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Theorem metuvalOLD 20249
Description: Value of the uniform structure generated by metric  D. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
metuvalOLD  |-  ( D  e.  ( *Met `  X )  ->  (metUnifOLD `  D
)  =  ( ( X  X.  X )
filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) ) )
Distinct variable groups:    D, a    X, a

Proof of Theorem metuvalOLD
Dummy variables  w  d  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-metuOLD 17934 . . 3  |- metUnifOLD  =  ( d  e.  U. ran  *Met  |->  ( ( dom  dom  d  X.  dom  dom  d
) filGen ran  ( a  e.  RR+  |->  ( `' d
" ( 0 [,) a ) ) ) ) )
21a1i 11 . 2  |-  ( D  e.  ( *Met `  X )  -> metUnifOLD  =  ( d  e.  U. ran  *Met  |->  ( ( dom  dom  d  X.  dom  dom  d
) filGen ran  ( a  e.  RR+  |->  ( `' d
" ( 0 [,) a ) ) ) ) ) )
3 simpr 461 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  d  =  D )
43dmeqd 5143 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  dom  d  =  dom  D )
54dmeqd 5143 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  dom  dom  D )
6 xmetdmdm 20035 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  X  =  dom  dom  D )
76adantr 465 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  X  =  dom  dom  D )
85, 7eqtr4d 2495 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
98, 8xpeq12d 4966 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ( dom  dom  d  X.  dom  dom  d )  =  ( X  X.  X ) )
10 simplr 754 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  d  =  D )  /\  a  e.  RR+ )  ->  d  =  D )
1110cnveqd 5116 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  d  =  D )  /\  a  e.  RR+ )  ->  `' d  =  `' D
)
1211imaeq1d 5269 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  d  =  D )  /\  a  e.  RR+ )  ->  ( `' d " (
0 [,) a ) )  =  ( `' D " ( 0 [,) a ) ) )
1312mpteq2dva 4479 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ( a  e.  RR+  |->  ( `' d
" ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )
1413rneqd 5168 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ran  ( a  e.  RR+  |->  ( `' d " ( 0 [,) a ) ) )  =  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )
159, 14oveq12d 6211 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ( ( dom  dom  d  X.  dom  dom  d ) filGen ran  (
a  e.  RR+  |->  ( `' d " ( 0 [,) a ) ) ) )  =  ( ( X  X.  X
) filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) ) )
16 elfvdm 5818 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  X  e.  dom  *Met )
17 fveq2 5792 . . . . . 6  |-  ( x  =  X  ->  ( *Met `  x )  =  ( *Met `  X ) )
1817eleq2d 2521 . . . . 5  |-  ( x  =  X  ->  ( D  e.  ( *Met `  x )  <->  D  e.  ( *Met `  X
) ) )
1918rspcev 3172 . . . 4  |-  ( ( X  e.  dom  *Met  /\  D  e.  ( *Met `  X
) )  ->  E. x  e.  dom  *Met D  e.  ( *Met `  x ) )
2016, 19mpancom 669 . . 3  |-  ( D  e.  ( *Met `  X )  ->  E. x  e.  dom  *Met D  e.  ( *Met `  x ) )
21 df-xmet 17928 . . . . 5  |-  *Met  =  ( x  e. 
_V  |->  { d  e.  ( RR*  ^m  (
x  X.  x ) )  |  A. y  e.  x  A. z  e.  x  ( (
( y d z )  =  0  <->  y  =  z )  /\  A. w  e.  x  ( y d z )  <_  ( ( w d y ) +e ( w d z ) ) ) } )
2221funmpt2 5556 . . . 4  |-  Fun  *Met
23 elunirn 6070 . . . 4  |-  ( Fun 
*Met  ->  ( D  e.  U. ran  *Met 
<->  E. x  e.  dom  *Met D  e.  ( *Met `  x
) ) )
2422, 23ax-mp 5 . . 3  |-  ( D  e.  U. ran  *Met 
<->  E. x  e.  dom  *Met D  e.  ( *Met `  x
) )
2520, 24sylibr 212 . 2  |-  ( D  e.  ( *Met `  X )  ->  D  e.  U. ran  *Met )
26 ovex 6218 . . 3  |-  ( ( X  X.  X )
filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )  e.  _V
2726a1i 11 . 2  |-  ( D  e.  ( *Met `  X )  ->  (
( X  X.  X
) filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )  e.  _V )
282, 15, 25, 27fvmptd 5881 1  |-  ( D  e.  ( *Met `  X )  ->  (metUnifOLD `  D
)  =  ( ( X  X.  X )
filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   {crab 2799   _Vcvv 3071   U.cuni 4192   class class class wbr 4393    |-> cmpt 4451    X. cxp 4939   `'ccnv 4940   dom cdm 4941   ran crn 4942   "cima 4944   Fun wfun 5513   ` cfv 5519  (class class class)co 6193    ^m cmap 7317   0cc0 9386   RR*cxr 9521    <_ cle 9523   RR+crp 11095   +ecxad 11191   [,)cico 11406   *Metcxmt 17919   filGencfg 17923  metUnifOLDcmetuOLD 17925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-map 7319  df-xr 9526  df-xmet 17928  df-metuOLD 17934
This theorem is referenced by:  metuustOLD  20271  cfilucfil2OLD  20273  metuelOLD  20277  metutopOLD  20282  metucnOLD  20288
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