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Theorem xblpnfps 21422
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.)
Assertion
Ref Expression
xblpnfps  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  ->  ( A  e.  ( P
( ball `  D ) +oo )  <->  ( A  e.  X  /\  ( P D A )  e.  RR ) ) )

Proof of Theorem xblpnfps
StepHypRef Expression
1 pnfxr 11419 . . 3  |- +oo  e.  RR*
2 elblps 21414 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\ +oo  e.  RR* )  ->  ( A  e.  ( P ( ball `  D ) +oo )  <->  ( A  e.  X  /\  ( P D A )  < +oo ) ) )
31, 2mp3an3 1355 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  ->  ( A  e.  ( P
( ball `  D ) +oo )  <->  ( A  e.  X  /\  ( P D A )  < +oo ) ) )
4 psmetcl 21335 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  A  e.  X )  ->  ( P D A )  e. 
RR* )
5 psmetge0 21340 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  A  e.  X )  ->  0  <_  ( P D A ) )
6 ge0nemnf 11475 . . . . . . . 8  |-  ( ( ( P D A )  e.  RR*  /\  0  <_  ( P D A ) )  ->  ( P D A )  =/= -oo )
74, 5, 6syl2anc 667 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  A  e.  X )  ->  ( P D A )  =/= -oo )
8 ngtmnft 11469 . . . . . . . . 9  |-  ( ( P D A )  e.  RR*  ->  ( ( P D A )  = -oo  <->  -. -oo  <  ( P D A ) ) )
94, 8syl 17 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  A  e.  X )  ->  (
( P D A )  = -oo  <->  -. -oo  <  ( P D A ) ) )
109necon2abid 2668 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  A  e.  X )  ->  ( -oo  <  ( P D A )  <->  ( P D A )  =/= -oo ) )
117, 10mpbird 236 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  A  e.  X )  -> -oo  <  ( P D A ) )
1211biantrurd 511 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  A  e.  X )  ->  (
( P D A )  < +oo  <->  ( -oo  <  ( P D A )  /\  ( P D A )  < +oo ) ) )
13 xrrebnd 11470 . . . . . 6  |-  ( ( P D A )  e.  RR*  ->  ( ( P D A )  e.  RR  <->  ( -oo  <  ( P D A )  /\  ( P D A )  < +oo ) ) )
144, 13syl 17 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  A  e.  X )  ->  (
( P D A )  e.  RR  <->  ( -oo  <  ( P D A )  /\  ( P D A )  < +oo ) ) )
1512, 14bitr4d 260 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X  /\  A  e.  X )  ->  (
( P D A )  < +oo  <->  ( P D A )  e.  RR ) )
16153expa 1209 . . 3  |-  ( ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  /\  A  e.  X )  ->  (
( P D A )  < +oo  <->  ( P D A )  e.  RR ) )
1716pm5.32da 647 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  ->  (
( A  e.  X  /\  ( P D A )  < +oo )  <->  ( A  e.  X  /\  ( P D A )  e.  RR ) ) )
183, 17bitrd 257 1  |-  ( ( D  e.  (PsMet `  X )  /\  P  e.  X )  ->  ( A  e.  ( P
( ball `  D ) +oo )  <->  ( A  e.  X  /\  ( P D A )  e.  RR ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624   class class class wbr 4405   ` cfv 5585  (class class class)co 6295   RRcr 9543   0cc0 9544   +oocpnf 9677   -oocmnf 9678   RR*cxr 9679    < clt 9680    <_ cle 9681  PsMetcpsmet 18966   ballcbl 18969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-po 4758  df-so 4759  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6798  df-2nd 6799  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-2 10675  df-rp 11310  df-xneg 11416  df-xadd 11417  df-xmul 11418  df-psmet 18974  df-bl 18977
This theorem is referenced by:  xblss2ps  21428
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