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Theorem tmsval 20852
Description: For any metric there is an associated metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
tmsval.m  |-  M  =  { <. ( Base `  ndx ) ,  X >. , 
<. ( dist `  ndx ) ,  D >. }
tmsval.k  |-  K  =  (toMetSp `  D )
Assertion
Ref Expression
tmsval  |-  ( D  e.  ( *Met `  X )  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  (
MetOpen `  D ) >.
) )

Proof of Theorem tmsval
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 tmsval.k . 2  |-  K  =  (toMetSp `  D )
2 df-tms 20693 . . . 4  |- toMetSp  =  ( d  e.  U. ran  *Met  |->  ( { <. (
Base `  ndx ) ,  dom  dom  d >. , 
<. ( dist `  ndx ) ,  d >. } sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  d ) >. ) )
32a1i 11 . . 3  |-  ( D  e.  ( *Met `  X )  -> toMetSp  =  ( d  e.  U. ran  *Met  |->  ( { <. (
Base `  ndx ) ,  dom  dom  d >. , 
<. ( dist `  ndx ) ,  d >. } sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  d ) >. ) ) )
4 dmeq 5209 . . . . . . . . 9  |-  ( d  =  D  ->  dom  d  =  dom  D )
54dmeqd 5211 . . . . . . . 8  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
6 xmetf 20700 . . . . . . . . . . 11  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
7 fdm 5741 . . . . . . . . . . 11  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
86, 7syl 16 . . . . . . . . . 10  |-  ( D  e.  ( *Met `  X )  ->  dom  D  =  ( X  X.  X ) )
98dmeqd 5211 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  dom  dom 
D  =  dom  ( X  X.  X ) )
10 dmxpid 5228 . . . . . . . . 9  |-  dom  ( X  X.  X )  =  X
119, 10syl6eq 2524 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  dom  dom 
D  =  X )
125, 11sylan9eqr 2530 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
1312opeq2d 4226 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  <. ( Base `  ndx ) ,  dom  dom  d >.  =  <. (
Base `  ndx ) ,  X >. )
14 simpr 461 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  d  =  D )
1514opeq2d 4226 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  <. ( dist `  ndx ) ,  d
>.  =  <. ( dist `  ndx ) ,  D >. )
1613, 15preq12d 4120 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  { <. ( Base `  ndx ) ,  dom  dom  d >. , 
<. ( dist `  ndx ) ,  d >. }  =  { <. ( Base `  ndx ) ,  X >. ,  <. ( dist `  ndx ) ,  D >. } )
17 tmsval.m . . . . 5  |-  M  =  { <. ( Base `  ndx ) ,  X >. , 
<. ( dist `  ndx ) ,  D >. }
1816, 17syl6eqr 2526 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  { <. ( Base `  ndx ) ,  dom  dom  d >. , 
<. ( dist `  ndx ) ,  d >. }  =  M )
1914fveq2d 5876 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ( MetOpen `  d )  =  (
MetOpen `  D ) )
2019opeq2d 4226 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  <. (TopSet `  ndx ) ,  ( MetOpen `  d ) >.  =  <. (TopSet `  ndx ) ,  (
MetOpen `  D ) >.
)
2118, 20oveq12d 6313 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ( { <. ( Base `  ndx ) ,  dom  dom  d >. ,  <. ( dist `  ndx ) ,  d >. } sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  d ) >. )  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. ) )
22 fvssunirn 5895 . . . 4  |-  ( *Met `  X ) 
C_  U. ran  *Met
2322sseli 3505 . . 3  |-  ( D  e.  ( *Met `  X )  ->  D  e.  U. ran  *Met )
24 ovex 6320 . . . 4  |-  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. )  e.  _V
2524a1i 11 . . 3  |-  ( D  e.  ( *Met `  X )  ->  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. )  e.  _V )
263, 21, 23, 25fvmptd 5962 . 2  |-  ( D  e.  ( *Met `  X )  ->  (toMetSp `  D )  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. )
)
271, 26syl5eq 2520 1  |-  ( D  e.  ( *Met `  X )  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  (
MetOpen `  D ) >.
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   {cpr 4035   <.cop 4039   U.cuni 4251    |-> cmpt 4511    X. cxp 5003   dom cdm 5005   ran crn 5006   -->wf 5590   ` cfv 5594  (class class class)co 6295   RR*cxr 9639   ndxcnx 14504   sSet csts 14505   Basecbs 14507  TopSetcts 14578   distcds 14581   *Metcxmt 18273   MetOpencmopn 18278  toMetSpctmt 20690
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-xr 9644  df-xmet 18282  df-tms 20693
This theorem is referenced by:  tmslem  20853
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