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Theorem blpnfctr 17982
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 2283 . . . . 5  |-  ( `' D " RR )  =  ( `' D " RR )
21xmeter 17979 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  ( `' D " RR )  Er  X )
323ad2ant1 976 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  -> 
( `' D " RR )  Er  X
)
4 simp3 957 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  ->  A  e.  ( P
( ball `  D )  +oo ) )
51xmetec 17980 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  [ P ] ( `' D " RR )  =  ( P ( ball `  D
)  +oo ) )
653adant3 975 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  ->  [ P ] ( `' D " RR )  =  ( P (
ball `  D )  +oo ) )
74, 6eleqtrrd 2360 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  ->  A  e.  [ P ] ( `' D " RR ) )
8 elecg 6698 . . . . . 6  |-  ( ( A  e.  ( P ( ball `  D
)  +oo )  /\  P  e.  X )  ->  ( A  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) A ) )
98ancoms 439 . . . . 5  |-  ( ( P  e.  X  /\  A  e.  ( P
( ball `  D )  +oo ) )  ->  ( A  e.  [ P ] ( `' D " RR )  <->  P ( `' D " RR ) A ) )
1093adant1 973 . . . 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 201 . . 3  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  ->  P ( `' D " RR ) A )
123, 11erthi 6706 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  ->  [ P ] ( `' D " RR )  =  [ A ]
( `' D " RR ) )
13 pnfxr 10455 . . . . . 6  |-  +oo  e.  RR*
14 blssm 17968 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  +oo  e.  RR* )  ->  ( P ( ball `  D )  +oo )  C_  X )
1513, 14mp3an3 1266 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X
)  ->  ( P
( ball `  D )  +oo )  C_  X )
1615sselda 3180 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  A  e.  ( P ( ball `  D )  +oo )
)  ->  A  e.  X )
171xmetec 17980 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X
)  ->  [ A ] ( `' D " RR )  =  ( A ( ball `  D
)  +oo ) )
1817adantlr 695 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  A  e.  X )  ->  [ A ] ( `' D " RR )  =  ( A ( ball `  D
)  +oo ) )
1916, 18syldan 456 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  P  e.  X )  /\  A  e.  ( P ( ball `  D )  +oo )
)  ->  [ A ] ( `' D " RR )  =  ( A ( ball `  D
)  +oo ) )
20193impa 1146 . 2  |-  ( ( D  e.  ( * Met `  X )  /\  P  e.  X  /\  A  e.  ( P ( ball `  D
)  +oo ) )  ->  [ A ] ( `' D " RR )  =  ( A (
ball `  D )  +oo ) )
2112, 6, 203eqtr3d 2323 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858    Er wer 6657   [cec 6658   RRcr 8736    +oocpnf 8864   RR*cxr 8866   * Metcxmt 16369   ballcbl 16371
This theorem is referenced by:  metdstri  18355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-ec 6662  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-xmet 16373  df-bl 16375
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