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Theorem blpnfctr 21382
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 2429 . . . . 5  |-  ( `' D " RR )  =  ( `' D " RR )
21xmeter 21379 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  ( `' D " RR )  Er  X )
323ad2ant1 1026 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  -> 
( `' D " RR )  Er  X
)
4 simp3 1007 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  A  e.  ( P
( ball `  D ) +oo ) )
51xmetec 21380 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X
)  ->  [ P ] ( `' D " RR )  =  ( P ( ball `  D
) +oo ) )
653adant3 1025 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  [ P ] ( `' D " RR )  =  ( P (
ball `  D ) +oo ) )
74, 6eleqtrrd 2520 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  A  e.  [ P ] ( `' D " RR ) )
8 elecg 7410 . . . . . 6  |-  ( ( A  e.  ( P ( ball `  D
) +oo )  /\  P  e.  X )  ->  ( A  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) A ) )
98ancoms 454 . . . . 5  |-  ( ( P  e.  X  /\  A  e.  ( P
( ball `  D ) +oo ) )  ->  ( A  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) A ) )
1093adant1 1023 . . . 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 213 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  P ( `' D " RR ) A )
123, 11erthi 7418 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  [ P ] ( `' D " RR )  =  [ A ]
( `' D " RR ) )
13 pnfxr 11412 . . . . . 6  |- +oo  e.  RR*
14 blssm 21364 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\ +oo  e.  RR* )  ->  ( P ( ball `  D ) +oo )  C_  X )
1513, 14mp3an3 1349 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X
)  ->  ( P
( ball `  D ) +oo )  C_  X )
1615sselda 3470 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  A  e.  ( P ( ball `  D ) +oo )
)  ->  A  e.  X )
171xmetec 21380 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X
)  ->  [ A ] ( `' D " RR )  =  ( A ( ball `  D
) +oo ) )
1817adantlr 719 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  A  e.  X )  ->  [ A ] ( `' D " RR )  =  ( A ( ball `  D
) +oo ) )
1916, 18syldan 472 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  P  e.  X )  /\  A  e.  ( P ( ball `  D ) +oo )
)  ->  [ A ] ( `' D " RR )  =  ( A ( ball `  D
) +oo ) )
20193impa 1200 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
) +oo ) )  ->  [ A ] ( `' D " RR )  =  ( A (
ball `  D ) +oo ) )
2112, 6, 203eqtr3d 2478 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 setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    C_ wss 3442   class class class wbr 4426   `'ccnv 4853   "cima 4857   ` cfv 5601  (class class class)co 6305    Er wer 7368   [cec 7369   RRcr 9537   +oocpnf 9671   RR*cxr 9673   *Metcxmt 18890   ballcbl 18892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7371  df-ec 7373  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-2 10668  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-psmet 18897  df-xmet 18898  df-bl 18900
This theorem is referenced by:  metdstri  21779
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