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Theorem tmsval 18401
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 18261 . . . 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 5010 . . . . . . . . 9  |-  ( d  =  D  ->  dom  d  =  dom  D )
54dmeqd 5012 . . . . . . . 8  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
6 xmetf 18268 . . . . . . . . . . 11  |-  ( D  e.  ( * Met `  X )  ->  D : ( X  X.  X ) --> RR* )
7 fdm 5535 . . . . . . . . . . 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 5012 . . . . . . . . 9  |-  ( D  e.  ( * Met `  X )  ->  dom  dom 
D  =  dom  ( X  X.  X ) )
10 dmxpid 5029 . . . . . . . . 9  |-  dom  ( X  X.  X )  =  X
119, 10syl6eq 2435 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  dom  dom 
D  =  X )
125, 11sylan9eqr 2441 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
1312opeq2d 3933 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  <. ( Base `  ndx ) ,  dom  dom  d >.  =  <. (
Base `  ndx ) ,  X >. )
14 simpr 448 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  d  =  D )
1514opeq2d 3933 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  <. ( dist `  ndx ) ,  d
>.  =  <. ( dist `  ndx ) ,  D >. )
1613, 15preq12d 3834 . . . . 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 2437 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  { <. ( Base `  ndx ) ,  dom  dom  d >. , 
<. ( dist `  ndx ) ,  d >. }  =  M )
1914fveq2d 5672 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( MetOpen `  d )  =  (
MetOpen `  D ) )
2019opeq2d 3933 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  <. (TopSet `  ndx ) ,  ( MetOpen `  d ) >.  =  <. (TopSet `  ndx ) ,  (
MetOpen `  D ) >.
)
2118, 20oveq12d 6038 . . 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 5694 . . . 4  |-  ( * Met `  X ) 
C_  U. ran  * Met
2322sseli 3287 . . 3  |-  ( D  e.  ( * Met `  X )  ->  D  e.  U. ran  * Met )
24 ovex 6045 . . . 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 5749 . 2  |-  ( D  e.  ( * Met `  X )  ->  (toMetSp `  D )  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. )
)
271, 26syl5eq 2431 1  |-  ( D  e.  ( * Met `  X )  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  (
MetOpen `  D ) >.
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899   {cpr 3758   <.cop 3760   U.cuni 3957    e. cmpt 4207    X. cxp 4816   dom cdm 4818   ran crn 4819   -->wf 5390   ` cfv 5394  (class class class)co 6020   RR*cxr 9052   ndxcnx 13393   sSet csts 13394   Basecbs 13396  TopSetcts 13462   distcds 13465   * Metcxmt 16612   MetOpencmopn 16617  toMetSpctmt 18258
This theorem is referenced by:  tmslem  18402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-map 6956  df-xr 9057  df-xmet 16619  df-tms 18261
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