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Theorem pstmval 26258
Description: Value of the metric induced by a pseudometric  D. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Hypothesis
Ref Expression
pstmval.1  |-  .~  =  (~Met `  D )
Assertion
Ref Expression
pstmval  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
)  =  ( a  e.  ( X /.  .~  ) ,  b  e.  ( X /.  .~  )  |->  U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x D y ) } ) )
Distinct variable groups:    a, b, x, y, z, D    X, a, b, x, y, z    .~ , a, b, x, y, z

Proof of Theorem pstmval
Dummy variables  c 
d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-pstm 26252 . . 3  |- pstoMet  =  ( d  e.  U. ran PsMet  |->  ( a  e.  ( dom  dom  d /. (~Met `  d ) ) ,  b  e.  ( dom  dom  d /. (~Met `  d ) ) 
|->  U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x d y ) } ) )
21a1i 11 . 2  |-  ( D  e.  (PsMet `  X
)  -> pstoMet  =  ( d  e.  U. ran PsMet  |->  ( a  e.  ( dom  dom  d /. (~Met `  d
) ) ,  b  e.  ( dom  dom  d /. (~Met `  d
) )  |->  U. {
z  |  E. x  e.  a  E. y  e.  b  z  =  ( x d y ) } ) ) )
3 psmetdmdm 19840 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )
43adantr 462 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  X  =  dom  dom  D )
5 dmeq 5036 . . . . . . . . 9  |-  ( d  =  D  ->  dom  d  =  dom  D )
65dmeqd 5038 . . . . . . . 8  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
76adantl 463 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  dom  dom  D )
84, 7eqtr4d 2476 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  X  =  dom  dom  d )
9 qseq1 7146 . . . . . 6  |-  ( X  =  dom  dom  d  ->  ( X /.  .~  )  =  ( dom  dom  d /.  .~  )
)
108, 9syl 16 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  ( X /.  .~  )  =  ( dom  dom  d /.  .~  ) )
11 fveq2 5688 . . . . . . . 8  |-  ( d  =  D  ->  (~Met `  d )  =  (~Met `  D ) )
12 pstmval.1 . . . . . . . 8  |-  .~  =  (~Met `  D )
1311, 12syl6reqr 2492 . . . . . . 7  |-  ( d  =  D  ->  .~  =  (~Met `  d ) )
14 qseq2 7147 . . . . . . 7  |-  (  .~  =  (~Met `  d )  ->  ( dom  dom  d /.  .~  )  =  ( dom  dom  d /. (~Met `  d ) ) )
1513, 14syl 16 . . . . . 6  |-  ( d  =  D  ->  ( dom  dom  d /.  .~  )  =  ( dom  dom  d /. (~Met `  d ) ) )
1615adantl 463 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  ( dom  dom  d /.  .~  )  =  ( dom  dom  d /. (~Met `  d ) ) )
1710, 16eqtr2d 2474 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  ( dom  dom  d /. (~Met `  d ) )  =  ( X /.  .~  ) )
18 mpt2eq12 6145 . . . 4  |-  ( ( ( dom  dom  d /. (~Met `  d )
)  =  ( X /.  .~  )  /\  ( dom  dom  d /. (~Met `  d ) )  =  ( X /.  .~  ) )  ->  (
a  e.  ( dom 
dom  d /. (~Met `  d ) ) ,  b  e.  ( dom 
dom  d /. (~Met `  d ) )  |->  U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x d y ) } )  =  ( a  e.  ( X /.  .~  ) ,  b  e.  ( X /.  .~  )  |-> 
U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x d y ) } ) )
1917, 17, 18syl2anc 656 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
a  e.  ( dom 
dom  d /. (~Met `  d ) ) ,  b  e.  ( dom 
dom  d /. (~Met `  d ) )  |->  U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x d y ) } )  =  ( a  e.  ( X /.  .~  ) ,  b  e.  ( X /.  .~  )  |-> 
U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x d y ) } ) )
20 simp1r 1008 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  ( X /.  .~  )  /\  b  e.  ( X /.  .~  )
)  ->  d  =  D )
2120oveqd 6107 . . . . . . . 8  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  ( X /.  .~  )  /\  b  e.  ( X /.  .~  )
)  ->  ( x
d y )  =  ( x D y ) )
2221eqeq2d 2452 . . . . . . 7  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  ( X /.  .~  )  /\  b  e.  ( X /.  .~  )
)  ->  ( z  =  ( x d y )  <->  z  =  ( x D y ) ) )
23222rexbidv 2756 . . . . . 6  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  ( X /.  .~  )  /\  b  e.  ( X /.  .~  )
)  ->  ( E. x  e.  a  E. y  e.  b  z  =  ( x d y )  <->  E. x  e.  a  E. y  e.  b  z  =  ( x D y ) ) )
2423abbidv 2555 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  ( X /.  .~  )  /\  b  e.  ( X /.  .~  )
)  ->  { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x d y ) }  =  { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x D y ) } )
2524unieqd 4098 . . . 4  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  ( X /.  .~  )  /\  b  e.  ( X /.  .~  )
)  ->  U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x d y ) }  =  U. {
z  |  E. x  e.  a  E. y  e.  b  z  =  ( x D y ) } )
2625mpt2eq3dva 6149 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
a  e.  ( X /.  .~  ) ,  b  e.  ( X /.  .~  )  |->  U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x d y ) } )  =  ( a  e.  ( X /.  .~  ) ,  b  e.  ( X /.  .~  )  |-> 
U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x D y ) } ) )
2719, 26eqtrd 2473 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
a  e.  ( dom 
dom  d /. (~Met `  d ) ) ,  b  e.  ( dom 
dom  d /. (~Met `  d ) )  |->  U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x d y ) } )  =  ( a  e.  ( X /.  .~  ) ,  b  e.  ( X /.  .~  )  |-> 
U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x D y ) } ) )
28 elfvdm 5713 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
29 fveq2 5688 . . . . . 6  |-  ( x  =  X  ->  (PsMet `  x )  =  (PsMet `  X ) )
3029eleq2d 2508 . . . . 5  |-  ( x  =  X  ->  ( D  e.  (PsMet `  x
)  <->  D  e.  (PsMet `  X ) ) )
3130rspcev 3070 . . . 4  |-  ( ( X  e.  dom PsMet  /\  D  e.  (PsMet `  X )
)  ->  E. x  e.  dom PsMet D  e.  (PsMet `  x ) )
3228, 31mpancom 664 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  E. x  e.  dom PsMet D  e.  (PsMet `  x ) )
33 df-psmet 17768 . . . . 5  |- PsMet  =  ( x  e.  _V  |->  { d  e.  ( RR*  ^m  ( x  X.  x
) )  |  A. a  e.  x  (
( a d a )  =  0  /\ 
A. b  e.  x  A. c  e.  x  ( a d b )  <_  ( (
c d a ) +e ( c d b ) ) ) } )
3433funmpt2 5452 . . . 4  |-  Fun PsMet
35 elunirn 5965 . . . 4  |-  ( Fun PsMet  ->  ( D  e.  U. ran PsMet  <->  E. x  e.  dom PsMet D  e.  (PsMet `  x
) ) )
3634, 35ax-mp 5 . . 3  |-  ( D  e.  U. ran PsMet  <->  E. x  e.  dom PsMet D  e.  (PsMet `  x ) )
3732, 36sylibr 212 . 2  |-  ( D  e.  (PsMet `  X
)  ->  D  e.  U.
ran PsMet )
38 elfvex 5714 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
39 qsexg 7154 . . . 4  |-  ( X  e.  _V  ->  ( X /.  .~  )  e. 
_V )
4038, 39syl 16 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( X /.  .~  )  e.  _V )
41 eqid 2441 . . . 4  |-  ( a  e.  ( X /.  .~  ) ,  b  e.  ( X /.  .~  )  |->  U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x D y ) } )  =  ( a  e.  ( X /.  .~  ) ,  b  e.  ( X /.  .~  )  |->  U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x D y ) } )
4241mpt2exg 6647 . . 3  |-  ( ( ( X /.  .~  )  e.  _V  /\  ( X /.  .~  )  e. 
_V )  ->  (
a  e.  ( X /.  .~  ) ,  b  e.  ( X /.  .~  )  |->  U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x D y ) } )  e.  _V )
4340, 40, 42syl2anc 656 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( a  e.  ( X /.  .~  ) ,  b  e.  ( X /.  .~  )  |-> 
U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x D y ) } )  e.  _V )
442, 27, 37, 43fvmptd 5776 1  |-  ( D  e.  (PsMet `  X
)  ->  (pstoMet `  D
)  =  ( a  e.  ( X /.  .~  ) ,  b  e.  ( X /.  .~  )  |->  U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x D y ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   {cab 2427   A.wral 2713   E.wrex 2714   {crab 2717   _Vcvv 2970   U.cuni 4088   class class class wbr 4289    e. cmpt 4347    X. cxp 4834   dom cdm 4836   ran crn 4837   Fun wfun 5409   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   /.cqs 7096    ^m cmap 7210   0cc0 9278   RR*cxr 9413    <_ cle 9415   +ecxad 11083  PsMetcpsmet 17759  ~Metcmetid 26249  pstoMetcpstm 26250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-ec 7099  df-qs 7103  df-map 7212  df-xr 9418  df-psmet 17768  df-pstm 26252
This theorem is referenced by:  pstmfval  26259  pstmxmet  26260
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