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| Mirrors > Home > MPE Home > Th. List > xblpnf | 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.) |
| Ref | Expression |
|---|---|
| xblpnf | β’ ((π· β (βMetβπ) 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 11267 | . . 3 β’ +β β β* | |
| 2 | elbl 24238 | . . 3 β’ ((π· β (βMetβπ) 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 (βMetβπ) β§ π β π) β (π΄ β (π(ballβπ·)+β) β (π΄ β π β§ (ππ·π΄) < +β))) |
| 4 | xmetcl 24181 | . . . . . . . 8 β’ ((π· β (βMetβπ) β§ π β π β§ π΄ β π) β (ππ·π΄) β β*) | |
| 5 | xmetge0 24194 | . . . . . . . 8 β’ ((π· β (βMetβπ) β§ π β π β§ π΄ β π) β 0 β€ (ππ·π΄)) | |
| 6 | ge0nemnf 13153 | . . . . . . . 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 (βMetβπ) β§ π β π β§ π΄ β π) β (ππ·π΄) β -β) |
| 8 | ngtmnft 13146 | . . . . . . . . 9 β’ ((ππ·π΄) β β* β ((ππ·π΄) = -β β Β¬ -β < (ππ·π΄))) | |
| 9 | 4, 8 | syl 17 | . . . . . . . 8 β’ ((π· β (βMetβπ) β§ π β π β§ π΄ β π) β ((ππ·π΄) = -β β Β¬ -β < (ππ·π΄))) |
| 10 | 9 | necon2abid 2975 | . . . . . . 7 β’ ((π· β (βMetβπ) 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 (βMetβπ) β§ π β π β§ π΄ β π) β -β < (ππ·π΄)) |
| 12 | 11 | biantrurd 532 | . . . . 5 β’ ((π· β (βMetβπ) β§ π β π β§ π΄ β π) β ((ππ·π΄) < +β β (-β < (ππ·π΄) β§ (ππ·π΄) < +β))) |
| 13 | xrrebnd 13148 | . . . . . 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 (βMetβπ) 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 (βMetβπ) 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 (βMetβπ) 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 (βMetβπ) 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 (βMetβπ) 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 2932 class class class wbr 5139 βcfv 6534 (class class class)co 7402 βcr 11106 0cc0 11107 +βcpnf 11244 -βcmnf 11245 β*cxr 11246 < clt 11247 β€ cle 11248 βMetcxmet 21219 ballcbl 21221 |
| 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 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
| 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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-2 12274 df-rp 12976 df-xneg 13093 df-xadd 13094 df-xmul 13095 df-psmet 21226 df-xmet 21227 df-bl 21229 |
| This theorem is referenced by: blpnf 24247 xmetec 24284 metdstri 24711 |
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