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Theorem pstmval 26322
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 26316 . . 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 19881 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )
43adantr 465 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  X  =  dom  dom  D )
5 dmeq 5040 . . . . . . . . 9  |-  ( d  =  D  ->  dom  d  =  dom  D )
65dmeqd 5042 . . . . . . . 8  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
76adantl 466 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  dom  dom  D )
84, 7eqtr4d 2478 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  X  =  dom  dom  d )
9 qseq1 7150 . . . . . 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 5691 . . . . . . . 8  |-  ( d  =  D  ->  (~Met `  d )  =  (~Met `  D ) )
12 pstmval.1 . . . . . . . 8  |-  .~  =  (~Met `  D )
1311, 12syl6reqr 2494 . . . . . . 7  |-  ( d  =  D  ->  .~  =  (~Met `  d ) )
14 qseq2 7151 . . . . . . 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 466 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  ( dom  dom  d /.  .~  )  =  ( dom  dom  d /. (~Met `  d ) ) )
1710, 16eqtr2d 2476 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  ( dom  dom  d /. (~Met `  d ) )  =  ( X /.  .~  ) )
18 mpt2eq12 6146 . . . 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 661 . . 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 1013 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  ( X /.  .~  )  /\  b  e.  ( X /.  .~  )
)  ->  d  =  D )
2120oveqd 6108 . . . . . . . 8  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  ( X /.  .~  )  /\  b  e.  ( X /.  .~  )
)  ->  ( x
d y )  =  ( x D y ) )
2221eqeq2d 2454 . . . . . . 7  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  ( X /.  .~  )  /\  b  e.  ( X /.  .~  )
)  ->  ( z  =  ( x d y )  <->  z  =  ( x D y ) ) )
23222rexbidv 2758 . . . . . 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 2557 . . . . 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 4101 . . . 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 6150 . . 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 2475 . 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 5716 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
29 fveq2 5691 . . . . . 6  |-  ( x  =  X  ->  (PsMet `  x )  =  (PsMet `  X ) )
3029eleq2d 2510 . . . . 5  |-  ( x  =  X  ->  ( D  e.  (PsMet `  x
)  <->  D  e.  (PsMet `  X ) ) )
3130rspcev 3073 . . . 4  |-  ( ( X  e.  dom PsMet  /\  D  e.  (PsMet `  X )
)  ->  E. x  e.  dom PsMet D  e.  (PsMet `  x ) )
3228, 31mpancom 669 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  E. x  e.  dom PsMet D  e.  (PsMet `  x ) )
33 df-psmet 17809 . . . . 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 5455 . . . 4  |-  Fun PsMet
35 elunirn 5968 . . . 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 5717 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
39 qsexg 7158 . . . 4  |-  ( X  e.  _V  ->  ( X /.  .~  )  e. 
_V )
4038, 39syl 16 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( X /.  .~  )  e.  _V )
41 eqid 2443 . . . 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 6648 . . 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 661 . 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 5779 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 965    = wceq 1369    e. wcel 1756   {cab 2429   A.wral 2715   E.wrex 2716   {crab 2719   _Vcvv 2972   U.cuni 4091   class class class wbr 4292    e. cmpt 4350    X. cxp 4838   dom cdm 4840   ran crn 4841   Fun wfun 5412   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   /.cqs 7100    ^m cmap 7214   0cc0 9282   RR*cxr 9417    <_ cle 9419   +ecxad 11087  PsMetcpsmet 17800  ~Metcmetid 26313  pstoMetcpstm 26314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-ec 7103  df-qs 7107  df-map 7216  df-xr 9422  df-psmet 17809  df-pstm 26316
This theorem is referenced by:  pstmfval  26323  pstmxmet  26324
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