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Theorem tmsval 18043
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 17903 . . . 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 4895 . . . . . . . . 9  |-  ( d  =  D  ->  dom  d  =  dom  D )
54dmeqd 4897 . . . . . . . 8  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
6 xmetf 17910 . . . . . . . . . . 11  |-  ( D  e.  ( * Met `  X )  ->  D : ( X  X.  X ) --> RR* )
7 fdm 5409 . . . . . . . . . . 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 4897 . . . . . . . . 9  |-  ( D  e.  ( * Met `  X )  ->  dom  dom 
D  =  dom  ( X  X.  X ) )
10 dmxpid 4914 . . . . . . . . 9  |-  dom  ( X  X.  X )  =  X
119, 10syl6eq 2344 . . . . . . . 8  |-  ( D  e.  ( * Met `  X )  ->  dom  dom 
D  =  X )
125, 11sylan9eqr 2350 . . . . . . 7  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
1312opeq2d 3819 . . . . . 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 3819 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  <. ( dist `  ndx ) ,  d
>.  =  <. ( dist `  ndx ) ,  D >. )
1613, 15preq12d 3727 . . . . 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 2346 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  { <. ( Base `  ndx ) ,  dom  dom  d >. , 
<. ( dist `  ndx ) ,  d >. }  =  M )
1914fveq2d 5545 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  ( MetOpen `  d )  =  (
MetOpen `  D ) )
2019opeq2d 3819 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  d  =  D )  ->  <. (TopSet `  ndx ) ,  ( MetOpen `  d ) >.  =  <. (TopSet `  ndx ) ,  (
MetOpen `  D ) >.
)
2118, 20oveq12d 5892 . . 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 5567 . . . 4  |-  ( * Met `  X ) 
C_  U. ran  * Met
2322sseli 3189 . . 3  |-  ( D  e.  ( * Met `  X )  ->  D  e.  U. ran  * Met )
24 ovex 5899 . . . 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 5622 . 2  |-  ( D  e.  ( * Met `  X )  ->  (toMetSp `  D )  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. )
)
271, 26syl5eq 2340 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 1632    e. wcel 1696   _Vcvv 2801   {cpr 3654   <.cop 3656   U.cuni 3843    e. cmpt 4093    X. cxp 4703   dom cdm 4705   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   RR*cxr 8882   ndxcnx 13161   sSet csts 13162   Basecbs 13164  TopSetcts 13230   distcds 13233   * Metcxmt 16385   MetOpencmopn 16388  toMetSpctmt 17900
This theorem is referenced by:  tmslem  18044
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-xr 8887  df-xmet 16389  df-tms 17903
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