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Theorem metuval 20785
Description: Value of the uniform structure generated by metric  D. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
metuval  |-  ( D  e.  (PsMet `  X
)  ->  (metUnif `  D
)  =  ( ( X  X.  X )
filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) ) )
Distinct variable groups:    D, a    X, a

Proof of Theorem metuval
Dummy variables  u  d  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-metu 18187 . . 3  |- metUnif  =  ( d  e.  U. ran PsMet  |->  ( ( dom  dom  d  X.  dom  dom  d
) filGen ran  ( a  e.  RR+  |->  ( `' d
" ( 0 [,) a ) ) ) ) )
21a1i 11 . 2  |-  ( D  e.  (PsMet `  X
)  -> metUnif  =  ( d  e.  U. ran PsMet  |->  ( ( dom  dom  d  X.  dom  dom  d ) filGen ran  ( a  e.  RR+  |->  ( `' d " (
0 [,) a ) ) ) ) ) )
3 simpr 461 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  d  =  D )
43dmeqd 5203 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  d  =  dom  D )
54dmeqd 5203 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  dom  dom  D )
6 psmetdmdm 20541 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )
76adantr 465 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  X  =  dom  dom  D )
85, 7eqtr4d 2511 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
98, 8xpeq12d 5024 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  ( dom  dom  d  X.  dom  dom  d )  =  ( X  X.  X ) )
10 simplr 754 . . . . . . 7  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  RR+ )  ->  d  =  D )
1110cnveqd 5176 . . . . . 6  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  RR+ )  ->  `' d  =  `' D
)
1211imaeq1d 5334 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  RR+ )  ->  ( `' d " (
0 [,) a ) )  =  ( `' D " ( 0 [,) a ) ) )
1312mpteq2dva 4533 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
a  e.  RR+  |->  ( `' d " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )
1413rneqd 5228 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  ran  ( a  e.  RR+  |->  ( `' d " (
0 [,) a ) ) )  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )
159, 14oveq12d 6300 . 2  |-  ( ( D  e.  (PsMet `  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 5890 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
17 fveq2 5864 . . . . . 6  |-  ( x  =  X  ->  (PsMet `  x )  =  (PsMet `  X ) )
1817eleq2d 2537 . . . . 5  |-  ( x  =  X  ->  ( D  e.  (PsMet `  x
)  <->  D  e.  (PsMet `  X ) ) )
1918rspcev 3214 . . . 4  |-  ( ( X  e.  dom PsMet  /\  D  e.  (PsMet `  X )
)  ->  E. x  e.  dom PsMet D  e.  (PsMet `  x ) )
2016, 19mpancom 669 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  E. x  e.  dom PsMet D  e.  (PsMet `  x ) )
21 df-psmet 18179 . . . . 5  |- PsMet  =  ( y  e.  _V  |->  { u  e.  ( RR*  ^m  ( y  X.  y
) )  |  A. z  e.  y  (
( z u z )  =  0  /\ 
A. w  e.  y 
A. v  e.  y  ( z u w )  <_  ( (
v u z ) +e ( v u w ) ) ) } )
2221funmpt2 5623 . . . 4  |-  Fun PsMet
23 elunirn 6149 . . . 4  |-  ( Fun PsMet  ->  ( D  e.  U. ran PsMet  <->  E. x  e.  dom PsMet D  e.  (PsMet `  x
) ) )
2422, 23ax-mp 5 . . 3  |-  ( D  e.  U. ran PsMet  <->  E. x  e.  dom PsMet D  e.  (PsMet `  x ) )
2520, 24sylibr 212 . 2  |-  ( D  e.  (PsMet `  X
)  ->  D  e.  U.
ran PsMet )
26 ovex 6307 . . 3  |-  ( ( X  X.  X )
filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) ) )  e.  _V
2726a1i 11 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ( ( X  X.  X ) filGen ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )  e.  _V )
282, 15, 25, 27fvmptd 5953 1  |-  ( D  e.  (PsMet `  X
)  ->  (metUnif `  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 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113   U.cuni 4245   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   Fun wfun 5580   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   0cc0 9488   RR*cxr 9623    <_ cle 9625   RR+crp 11216   +ecxad 11312   [,)cico 11527  PsMetcpsmet 18170   filGencfg 18175  metUnifcmetu 18178
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545
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-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-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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-xr 9628  df-psmet 18179  df-metu 18187
This theorem is referenced by:  metuust  20807  cfilucfil2  20809  metuel  20813  psmetutop  20818  restmetu  20822  metucn  20824
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