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Theorem xblpnf 21404
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.)
Assertion
Ref Expression
xblpnf  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X
)  ->  ( A  e.  ( P ( ball `  D ) +oo )  <->  ( A  e.  X  /\  ( P D A )  e.  RR ) ) )

Proof of Theorem xblpnf
StepHypRef Expression
1 pnfxr 11409 . . 3  |- +oo  e.  RR*
2 elbl 21396 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\ +oo  e.  RR* )  ->  ( A  e.  ( P ( ball `  D
) +oo )  <->  ( A  e.  X  /\  ( P D A )  < +oo ) ) )
31, 2mp3an3 1352 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X
)  ->  ( A  e.  ( P ( ball `  D ) +oo )  <->  ( A  e.  X  /\  ( P D A )  < +oo ) ) )
4 xmetcl 21339 . . . . . . . 8  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  X
)  ->  ( P D A )  e.  RR* )
5 xmetge0 21352 . . . . . . . 8  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  X
)  ->  0  <_  ( P D A ) )
6 ge0nemnf 11465 . . . . . . . 8  |-  ( ( ( P D A )  e.  RR*  /\  0  <_  ( P D A ) )  ->  ( P D A )  =/= -oo )
74, 5, 6syl2anc 666 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  X
)  ->  ( P D A )  =/= -oo )
8 ngtmnft 11459 . . . . . . . . 9  |-  ( ( P D A )  e.  RR*  ->  ( ( P D A )  = -oo  <->  -. -oo  <  ( P D A ) ) )
94, 8syl 17 . . . . . . . 8  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  X
)  ->  ( ( P D A )  = -oo  <->  -. -oo  <  ( P D A ) ) )
109necon2abid 2665 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  X
)  ->  ( -oo  <  ( P D A )  <->  ( P D A )  =/= -oo ) )
117, 10mpbird 236 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  X
)  -> -oo  <  ( P D A ) )
1211biantrurd 511 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  X
)  ->  ( ( P D A )  < +oo 
<->  ( -oo  <  ( P D A )  /\  ( P D A )  < +oo ) ) )
13 xrrebnd 11460 . . . . . 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.  ( *Met `  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.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  X
)  ->  ( ( P D A )  < +oo 
<->  ( P D A )  e.  RR ) )
16153expa 1207 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  A  e.  X )  ->  (
( P D A )  < +oo  <->  ( P D A )  e.  RR ) )
1716pm5.32da 646 . 2  |-  ( ( D  e.  ( *Met `  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.  ( *Met `  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 984    = wceq 1443    e. wcel 1886    =/= wne 2621   class class class wbr 4401   ` cfv 5581  (class class class)co 6288   RRcr 9535   0cc0 9536   +oocpnf 9669   -oocmnf 9670   RR*cxr 9671    < clt 9672    <_ cle 9673   *Metcxmt 18948   ballcbl 18950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-er 7360  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-2 10665  df-rp 11300  df-xneg 11406  df-xadd 11407  df-xmul 11408  df-psmet 18955  df-xmet 18956  df-bl 18958
This theorem is referenced by:  blpnf  21405  xmetec  21442  metdstri  21861  metdstriOLD  21876
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