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Theorem pstmval 28772
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 28766 . . 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 21399 . . . . . . . 8  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )
43adantr 472 . . . . . . 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 473 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  dom  dom  D )
84, 7eqtr4d 2508 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  X  =  dom  dom  d )
9 qseq1 7431 . . . . . 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 5879 . . . . . . . 8  |-  ( d  =  D  ->  (~Met `  d )  =  (~Met `  D ) )
12 pstmval.1 . . . . . . . 8  |-  .~  =  (~Met `  D )
1311, 12syl6reqr 2524 . . . . . . 7  |-  ( d  =  D  ->  .~  =  (~Met `  d ) )
14 qseq2 7432 . . . . . . 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 473 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  ( dom  dom  d /.  .~  )  =  ( dom  dom  d /. (~Met `  d ) ) )
1710, 16eqtr2d 2506 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  ( dom  dom  d /. (~Met `  d ) )  =  ( X /.  .~  ) )
18 mpt2eq12 6370 . . . 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 673 . . 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 1055 . . . . . . . . 9  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  ( X /.  .~  )  /\  b  e.  ( X /.  .~  )
)  ->  d  =  D )
2120oveqd 6325 . . . . . . . 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 2897 . . . . . 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 2589 . . . . 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 4200 . . . 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 6374 . . 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 2505 . 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 5905 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
29 fveq2 5879 . . . . . 6  |-  ( x  =  X  ->  (PsMet `  x )  =  (PsMet `  X ) )
3029eleq2d 2534 . . . . 5  |-  ( x  =  X  ->  ( D  e.  (PsMet `  x
)  <->  D  e.  (PsMet `  X ) ) )
3130rspcev 3136 . . . 4  |-  ( ( X  e.  dom PsMet  /\  D  e.  (PsMet `  X )
)  ->  E. x  e.  dom PsMet D  e.  (PsMet `  x ) )
3228, 31mpancom 682 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  E. x  e.  dom PsMet D  e.  (PsMet `  x ) )
33 df-psmet 19039 . . . . 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 5626 . . . 4  |-  Fun PsMet
35 elunirn 6174 . . . 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 217 . 2  |-  ( D  e.  (PsMet `  X
)  ->  D  e.  U.
ran PsMet )
38 elfvex 5906 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  _V )
39 qsexg 7439 . . . 4  |-  ( X  e.  _V  ->  ( X /.  .~  )  e. 
_V )
4038, 39syl 17 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  ( X /.  .~  )  e.  _V )
41 mpt2exga 6888 . . 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 673 . 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 5969 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   {cab 2457   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031   U.cuni 4190   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   dom cdm 4839   ran crn 4840   Fun wfun 5583   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   /.cqs 7380    ^m cmap 7490   0cc0 9557   RR*cxr 9692    <_ cle 9694   +ecxad 11430  PsMetcpsmet 19031  ~Metcmetid 28763  pstoMetcpstm 28764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-ec 7383  df-qs 7387  df-map 7492  df-xr 9697  df-psmet 19039  df-pstm 28766
This theorem is referenced by:  pstmfval  28773  pstmxmet  28774
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