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Theorem mspropd 18233
Description: Property deduction for a 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
mspropd  |-  ( ph  ->  ( K  e.  MetSp  <->  L  e.  MetSp ) )

Proof of Theorem mspropd
StepHypRef Expression
1 xmspropd.1 . . . 4  |-  ( ph  ->  B  =  ( Base `  K ) )
2 xmspropd.2 . . . 4  |-  ( ph  ->  B  =  ( Base `  L ) )
3 xmspropd.3 . . . 4  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
4 xmspropd.4 . . . 4  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
51, 2, 3, 4xmspropd 18232 . . 3  |-  ( ph  ->  ( K  e.  * MetSp  <-> 
L  e.  * MetSp ) )
61, 1xpeq12d 4817 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
76reseq2d 5058 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
83, 7eqtr3d 2400 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
92, 2xpeq12d 4817 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
109reseq2d 5058 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
118, 10eqtr3d 2400 . . . 4  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
121, 2eqtr3d 2400 . . . . 5  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
1312fveq2d 5636 . . . 4  |-  ( ph  ->  ( Met `  ( Base `  K ) )  =  ( Met `  ( Base `  L ) ) )
1411, 13eleq12d 2434 . . 3  |-  ( ph  ->  ( ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  e.  ( Met `  ( Base `  K ) )  <-> 
( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) )  e.  ( Met `  ( Base `  L
) ) ) )
155, 14anbi12d 691 . 2  |-  ( ph  ->  ( ( K  e. 
* MetSp  /\  ( ( dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  e.  ( Met `  ( Base `  K ) ) )  <->  ( L  e. 
* MetSp  /\  ( ( dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  e.  ( Met `  ( Base `  L ) ) ) ) )
16 eqid 2366 . . 3  |-  ( TopOpen `  K )  =  (
TopOpen `  K )
17 eqid 2366 . . 3  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2366 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
1916, 17, 18isms 18208 . 2  |-  ( K  e.  MetSp 
<->  ( K  e.  * MetSp  /\  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  e.  ( Met `  ( Base `  K ) ) ) )
20 eqid 2366 . . 3  |-  ( TopOpen `  L )  =  (
TopOpen `  L )
21 eqid 2366 . . 3  |-  ( Base `  L )  =  (
Base `  L )
22 eqid 2366 . . 3  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
2320, 21, 22isms 18208 . 2  |-  ( L  e.  MetSp 
<->  ( L  e.  * MetSp  /\  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )  e.  ( Met `  ( Base `  L ) ) ) )
2415, 19, 233bitr4g 279 1  |-  ( ph  ->  ( K  e.  MetSp  <->  L  e.  MetSp ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715    X. cxp 4790    |` cres 4794   ` cfv 5358   Basecbs 13356   distcds 13425   TopOpenctopn 13536   Metcme 16580   *
MetSpcxme 18095   MetSpcmt 18096
This theorem is referenced by:  ngppropd  18366  cmspropd  18986  zhmnrg  23825
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-res 4804  df-iota 5322  df-fun 5360  df-fv 5366  df-top 16853  df-topon 16856  df-topsp 16857  df-xms 18098  df-ms 18099
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