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Theorem isxms 18469
Description: Express the predicate " <. X ,  D >. is an extended metric space" with underlying set  X and distance function  D. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypotheses
Ref Expression
isms.j  |-  J  =  ( TopOpen `  K )
isms.x  |-  X  =  ( Base `  K
)
isms.d  |-  D  =  ( ( dist `  K
)  |`  ( X  X.  X ) )
Assertion
Ref Expression
isxms  |-  ( K  e.  * MetSp  <->  ( K  e.  TopSp  /\  J  =  ( MetOpen `  D )
) )

Proof of Theorem isxms
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 fveq2 5720 . . . 4  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  ( TopOpen `  K )
)
2 isms.j . . . 4  |-  J  =  ( TopOpen `  K )
31, 2syl6eqr 2485 . . 3  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  J )
4 fveq2 5720 . . . . . 6  |-  ( f  =  K  ->  ( dist `  f )  =  ( dist `  K
) )
5 fveq2 5720 . . . . . . . 8  |-  ( f  =  K  ->  ( Base `  f )  =  ( Base `  K
) )
6 isms.x . . . . . . . 8  |-  X  =  ( Base `  K
)
75, 6syl6eqr 2485 . . . . . . 7  |-  ( f  =  K  ->  ( Base `  f )  =  X )
87, 7xpeq12d 4895 . . . . . 6  |-  ( f  =  K  ->  (
( Base `  f )  X.  ( Base `  f
) )  =  ( X  X.  X ) )
94, 8reseq12d 5139 . . . . 5  |-  ( f  =  K  ->  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) )  =  ( ( dist `  K )  |`  ( X  X.  X ) ) )
10 isms.d . . . . 5  |-  D  =  ( ( dist `  K
)  |`  ( X  X.  X ) )
119, 10syl6eqr 2485 . . . 4  |-  ( f  =  K  ->  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) )  =  D )
1211fveq2d 5724 . . 3  |-  ( f  =  K  ->  ( MetOpen
`  ( ( dist `  f )  |`  (
( Base `  f )  X.  ( Base `  f
) ) ) )  =  ( MetOpen `  D
) )
133, 12eqeq12d 2449 . 2  |-  ( f  =  K  ->  (
( TopOpen `  f )  =  ( MetOpen `  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) ) )  <->  J  =  ( MetOpen
`  D ) ) )
14 df-xms 18342 . 2  |-  * MetSp  =  { f  e.  TopSp  |  ( TopOpen `  f )  =  ( MetOpen `  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) ) ) }
1513, 14elrab2 3086 1  |-  ( K  e.  * MetSp  <->  ( K  e.  TopSp  /\  J  =  ( MetOpen `  D )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    X. cxp 4868    |` cres 4872   ` cfv 5446   Basecbs 13461   distcds 13530   TopOpenctopn 13641   MetOpencmopn 16683   TopSpctps 16953   *
MetSpcxme 18339
This theorem is referenced by:  isxms2  18470  xmstopn  18473  xmstps  18475  xmspropd  18495
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-xp 4876  df-res 4882  df-iota 5410  df-fv 5454  df-xms 18342
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