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Theorem isxms 17993
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 5525 . . . 4  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  ( TopOpen `  K )
)
2 isms.j . . . 4  |-  J  =  ( TopOpen `  K )
31, 2syl6eqr 2333 . . 3  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  J )
4 fveq2 5525 . . . . . 6  |-  ( f  =  K  ->  ( dist `  f )  =  ( dist `  K
) )
5 fveq2 5525 . . . . . . . 8  |-  ( f  =  K  ->  ( Base `  f )  =  ( Base `  K
) )
6 isms.x . . . . . . . 8  |-  X  =  ( Base `  K
)
75, 6syl6eqr 2333 . . . . . . 7  |-  ( f  =  K  ->  ( Base `  f )  =  X )
87, 7xpeq12d 4714 . . . . . 6  |-  ( f  =  K  ->  (
( Base `  f )  X.  ( Base `  f
) )  =  ( X  X.  X ) )
94, 8reseq12d 4956 . . . . 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 2333 . . . 4  |-  ( f  =  K  ->  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) )  =  D )
1211fveq2d 5529 . . 3  |-  ( f  =  K  ->  ( MetOpen
`  ( ( dist `  f )  |`  (
( Base `  f )  X.  ( Base `  f
) ) ) )  =  ( MetOpen `  D
) )
133, 12eqeq12d 2297 . 2  |-  ( f  =  K  ->  (
( TopOpen `  f )  =  ( MetOpen `  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) ) )  <->  J  =  ( MetOpen
`  D ) ) )
14 df-xms 17885 . 2  |-  * MetSp  =  { f  e.  TopSp  |  ( TopOpen `  f )  =  ( MetOpen `  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) ) ) }
1513, 14elrab2 2925 1  |-  ( K  e.  * MetSp  <->  ( K  e.  TopSp  /\  J  =  ( MetOpen `  D )
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    X. cxp 4687    |` cres 4691   ` cfv 5255   Basecbs 13148   distcds 13217   TopOpenctopn 13326   MetOpencmopn 16372   TopSpctps 16634   *
MetSpcxme 17882
This theorem is referenced by:  isxms2  17994  xmstopn  17997  xmstps  17999  xmspropd  18019
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-res 4701  df-iota 5219  df-fv 5263  df-xms 17885
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