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Theorem tmsval 18027
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 17887 . . . 4  |- toMetSp  =  ( d  e.  U. ran  * Met  |->  ( { <. (
Base `  ndx ) ,  dom  dom  d >. , 
<. ( dist `  ndx ) ,  d >. } sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  d ) >. ) )
32a1i 10 . . 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 4879 . . . . . . . . 9  |-  ( d  =  D  ->  dom  d  =  dom  D )
54dmeqd 4881 . . . . . . . 8  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
6 xmetf 17894 . . . . . . . . . . 11  |-  ( D  e.  ( * Met `  X )  ->  D : ( X  X.  X ) --> RR* )
7 fdm 5393 . . . . . . . . . . 11  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
86, 7syl 15 . . . . . . . . . 10  |-  ( D  e.  ( * Met `  X )  ->  dom  D  =  ( X  X.  X ) )
98dmeqd 4881 . . . . . . . . 9  |-  ( D  e.  ( * Met `  X )  ->  dom  dom 
D  =  dom  ( X  X.  X ) )
10 dmxpid 4898 . . . . . . . . 9  |-  dom  ( X  X.  X )  =  X
119, 10syl6eq 2331 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  dom  dom 
D  =  X )
125, 11sylan9eqr 2337 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
1312opeq2d 3803 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  <. ( Base `  ndx ) ,  dom  dom  d >.  =  <. (
Base `  ndx ) ,  X >. )
14 simpr 447 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  d  =  D )
1514opeq2d 3803 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  <. ( dist `  ndx ) ,  d
>.  =  <. ( dist `  ndx ) ,  D >. )
1613, 15preq12d 3714 . . . . 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 2333 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  { <. ( Base `  ndx ) ,  dom  dom  d >. , 
<. ( dist `  ndx ) ,  d >. }  =  M )
1914fveq2d 5529 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( MetOpen `  d )  =  (
MetOpen `  D ) )
2019opeq2d 3803 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  <. (TopSet `  ndx ) ,  ( MetOpen `  d ) >.  =  <. (TopSet `  ndx ) ,  (
MetOpen `  D ) >.
)
2118, 20oveq12d 5876 . . 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 5551 . . . 4  |-  ( * Met `  X ) 
C_  U. ran  * Met
2322sseli 3176 . . 3  |-  ( D  e.  ( * Met `  X )  ->  D  e.  U. ran  * Met )
24 ovex 5883 . . . 4  |-  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. )  e.  _V
2524a1i 10 . . 3  |-  ( D  e.  ( * Met `  X )  ->  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. )  e.  _V )
263, 21, 23, 25fvmptd 5606 . 2  |-  ( D  e.  ( * Met `  X )  ->  (toMetSp `  D )  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. )
)
271, 26syl5eq 2327 1  |-  ( D  e.  ( * Met `  X )  ->  K  =  ( M sSet  <. (TopSet `  ndx ) ,  (
MetOpen `  D ) >.
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   {cpr 3641   <.cop 3643   U.cuni 3827    e. cmpt 4077    X. cxp 4687   dom cdm 4689   ran crn 4690   -->wf 5251   ` cfv 5255  (class class class)co 5858   RR*cxr 8866   ndxcnx 13145   sSet csts 13146   Basecbs 13148  TopSetcts 13214   distcds 13217   * Metcxmt 16369   MetOpencmopn 16372  toMetSpctmt 17884
This theorem is referenced by:  tmslem  18028
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-xr 8871  df-xmet 16373  df-tms 17887
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