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Theorem tmsval 18503
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 18344 . . . 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 5062 . . . . . . . . 9  |-  ( d  =  D  ->  dom  d  =  dom  D )
54dmeqd 5064 . . . . . . . 8  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
6 xmetf 18351 . . . . . . . . . . 11  |-  ( D  e.  ( * Met `  X )  ->  D : ( X  X.  X ) --> RR* )
7 fdm 5587 . . . . . . . . . . 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 5064 . . . . . . . . 9  |-  ( D  e.  ( * Met `  X )  ->  dom  dom 
D  =  dom  ( X  X.  X ) )
10 dmxpid 5081 . . . . . . . . 9  |-  dom  ( X  X.  X )  =  X
119, 10syl6eq 2483 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  dom  dom 
D  =  X )
125, 11sylan9eqr 2489 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
1312opeq2d 3983 . . . . . 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 3983 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  <. ( dist `  ndx ) ,  d
>.  =  <. ( dist `  ndx ) ,  D >. )
1613, 15preq12d 3883 . . . . 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 2485 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  { <. ( Base `  ndx ) ,  dom  dom  d >. , 
<. ( dist `  ndx ) ,  d >. }  =  M )
1914fveq2d 5724 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( MetOpen `  d )  =  (
MetOpen `  D ) )
2019opeq2d 3983 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  <. (TopSet `  ndx ) ,  ( MetOpen `  d ) >.  =  <. (TopSet `  ndx ) ,  (
MetOpen `  D ) >.
)
2118, 20oveq12d 6091 . . 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 5746 . . . 4  |-  ( * Met `  X ) 
C_  U. ran  * Met
2322sseli 3336 . . 3  |-  ( D  e.  ( * Met `  X )  ->  D  e.  U. ran  * Met )
24 ovex 6098 . . . 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 5802 . 2  |-  ( D  e.  ( * Met `  X )  ->  (toMetSp `  D )  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. )
)
271, 26syl5eq 2479 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 1652    e. wcel 1725   _Vcvv 2948   {cpr 3807   <.cop 3809   U.cuni 4007    e. cmpt 4258    X. cxp 4868   dom cdm 4870   ran crn 4871   -->wf 5442   ` cfv 5446  (class class class)co 6073   RR*cxr 9111   ndxcnx 13458   sSet csts 13459   Basecbs 13461  TopSetcts 13527   distcds 13530   * Metcxmt 16678   MetOpencmopn 16683  toMetSpctmt 18341
This theorem is referenced by:  tmslem  18504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-xr 9116  df-xmet 16687  df-tms 18344
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