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Theorem blpnfctr 14342
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 2189 . . . . 5  |-  ( `' D " RR )  =  ( `' D " RR )
21xmeter 14339 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  ( `' D " RR )  Er  X )
323ad2ant1 1020 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  -> 
( `' D " RR )  Er  X
)
4 simp3 1001 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  A  e.  ( P
( ball `  D ) +oo ) )
51xmetec 14340 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X
)  ->  [ P ] ( `' D " RR )  =  ( P ( ball `  D
) +oo ) )
653adant3 1019 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  [ P ] ( `' D " RR )  =  ( P (
ball `  D ) +oo ) )
74, 6eleqtrrd 2269 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  A  e.  [ P ] ( `' D " RR ) )
8 elecg 6591 . . . . . 6  |-  ( ( A  e.  ( P ( ball `  D
) +oo )  /\  P  e.  X )  ->  ( A  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) A ) )
98ancoms 268 . . . . 5  |-  ( ( P  e.  X  /\  A  e.  ( P
( ball `  D ) +oo ) )  ->  ( A  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) A ) )
1093adant1 1017 . . . 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 147 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  P ( `' D " RR ) A )
123, 11erthi 6599 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  [ P ] ( `' D " RR )  =  [ A ]
( `' D " RR ) )
13 pnfxr 8029 . . . . . 6  |- +oo  e.  RR*
14 blssm 14324 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\ +oo  e.  RR* )  ->  ( P ( ball `  D ) +oo )  C_  X )
1513, 14mp3an3 1337 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X
)  ->  ( P
( ball `  D ) +oo )  C_  X )
1615sselda 3170 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  A  e.  ( P ( ball `  D ) +oo )
)  ->  A  e.  X )
171xmetec 14340 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X
)  ->  [ A ] ( `' D " RR )  =  ( A ( ball `  D
) +oo ) )
1817adantlr 477 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  A  e.  X )  ->  [ A ] ( `' D " RR )  =  ( A ( ball `  D
) +oo ) )
1916, 18syldan 282 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  A  e.  ( P ( ball `  D ) +oo )
)  ->  [ A ] ( `' D " RR )  =  ( A ( ball `  D
) +oo ) )
20193impa 1196 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  [ A ] ( `' D " RR )  =  ( A (
ball `  D ) +oo ) )
2112, 6, 203eqtr3d 2230 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    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160    C_ wss 3144   class class class wbr 4018   `'ccnv 4640   "cima 4644   ` cfv 5231  (class class class)co 5891    Er wer 6550   [cec 6551   RRcr 7829   +oocpnf 8008   RR*cxr 8010   *Metcxmet 13816   ballcbl 13818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-po 4311  df-iso 4312  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-er 6553  df-ec 6555  df-map 6668  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-2 8997  df-xadd 9792  df-psmet 13823  df-xmet 13824  df-bl 13826
This theorem is referenced by: (None)
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