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Theorem metuvalOLD 21218
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 18613 . . 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 459 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  d  =  D )
43dmeqd 5194 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  dom  d  =  dom  D )
54dmeqd 5194 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  dom  dom  D )
6 xmetdmdm 21004 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  X  =  dom  dom  D )
76adantr 463 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  X  =  dom  dom  D )
85, 7eqtr4d 2498 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
98, 8xpeq12d 5013 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ( dom  dom  d  X.  dom  dom  d )  =  ( X  X.  X ) )
10 simplr 753 . . . . . . 7  |-  ( ( ( D  e.  ( *Met `  X
)  /\  d  =  D )  /\  a  e.  RR+ )  ->  d  =  D )
1110cnveqd 5167 . . . . . 6  |-  ( ( ( D  e.  ( *Met `  X
)  /\  d  =  D )  /\  a  e.  RR+ )  ->  `' d  =  `' D
)
1211imaeq1d 5324 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  d  =  D )  /\  a  e.  RR+ )  ->  ( `' d " (
0 [,) a ) )  =  ( `' D " ( 0 [,) a ) ) )
1312mpteq2dva 4525 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ( a  e.  RR+  |->  ( `' d
" ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )
1413rneqd 5219 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ran  ( a  e.  RR+  |->  ( `' d " ( 0 [,) a ) ) )  =  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )
159, 14oveq12d 6288 . 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 5874 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  X  e.  dom  *Met )
17 fveq2 5848 . . . . . 6  |-  ( x  =  X  ->  ( *Met `  x )  =  ( *Met `  X ) )
1817eleq2d 2524 . . . . 5  |-  ( x  =  X  ->  ( D  e.  ( *Met `  x )  <->  D  e.  ( *Met `  X
) ) )
1918rspcev 3207 . . . 4  |-  ( ( X  e.  dom  *Met  /\  D  e.  ( *Met `  X
) )  ->  E. x  e.  dom  *Met D  e.  ( *Met `  x ) )
2016, 19mpancom 667 . . 3  |-  ( D  e.  ( *Met `  X )  ->  E. x  e.  dom  *Met D  e.  ( *Met `  x ) )
21 df-xmet 18607 . . . . 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 5607 . . . 4  |-  Fun  *Met
23 elunirn 6138 . . . 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 6298 . . 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 5936 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 367    = wceq 1398    e. wcel 1823   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106   U.cuni 4235   class class class wbr 4439    |-> cmpt 4497    X. cxp 4986   `'ccnv 4987   dom cdm 4988   ran crn 4989   "cima 4991   Fun wfun 5564   ` cfv 5570  (class class class)co 6270    ^m cmap 7412   0cc0 9481   RR*cxr 9616    <_ cle 9618   RR+crp 11221   +ecxad 11319   [,)cico 11534   *Metcxmt 18598   filGencfg 18602  metUnifOLDcmetuOLD 18604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-xr 9621  df-xmet 18607  df-metuOLD 18613
This theorem is referenced by:  metuustOLD  21240  cfilucfil2OLD  21242  metuelOLD  21246  metutopOLD  21251  metucnOLD  21257
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