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Theorem cmspropd 18787
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 18036 . . 3  |-  ( ph  ->  ( K  e.  MetSp  <->  L  e.  MetSp ) )
61, 1xpeq12d 4730 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
76reseq2d 4971 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
83, 7eqtr3d 2330 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
92, 2xpeq12d 4730 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
109reseq2d 4971 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
118, 10eqtr3d 2330 . . . 4  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
121, 2eqtr3d 2330 . . . . 5  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
1312fveq2d 5545 . . . 4  |-  ( ph  ->  ( CMet `  ( Base `  K ) )  =  ( CMet `  ( Base `  L ) ) )
1411, 13eleq12d 2364 . . 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 2296 . . 3  |-  ( Base `  K )  =  (
Base `  K )
17 eqid 2296 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
1816, 17iscms 18783 . 2  |-  ( K  e. CMetSp 
<->  ( K  e.  MetSp  /\  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( CMet `  ( Base `  K
) ) ) )
19 eqid 2296 . . 3  |-  ( Base `  L )  =  (
Base `  L )
20 eqid 2296 . . 3  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
2119, 20iscms 18783 . 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 1632    e. wcel 1696    X. cxp 4703    |` cres 4707   ` cfv 5271   Basecbs 13164   distcds 13233   TopOpenctopn 13342   MetSpcmt 17899   CMetcms 18696  CMetSpccms 18770
This theorem is referenced by:  srabn  18793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-top 16652  df-topon 16655  df-topsp 16656  df-xms 17901  df-ms 17902  df-cms 18773
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