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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 11219 | . . 3 β’ +β β β* | |
| 2 | elblps 23778 | . . 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 23698 | . . . . . . . 8 β’ ((π· β (PsMetβπ) β§ π β π β§ π΄ β π) β (ππ·π΄) β β*) | |
| 5 | psmetge0 23703 | . . . . . . . 8 β’ ((π· β (PsMetβπ) β§ π β π β§ π΄ β π) β 0 β€ (ππ·π΄)) | |
| 6 | ge0nemnf 13103 | . . . . . . . 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 13096 | . . . . . . . . 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 2983 | . . . . . . 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 13098 | . . . . . 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 2940 class class class wbr 5111 βcfv 6502 (class class class)co 7363 βcr 11060 0cc0 11061 +βcpnf 11196 -βcmnf 11197 β*cxr 11198 < clt 11199 β€ cle 11200 PsMetcpsmet 20818 ballcbl 20821 |
| 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 2703 ax-sep 5262 ax-nul 5269 ax-pow 5326 ax-pr 5390 ax-un 7678 ax-cnex 11117 ax-resscn 11118 ax-1cn 11119 ax-icn 11120 ax-addcl 11121 ax-addrcl 11122 ax-mulcl 11123 ax-mulrcl 11124 ax-mulcom 11125 ax-addass 11126 ax-mulass 11127 ax-distr 11128 ax-i2m1 11129 ax-1ne0 11130 ax-1rid 11131 ax-rnegex 11132 ax-rrecex 11133 ax-cnre 11134 ax-pre-lttri 11135 ax-pre-lttrn 11136 ax-pre-ltadd 11137 ax-pre-mulgt0 11138 |
| 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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4289 df-if 4493 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4872 df-iun 4962 df-br 5112 df-opab 5174 df-mpt 5195 df-id 5537 df-po 5551 df-so 5552 df-xp 5645 df-rel 5646 df-cnv 5647 df-co 5648 df-dm 5649 df-rn 5650 df-res 5651 df-ima 5652 df-iota 6454 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7319 df-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7927 df-2nd 7928 df-er 8656 df-map 8775 df-en 8892 df-dom 8893 df-sdom 8894 df-pnf 11201 df-mnf 11202 df-xr 11203 df-ltxr 11204 df-le 11205 df-sub 11397 df-neg 11398 df-div 11823 df-2 12226 df-rp 12926 df-xneg 13043 df-xadd 13044 df-xmul 13045 df-psmet 20826 df-bl 20829 |
| This theorem is referenced by: xblss2ps 23792 |
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