)co 7404 βcr 11108 0cc0 11109 +βcpnf 11246 -βcmnf 11247 β*cxr 11248 < clt 11249 β€ cle 11250 PsMetcpsmet 21220 ballcbl 21223 |
| 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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-2 12276 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-psmet 21228 df-bl 21231 |
| This theorem is referenced by: xblss2ps 24258 |
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