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Theorem xmspropd 20844
Description: Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
Hypotheses
Ref Expression
xmspropd.1  |-  ( ph  ->  B  =  ( Base `  K ) )
xmspropd.2  |-  ( ph  ->  B  =  ( Base `  L ) )
xmspropd.3  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
xmspropd.4  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
Assertion
Ref Expression
xmspropd  |-  ( ph  ->  ( K  e.  *MetSp  <-> 
L  e.  *MetSp ) )

Proof of Theorem xmspropd
StepHypRef Expression
1 xmspropd.1 . . . . 5  |-  ( ph  ->  B  =  ( Base `  K ) )
2 xmspropd.2 . . . . 5  |-  ( ph  ->  B  =  ( Base `  L ) )
31, 2eqtr3d 2510 . . . 4  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
4 xmspropd.4 . . . 4  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
53, 4tpspropd 19310 . . 3  |-  ( ph  ->  ( K  e.  TopSp  <->  L  e.  TopSp ) )
6 xmspropd.3 . . . . . . 7  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
71, 1xpeq12d 5030 . . . . . . . 8  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
87reseq2d 5279 . . . . . . 7  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
96, 8eqtr3d 2510 . . . . . 6  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
102, 2xpeq12d 5030 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
1110reseq2d 5279 . . . . . 6  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
129, 11eqtr3d 2510 . . . . 5  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
1312fveq2d 5876 . . . 4  |-  ( ph  ->  ( MetOpen `  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) )  =  ( MetOpen `  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) ) )
144, 13eqeq12d 2489 . . 3  |-  ( ph  ->  ( ( TopOpen `  K
)  =  ( MetOpen `  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )  <->  ( TopOpen `  L )  =  (
MetOpen `  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ) ) )
155, 14anbi12d 710 . 2  |-  ( ph  ->  ( ( K  e. 
TopSp  /\  ( TopOpen `  K
)  =  ( MetOpen `  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) ) )  <->  ( L  e.  TopSp  /\  ( TopOpen `  L )  =  (
MetOpen `  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ) ) ) )
16 eqid 2467 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
17 eqid 2467 . . 3  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2467 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
1916, 17, 18isxms 20818 . 2  |-  ( K  e.  *MetSp  <->  ( K  e.  TopSp  /\  ( TopOpen `  K )  =  (
MetOpen `  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) ) ) ) )
20 eqid 2467 . . 3  |-  ( TopOpen `  L )  =  (
TopOpen `  L )
21 eqid 2467 . . 3  |-  ( Base `  L )  =  (
Base `  L )
22 eqid 2467 . . 3  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
2320, 21, 22isxms 20818 . 2  |-  ( L  e.  *MetSp  <->  ( L  e.  TopSp  /\  ( TopOpen `  L )  =  (
MetOpen `  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) ) ) ) )
2415, 19, 233bitr4g 288 1  |-  ( ph  ->  ( K  e.  *MetSp  <-> 
L  e.  *MetSp ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    X. cxp 5003    |` cres 5007   ` cfv 5594   Basecbs 14507   distcds 14581   TopOpenctopn 14694   MetOpencmopn 18278   TopSpctps 19266   *MetSpcxme 20688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-res 5017  df-iota 5557  df-fun 5596  df-fv 5602  df-top 19268  df-topon 19271  df-topsp 19272  df-xms 20691
This theorem is referenced by:  mspropd  20845
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