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Theorem metuval 20123
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 17815 . . 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 5040 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  d  =  dom  D )
54dmeqd 5040 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  dom  dom  D )
6 psmetdmdm 19879 . . . . . 6  |-  ( D  e.  (PsMet `  X
)  ->  X  =  dom  dom  D )
76adantr 465 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  X  =  dom  dom  D )
85, 7eqtr4d 2476 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
98, 8xpeq12d 4863 . . 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 5013 . . . . . 6  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  RR+ )  ->  `' d  =  `' D
)
1211imaeq1d 5166 . . . . 5  |-  ( ( ( D  e.  (PsMet `  X )  /\  d  =  D )  /\  a  e.  RR+ )  ->  ( `' d " (
0 [,) a ) )  =  ( `' D " ( 0 [,) a ) ) )
1312mpteq2dva 4376 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  (
a  e.  RR+  |->  ( `' d " ( 0 [,) a ) ) )  =  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )
1413rneqd 5065 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  d  =  D )  ->  ran  ( a  e.  RR+  |->  ( `' d " (
0 [,) a ) ) )  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) ) )
159, 14oveq12d 6107 . 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 5714 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  X  e.  dom PsMet )
17 fveq2 5689 . . . . . 6  |-  ( x  =  X  ->  (PsMet `  x )  =  (PsMet `  X ) )
1817eleq2d 2508 . . . . 5  |-  ( x  =  X  ->  ( D  e.  (PsMet `  x
)  <->  D  e.  (PsMet `  X ) ) )
1918rspcev 3071 . . . 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 17807 . . . . 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 5453 . . . 4  |-  Fun PsMet
23 elunirn 5966 . . . 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 6114 . . 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 5777 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 1369    e. wcel 1756   A.wral 2713   E.wrex 2714   {crab 2717   _Vcvv 2970   U.cuni 4089   class class class wbr 4290    e. cmpt 4348    X. cxp 4836   `'ccnv 4837   dom cdm 4838   ran crn 4839   "cima 4841   Fun wfun 5410   ` cfv 5416  (class class class)co 6089    ^m cmap 7212   0cc0 9280   RR*cxr 9415    <_ cle 9417   RR+crp 10989   +ecxad 11085   [,)cico 11300  PsMetcpsmet 17798   filGencfg 17803  metUnifcmetu 17806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-map 7214  df-xr 9420  df-psmet 17807  df-metu 17815
This theorem is referenced by:  metuust  20145  cfilucfil2  20147  metuel  20151  psmetutop  20156  restmetu  20160  metucn  20162
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