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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 11299 | . . 3 β’ +β β β* | |
| 2 | elbl 24307 | . . 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 1447 | . 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 24250 | . . . . . . . 8 β’ ((π· β (βMetβπ) β§ π β π β§ π΄ β π) β (ππ·π΄) β β*) | |
| 5 | xmetge0 24263 | . . . . . . . 8 β’ ((π· β (βMetβπ) β§ π β π β§ π΄ β π) β 0 β€ (ππ·π΄)) | |
| 6 | ge0nemnf 13185 | . . . . . . . 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 13178 | . . . . . . . . 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 2980 | . . . . . . 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 13180 | . . . . . 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 1116 | . . 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 1085 = wceq 1534 β wcel 2099 β wne 2937 class class class wbr 5148 βcfv 6548 (class class class)co 7420 βcr 11138 0cc0 11139 +βcpnf 11276 -βcmnf 11277 β*cxr 11278 < clt 11279 β€ cle 11280 βMetcxmet 21264 ballcbl 21266 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-2 12306 df-rp 13008 df-xneg 13125 df-xadd 13126 df-xmul 13127 df-psmet 21271 df-xmet 21272 df-bl 21274 |
| This theorem is referenced by: blpnf 24316 xmetec 24353 metdstri 24780 |
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