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Theorem blpnfctr 18356
Description: The infinity ball in an extended metric acts like an ultrametric ball in that every point in the ball is also its center. (Contributed by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
blpnfctr  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  -> 
( P ( ball `  D )  +oo )  =  ( A (
ball `  D )  +oo ) )

Proof of Theorem blpnfctr
StepHypRef Expression
1 eqid 2387 . . . . 5  |-  ( `' D " RR )  =  ( `' D " RR )
21xmeter 18353 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  ( `' D " RR )  Er  X )
323ad2ant1 978 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  -> 
( `' D " RR )  Er  X
)
4 simp3 959 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  ->  A  e.  ( P
( ball `  D )  +oo ) )
51xmetec 18354 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  [ P ] ( `' D " RR )  =  ( P ( ball `  D
)  +oo ) )
653adant3 977 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  ->  [ P ] ( `' D " RR )  =  ( P (
ball `  D )  +oo ) )
74, 6eleqtrrd 2464 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  ->  A  e.  [ P ] ( `' D " RR ) )
8 elecg 6879 . . . . . 6  |-  ( ( A  e.  ( P ( ball `  D
)  +oo )  /\  P  e.  X )  ->  ( A  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) A ) )
98ancoms 440 . . . . 5  |-  ( ( P  e.  X  /\  A  e.  ( P
( ball `  D )  +oo ) )  ->  ( A  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) A ) )
1093adant1 975 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  -> 
( A  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) A ) )
117, 10mpbid 202 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  ->  P ( `' D " RR ) A )
123, 11erthi 6887 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  ->  [ P ] ( `' D " RR )  =  [ A ]
( `' D " RR ) )
13 pnfxr 10645 . . . . . 6  |-  +oo  e.  RR*
14 blssm 18342 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  +oo  e.  RR* )  ->  ( P ( ball `  D )  +oo )  C_  X )
1513, 14mp3an3 1268 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  ( P
( ball `  D )  +oo )  C_  X )
1615sselda 3291 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  A  e.  ( P ( ball `  D )  +oo )
)  ->  A  e.  X )
171xmetec 18354 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X
)  ->  [ A ] ( `' D " RR )  =  ( A ( ball `  D
)  +oo ) )
1817adantlr 696 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  A  e.  X )  ->  [ A ] ( `' D " RR )  =  ( A ( ball `  D
)  +oo ) )
1916, 18syldan 457 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  A  e.  ( P ( ball `  D )  +oo )
)  ->  [ A ] ( `' D " RR )  =  ( A ( ball `  D
)  +oo ) )
20193impa 1148 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  ->  [ A ] ( `' D " RR )  =  ( A (
ball `  D )  +oo ) )
2112, 6, 203eqtr3d 2427 1  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  -> 
( P ( ball `  D )  +oo )  =  ( A (
ball `  D )  +oo ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3263   class class class wbr 4153   `'ccnv 4817   "cima 4821   ` cfv 5394  (class class class)co 6020    Er wer 6838   [cec 6839   RRcr 8922    +oocpnf 9050   RR*cxr 9052   * Metcxmt 16612   ballcbl 16614
This theorem is referenced by:  metdstri  18752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-po 4444  df-so 4445  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-er 6841  df-ec 6843  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-2 9990  df-rp 10545  df-xneg 10642  df-xadd 10643  df-xmul 10644  df-xmet 16619  df-bl 16621
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