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Theorem cmspropd 18771
Description: Property deduction for a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Hypotheses
Ref Expression
cmspropd.1  |-  ( ph  ->  B  =  ( Base `  K ) )
cmspropd.2  |-  ( ph  ->  B  =  ( Base `  L ) )
cmspropd.3  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
cmspropd.4  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
Assertion
Ref Expression
cmspropd  |-  ( ph  ->  ( K  e. CMetSp  <->  L  e. CMetSp ) )

Proof of Theorem cmspropd
StepHypRef Expression
1 cmspropd.1 . . . 4  |-  ( ph  ->  B  =  ( Base `  K ) )
2 cmspropd.2 . . . 4  |-  ( ph  ->  B  =  ( Base `  L ) )
3 cmspropd.3 . . . 4  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( B  X.  B ) ) )
4 cmspropd.4 . . . 4  |-  ( ph  ->  ( TopOpen `  K )  =  ( TopOpen `  L
) )
51, 2, 3, 4mspropd 18020 . . 3  |-  ( ph  ->  ( K  e.  MetSp  <->  L  e.  MetSp ) )
61, 1xpeq12d 4714 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
76reseq2d 4955 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
83, 7eqtr3d 2317 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
92, 2xpeq12d 4714 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
109reseq2d 4955 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
118, 10eqtr3d 2317 . . . 4  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
121, 2eqtr3d 2317 . . . . 5  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
1312fveq2d 5529 . . . 4  |-  ( ph  ->  ( CMet `  ( Base `  K ) )  =  ( CMet `  ( Base `  L ) ) )
1411, 13eleq12d 2351 . . 3  |-  ( ph  ->  ( ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  e.  ( CMet `  ( Base `  K ) )  <-> 
( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) )  e.  ( CMet `  ( Base `  L
) ) ) )
155, 14anbi12d 691 . 2  |-  ( ph  ->  ( ( K  e. 
MetSp  /\  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  e.  ( CMet `  ( Base `  K ) ) )  <->  ( L  e. 
MetSp  /\  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )  e.  ( CMet `  ( Base `  L ) ) ) ) )
16 eqid 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
17 eqid 2283 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
1816, 17iscms 18767 . 2  |-  ( K  e. CMetSp 
<->  ( K  e.  MetSp  /\  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( CMet `  ( Base `  K
) ) ) )
19 eqid 2283 . . 3  |-  ( Base `  L )  =  (
Base `  L )
20 eqid 2283 . . 3  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
2119, 20iscms 18767 . 2  |-  ( L  e. CMetSp 
<->  ( L  e.  MetSp  /\  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) )  e.  ( CMet `  ( Base `  L
) ) ) )
2215, 18, 213bitr4g 279 1  |-  ( ph  ->  ( K  e. CMetSp  <->  L  e. CMetSp ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    X. cxp 4687    |` cres 4691   ` cfv 5255   Basecbs 13148   distcds 13217   TopOpenctopn 13326   MetSpcmt 17883   CMetcms 18680  CMetSpccms 18754
This theorem is referenced by:  srabn  18777
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-top 16636  df-topon 16639  df-topsp 16640  df-xms 17885  df-ms 17886  df-cms 18757
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