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Theorem isxms 14403
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 5534 . . . 4  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  ( TopOpen `  K )
)
2 isms.j . . . 4  |-  J  =  ( TopOpen `  K )
31, 2eqtr4di 2240 . . 3  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  J )
4 fveq2 5534 . . . . . 6  |-  ( f  =  K  ->  ( dist `  f )  =  ( dist `  K
) )
5 fveq2 5534 . . . . . . . 8  |-  ( f  =  K  ->  ( Base `  f )  =  ( Base `  K
) )
6 isms.x . . . . . . . 8  |-  X  =  ( Base `  K
)
75, 6eqtr4di 2240 . . . . . . 7  |-  ( f  =  K  ->  ( Base `  f )  =  X )
87sqxpeqd 4670 . . . . . 6  |-  ( f  =  K  ->  (
( Base `  f )  X.  ( Base `  f
) )  =  ( X  X.  X ) )
94, 8reseq12d 4926 . . . . 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, 10eqtr4di 2240 . . . 4  |-  ( f  =  K  ->  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) )  =  D )
1211fveq2d 5538 . . 3  |-  ( f  =  K  ->  ( MetOpen
`  ( ( dist `  f )  |`  (
( Base `  f )  X.  ( Base `  f
) ) ) )  =  ( MetOpen `  D
) )
133, 12eqeq12d 2204 . 2  |-  ( f  =  K  ->  (
( TopOpen `  f )  =  ( MetOpen `  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) ) )  <->  J  =  ( MetOpen
`  D ) ) )
14 df-xms 14291 . 2  |-  *MetSp  =  { f  e.  TopSp  |  ( TopOpen `  f )  =  ( MetOpen `  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) ) ) }
1513, 14elrab2 2911 1  |-  ( K  e.  *MetSp  <->  ( K  e.  TopSp  /\  J  =  ( MetOpen `  D )
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2160    X. cxp 4642    |` cres 4646   ` cfv 5235   Basecbs 12511   distcds 12595   TopOpenctopn 12742   MetOpencmopn 13851   TopSpctps 13982   *MetSpcxms 14288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-rab 2477  df-v 2754  df-un 3148  df-in 3150  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-res 4656  df-iota 5196  df-fv 5243  df-xms 14291
This theorem is referenced by:  isxms2  14404  xmstopn  14407  xmstps  14409  xmspropd  14429
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