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Theorem blpnfctr 12367
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 2100 . . . . 5  |-  ( `' D " RR )  =  ( `' D " RR )
21xmeter 12364 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  ( `' D " RR )  Er  X )
323ad2ant1 970 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  -> 
( `' D " RR )  Er  X
)
4 simp3 951 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  A  e.  ( P
( ball `  D ) +oo ) )
51xmetec 12365 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X
)  ->  [ P ] ( `' D " RR )  =  ( P ( ball `  D
) +oo ) )
653adant3 969 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  [ P ] ( `' D " RR )  =  ( P (
ball `  D ) +oo ) )
74, 6eleqtrrd 2179 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  A  e.  [ P ] ( `' D " RR ) )
8 elecg 6397 . . . . . 6  |-  ( ( A  e.  ( P ( ball `  D
) +oo )  /\  P  e.  X )  ->  ( A  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) A ) )
98ancoms 266 . . . . 5  |-  ( ( P  e.  X  /\  A  e.  ( P
( ball `  D ) +oo ) )  ->  ( A  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) A ) )
1093adant1 967 . . . 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 146 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  P ( `' D " RR ) A )
123, 11erthi 6405 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  [ P ] ( `' D " RR )  =  [ A ]
( `' D " RR ) )
13 pnfxr 7690 . . . . . 6  |- +oo  e.  RR*
14 blssm 12349 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\ +oo  e.  RR* )  ->  ( P ( ball `  D ) +oo )  C_  X )
1513, 14mp3an3 1272 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X
)  ->  ( P
( ball `  D ) +oo )  C_  X )
1615sselda 3047 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  A  e.  ( P ( ball `  D ) +oo )
)  ->  A  e.  X )
171xmetec 12365 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X
)  ->  [ A ] ( `' D " RR )  =  ( A ( ball `  D
) +oo ) )
1817adantlr 464 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  A  e.  X )  ->  [ A ] ( `' D " RR )  =  ( A ( ball `  D
) +oo ) )
1916, 18syldan 278 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  A  e.  ( P ( ball `  D ) +oo )
)  ->  [ A ] ( `' D " RR )  =  ( A ( ball `  D
) +oo ) )
20193impa 1144 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  [ A ] ( `' D " RR )  =  ( A (
ball `  D ) +oo ) )
2112, 6, 203eqtr3d 2140 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 103    <-> wb 104    /\ w3a 930    = wceq 1299    e. wcel 1448    C_ wss 3021   class class class wbr 3875   `'ccnv 4476   "cima 4480   ` cfv 5059  (class class class)co 5706    Er wer 6356   [cec 6357   RRcr 7499   +oocpnf 7669   RR*cxr 7671   *Metcxmet 11931   ballcbl 11933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-po 4156  df-iso 4157  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-er 6359  df-ec 6361  df-map 6474  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-2 8637  df-xadd 9401  df-psmet 11938  df-xmet 11939  df-bl 11941
This theorem is referenced by: (None)
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