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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 11268 | . . 3 β’ +β β β* | |
| 2 | elblps 23893 | . . 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 1451 | . 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 23813 | . . . . . . . 8 β’ ((π· β (PsMetβπ) β§ π β π β§ π΄ β π) β (ππ·π΄) β β*) | |
| 5 | psmetge0 23818 | . . . . . . . 8 β’ ((π· β (PsMetβπ) β§ π β π β§ π΄ β π) β 0 β€ (ππ·π΄)) | |
| 6 | ge0nemnf 13152 | . . . . . . . 8 β’ (((ππ·π΄) β β* β§ 0 β€ (ππ·π΄)) β (ππ·π΄) β -β) | |
| 7 | 4, 5, 6 | syl2anc 585 | . . . . . . 7 β’ ((π· β (PsMetβπ) β§ π β π β§ π΄ β π) β (ππ·π΄) β -β) |
| 8 | ngtmnft 13145 | . . . . . . . . 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 2984 | . . . . . . 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 534 | . . . . 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 13147 | . . . . . 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 1119 | . . 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 580 | . 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2941 class class class wbr 5149 βcfv 6544 (class class class)co 7409 βcr 11109 0cc0 11110 +βcpnf 11245 -βcmnf 11246 β*cxr 11247 < clt 11248 β€ cle 11249 PsMetcpsmet 20928 ballcbl 20931 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-2 12275 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-psmet 20936 df-bl 20939 |
| This theorem is referenced by: xblss2ps 23907 |
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