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Theorem tmsval 20056
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 19897 . . . 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 5040 . . . . . . . . 9  |-  ( d  =  D  ->  dom  d  =  dom  D )
54dmeqd 5042 . . . . . . . 8  |-  ( d  =  D  ->  dom  dom  d  =  dom  dom  D )
6 xmetf 19904 . . . . . . . . . . 11  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
7 fdm 5563 . . . . . . . . . . 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 5042 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  dom  dom 
D  =  dom  ( X  X.  X ) )
10 dmxpid 5059 . . . . . . . . 9  |-  dom  ( X  X.  X )  =  X
119, 10syl6eq 2491 . . . . . . . 8  |-  ( D  e.  ( *Met `  X )  ->  dom  dom 
D  =  X )
125, 11sylan9eqr 2497 . . . . . . 7  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  dom  dom  d  =  X )
1312opeq2d 4066 . . . . . 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 4066 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  <. ( dist `  ndx ) ,  d
>.  =  <. ( dist `  ndx ) ,  D >. )
1613, 15preq12d 3962 . . . . 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 2493 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  { <. ( Base `  ndx ) ,  dom  dom  d >. , 
<. ( dist `  ndx ) ,  d >. }  =  M )
1914fveq2d 5695 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  ( MetOpen `  d )  =  (
MetOpen `  D ) )
2019opeq2d 4066 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  d  =  D )  ->  <. (TopSet `  ndx ) ,  ( MetOpen `  d ) >.  =  <. (TopSet `  ndx ) ,  (
MetOpen `  D ) >.
)
2118, 20oveq12d 6109 . . 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 5713 . . . 4  |-  ( *Met `  X ) 
C_  U. ran  *Met
2322sseli 3352 . . 3  |-  ( D  e.  ( *Met `  X )  ->  D  e.  U. ran  *Met )
24 ovex 6116 . . . 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 5779 . 2  |-  ( D  e.  ( *Met `  X )  ->  (toMetSp `  D )  =  ( M sSet  <. (TopSet `  ndx ) ,  ( MetOpen `  D ) >. )
)
271, 26syl5eq 2487 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 1369    e. wcel 1756   _Vcvv 2972   {cpr 3879   <.cop 3883   U.cuni 4091    e. cmpt 4350    X. cxp 4838   dom cdm 4840   ran crn 4841   -->wf 5414   ` cfv 5418  (class class class)co 6091   RR*cxr 9417   ndxcnx 14171   sSet csts 14172   Basecbs 14174  TopSetcts 14244   distcds 14247   *Metcxmt 17801   MetOpencmopn 17806  toMetSpctmt 19894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-map 7216  df-xr 9422  df-xmet 17810  df-tms 19897
This theorem is referenced by:  tmslem  20057
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