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Theorem cmspropd 19302
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 18504 . . 3  |-  ( ph  ->  ( K  e.  MetSp  <->  L  e.  MetSp ) )
61, 1xpeq12d 4903 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
76reseq2d 5146 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
83, 7eqtr3d 2470 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
92, 2xpeq12d 4903 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
109reseq2d 5146 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
118, 10eqtr3d 2470 . . . 4  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
121, 2eqtr3d 2470 . . . . 5  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
1312fveq2d 5732 . . . 4  |-  ( ph  ->  ( CMet `  ( Base `  K ) )  =  ( CMet `  ( Base `  L ) ) )
1411, 13eleq12d 2504 . . 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 692 . 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 2436 . . 3  |-  ( Base `  K )  =  (
Base `  K )
17 eqid 2436 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
1816, 17iscms 19298 . 2  |-  ( K  e. CMetSp 
<->  ( K  e.  MetSp  /\  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( CMet `  ( Base `  K
) ) ) )
19 eqid 2436 . . 3  |-  ( Base `  L )  =  (
Base `  L )
20 eqid 2436 . . 3  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
2119, 20iscms 19298 . 2  |-  ( L  e. CMetSp 
<->  ( L  e.  MetSp  /\  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) )  e.  ( CMet `  ( Base `  L
) ) ) )
2215, 18, 213bitr4g 280 1  |-  ( ph  ->  ( K  e. CMetSp  <->  L  e. CMetSp ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    X. cxp 4876    |` cres 4880   ` cfv 5454   Basecbs 13469   distcds 13538   TopOpenctopn 13649   MetSpcmt 18348   CMetcms 19207  CMetSpccms 19285
This theorem is referenced by:  srabn  19314
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-res 4890  df-iota 5418  df-fun 5456  df-fv 5462  df-top 16963  df-topon 16966  df-topsp 16967  df-xms 18350  df-ms 18351  df-cms 19288
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