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Theorem xmspropd 20023
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 2472 . . . 4  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
4 xmspropd.4 . . . 4  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
53, 4tpspropd 18520 . . 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 4860 . . . . . . . 8  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
87reseq2d 5105 . . . . . . 7  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
96, 8eqtr3d 2472 . . . . . 6  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
102, 2xpeq12d 4860 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
1110reseq2d 5105 . . . . . 6  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
129, 11eqtr3d 2472 . . . . 5  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
1312fveq2d 5690 . . . 4  |-  ( ph  ->  ( MetOpen `  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) ) )  =  ( MetOpen `  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) ) )
144, 13eqeq12d 2452 . . 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 2438 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
17 eqid 2438 . . 3  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2438 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
1916, 17, 18isxms 19997 . 2  |-  ( K  e.  *MetSp  <->  ( K  e.  TopSp  /\  ( TopOpen `  K )  =  (
MetOpen `  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) ) ) ) )
20 eqid 2438 . . 3  |-  ( TopOpen `  L )  =  (
TopOpen `  L )
21 eqid 2438 . . 3  |-  ( Base `  L )  =  (
Base `  L )
22 eqid 2438 . . 3  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
2320, 21, 22isxms 19997 . 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 1369    e. wcel 1756    X. cxp 4833    |` cres 4837   ` cfv 5413   Basecbs 14166   distcds 14239   TopOpenctopn 14352   MetOpencmopn 17781   TopSpctps 18476   *MetSpcxme 19867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-res 4847  df-iota 5376  df-fun 5415  df-fv 5421  df-top 18478  df-topon 18481  df-topsp 18482  df-xms 19870
This theorem is referenced by:  mspropd  20024
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