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Theorem isxms 12634
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 5421 . . . 4  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  ( TopOpen `  K )
)
2 isms.j . . . 4  |-  J  =  ( TopOpen `  K )
31, 2eqtr4di 2190 . . 3  |-  ( f  =  K  ->  ( TopOpen
`  f )  =  J )
4 fveq2 5421 . . . . . 6  |-  ( f  =  K  ->  ( dist `  f )  =  ( dist `  K
) )
5 fveq2 5421 . . . . . . . 8  |-  ( f  =  K  ->  ( Base `  f )  =  ( Base `  K
) )
6 isms.x . . . . . . . 8  |-  X  =  ( Base `  K
)
75, 6eqtr4di 2190 . . . . . . 7  |-  ( f  =  K  ->  ( Base `  f )  =  X )
87sqxpeqd 4565 . . . . . 6  |-  ( f  =  K  ->  (
( Base `  f )  X.  ( Base `  f
) )  =  ( X  X.  X ) )
94, 8reseq12d 4820 . . . . 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 2190 . . . 4  |-  ( f  =  K  ->  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) )  =  D )
1211fveq2d 5425 . . 3  |-  ( f  =  K  ->  ( MetOpen
`  ( ( dist `  f )  |`  (
( Base `  f )  X.  ( Base `  f
) ) ) )  =  ( MetOpen `  D
) )
133, 12eqeq12d 2154 . 2  |-  ( f  =  K  ->  (
( TopOpen `  f )  =  ( MetOpen `  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) ) )  <->  J  =  ( MetOpen
`  D ) ) )
14 df-xms 12522 . 2  |-  *MetSp  =  { f  e.  TopSp  |  ( TopOpen `  f )  =  ( MetOpen `  (
( dist `  f )  |`  ( ( Base `  f
)  X.  ( Base `  f ) ) ) ) }
1513, 14elrab2 2843 1  |-  ( K  e.  *MetSp  <->  ( K  e.  TopSp  /\  J  =  ( MetOpen `  D )
) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480    X. cxp 4537    |` cres 4541   ` cfv 5123   Basecbs 11973   distcds 12044   TopOpenctopn 12135   MetOpencmopn 12168   TopSpctps 12211   *MetSpcxms 12519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-rab 2425  df-v 2688  df-un 3075  df-in 3077  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-res 4551  df-iota 5088  df-fv 5131  df-xms 12522
This theorem is referenced by:  isxms2  12635  xmstopn  12638  xmstps  12640  xmspropd  12660
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