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Theorem pstmval 27526
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 27520 . . 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 20560 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )
43adantr 465 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  X  =  dom  dom  D )
5 dmeq 5202 . . . . . . . . 9  |-  ( d  =  D  ->  dom  d  =  dom  D )
65dmeqd 5204 . . . . . . . 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 2511 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  X  =  dom  dom  d )
9 qseq1 7361 . . . . . 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 5865 . . . . . . . 8  |-  ( d  =  D  ->  (~Met `  d )  =  (~Met `  D ) )
12 pstmval.1 . . . . . . . 8  |-  .~  =  (~Met `  D )
1311, 12syl6reqr 2527 . . . . . . 7  |-  ( d  =  D  ->  .~  =  (~Met `  d ) )
14 qseq2 7362 . . . . . . 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 2509 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  ( dom  dom  d /. (~Met `  d ) )  =  ( X /.  .~  ) )
18 mpt2eq12 6340 . . . 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 1021 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  ( X /.  .~  )  /\  b  e.  ( X /.  .~  )
)  ->  d  =  D )
2120oveqd 6300 . . . . . . . 8  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  ( X /.  .~  )  /\  b  e.  ( X /.  .~  )
)  ->  ( x
d y )  =  ( x D y ) )
2221eqeq2d 2481 . . . . . . 7  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  ( X /.  .~  )  /\  b  e.  ( X /.  .~  )
)  ->  ( z  =  ( x d y )  <->  z  =  ( x D y ) ) )
23222rexbidv 2980 . . . . . 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 2603 . . . . 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 4255 . . . 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 6344 . . 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 2508 . 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 5891 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
29 fveq2 5865 . . . . . 6  |-  ( x  =  X  ->  (PsMet `  x )  =  (PsMet `  X ) )
3029eleq2d 2537 . . . . 5  |-  ( x  =  X  ->  ( D  e.  (PsMet `  x
)  <->  D  e.  (PsMet `  X ) ) )
3130rspcev 3214 . . . 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 18198 . . . . 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 5624 . . . 4  |-  Fun PsMet
35 elunirn 6150 . . . 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 5892 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
39 qsexg 7369 . . . 4  |-  ( X  e.  _V  ->  ( X /.  .~  )  e. 
_V )
4038, 39syl 16 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( X /.  .~  )  e.  _V )
41 eqid 2467 . . . 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 6858 . . 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 5954 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 973    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113   U.cuni 4245   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   dom cdm 4999   ran crn 5000   Fun wfun 5581   ` cfv 5587  (class class class)co 6283    |-> cmpt2 6285   /.cqs 7310    ^m cmap 7420   0cc0 9491   RR*cxr 9626    <_ cle 9628   +ecxad 11315  PsMetcpsmet 18189  ~Metcmetid 27517  pstoMetcpstm 27518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-1st 6784  df-2nd 6785  df-ec 7313  df-qs 7317  df-map 7422  df-xr 9631  df-psmet 18198  df-pstm 27520
This theorem is referenced by:  pstmfval  27527  pstmxmet  27528
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