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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 11210 | . . 3 β’ +β β β* | |
2 | elbl 23744 | . . 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 1451 | . 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 23687 | . . . . . . . 8 β’ ((π· β (βMetβπ) β§ π β π β§ π΄ β π) β (ππ·π΄) β β*) | |
5 | xmetge0 23700 | . . . . . . . 8 β’ ((π· β (βMetβπ) β§ π β π β§ π΄ β π) β 0 β€ (ππ·π΄)) | |
6 | ge0nemnf 13093 | . . . . . . . 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 (βMetβπ) β§ π β π β§ π΄ β π) β (ππ·π΄) β -β) |
8 | ngtmnft 13086 | . . . . . . . . 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 2987 | . . . . . . 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 534 | . . . . 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 13088 | . . . . . 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 1119 | . . 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 580 | . 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wne 2944 class class class wbr 5106 βcfv 6497 (class class class)co 7358 βcr 11051 0cc0 11052 +βcpnf 11187 -βcmnf 11188 β*cxr 11189 < clt 11190 β€ cle 11191 βMetcxmet 20784 ballcbl 20786 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8649 df-map 8768 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-2 12217 df-rp 12917 df-xneg 13034 df-xadd 13035 df-xmul 13036 df-psmet 20791 df-xmet 20792 df-bl 20794 |
This theorem is referenced by: blpnf 23753 xmetec 23790 metdstri 24217 |
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