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Theorem pstmval 28537
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 28531 . . 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 21252 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )
43adantr 466 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  X  =  dom  dom  D )
5 dmeq 5055 . . . . . . . . 9  |-  ( d  =  D  ->  dom  d  =  dom  D )
65dmeqd 5057 . . . . . . . 8  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
76adantl 467 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  dom  dom  D )
84, 7eqtr4d 2473 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  X  =  dom  dom  d )
9 qseq1 7421 . . . . . 6  |-  ( X  =  dom  dom  d  ->  ( X /.  .~  )  =  ( dom  dom  d /.  .~  )
)
108, 9syl 17 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  ( X /.  .~  )  =  ( dom  dom  d /.  .~  ) )
11 fveq2 5881 . . . . . . . 8  |-  ( d  =  D  ->  (~Met `  d )  =  (~Met `  D ) )
12 pstmval.1 . . . . . . . 8  |-  .~  =  (~Met `  D )
1311, 12syl6reqr 2489 . . . . . . 7  |-  ( d  =  D  ->  .~  =  (~Met `  d ) )
14 qseq2 7422 . . . . . . 7  |-  (  .~  =  (~Met `  d )  ->  ( dom  dom  d /.  .~  )  =  ( dom  dom  d /. (~Met `  d ) ) )
1513, 14syl 17 . . . . . 6  |-  ( d  =  D  ->  ( dom  dom  d /.  .~  )  =  ( dom  dom  d /. (~Met `  d ) ) )
1615adantl 467 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  ( dom  dom  d /.  .~  )  =  ( dom  dom  d /. (~Met `  d ) ) )
1710, 16eqtr2d 2471 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  ( dom  dom  d /. (~Met `  d ) )  =  ( X /.  .~  ) )
18 mpt2eq12 6365 . . . 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 665 . . 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 1030 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  ( X /.  .~  )  /\  b  e.  ( X /.  .~  )
)  ->  d  =  D )
2120oveqd 6322 . . . . . . . 8  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  ( X /.  .~  )  /\  b  e.  ( X /.  .~  )
)  ->  ( x
d y )  =  ( x D y ) )
2221eqeq2d 2443 . . . . . . 7  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  ( X /.  .~  )  /\  b  e.  ( X /.  .~  )
)  ->  ( z  =  ( x d y )  <->  z  =  ( x D y ) ) )
23222rexbidv 2953 . . . . . 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 2565 . . . . 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 4232 . . . 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 6369 . . 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 2470 . 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 5907 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
29 fveq2 5881 . . . . . 6  |-  ( x  =  X  ->  (PsMet `  x )  =  (PsMet `  X ) )
3029eleq2d 2499 . . . . 5  |-  ( x  =  X  ->  ( D  e.  (PsMet `  x
)  <->  D  e.  (PsMet `  X ) ) )
3130rspcev 3188 . . . 4  |-  ( ( X  e.  dom PsMet  /\  D  e.  (PsMet `  X )
)  ->  E. x  e.  dom PsMet D  e.  (PsMet `  x ) )
3228, 31mpancom 673 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  E. x  e.  dom PsMet D  e.  (PsMet `  x ) )
33 df-psmet 18897 . . . . 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 5638 . . . 4  |-  Fun PsMet
35 elunirn 6171 . . . 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 215 . 2  |-  ( D  e.  (PsMet `  X
)  ->  D  e.  U.
ran PsMet )
38 elfvex 5908 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
39 qsexg 7429 . . . 4  |-  ( X  e.  _V  ->  ( X /.  .~  )  e. 
_V )
4038, 39syl 17 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( X /.  .~  )  e.  _V )
41 mpt2exga 6883 . . 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 )
4240, 40, 41syl2anc 665 . 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 )
432, 27, 37, 42fvmptd 5970 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   {cab 2414   A.wral 2782   E.wrex 2783   {crab 2786   _Vcvv 3087   U.cuni 4222   class class class wbr 4426    |-> cmpt 4484    X. cxp 4852   dom cdm 4854   ran crn 4855   Fun wfun 5595   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   /.cqs 7370    ^m cmap 7480   0cc0 9538   RR*cxr 9673    <_ cle 9675   +ecxad 11407  PsMetcpsmet 18889  ~Metcmetid 28528  pstoMetcpstm 28529
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-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  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-ral 2787  df-rex 2788  df-reu 2789  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-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-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-ec 7373  df-qs 7377  df-map 7482  df-xr 9678  df-psmet 18897  df-pstm 28531
This theorem is referenced by:  pstmfval  28538  pstmxmet  28539
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