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Theorem cmspropd 20985
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 20174 . . 3  |-  ( ph  ->  ( K  e.  MetSp  <->  L  e.  MetSp ) )
61, 1xpeq12d 4966 . . . . . . 7  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  K )  X.  ( Base `  K
) ) )
76reseq2d 5211 . . . . . 6  |-  ( ph  ->  ( ( dist `  K
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
83, 7eqtr3d 2494 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
92, 2xpeq12d 4966 . . . . . 6  |-  ( ph  ->  ( B  X.  B
)  =  ( (
Base `  L )  X.  ( Base `  L
) ) )
109reseq2d 5211 . . . . 5  |-  ( ph  ->  ( ( dist `  L
)  |`  ( B  X.  B ) )  =  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) ) )
118, 10eqtr3d 2494 . . . 4  |-  ( ph  ->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  =  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) ) )
121, 2eqtr3d 2494 . . . . 5  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
1312fveq2d 5796 . . . 4  |-  ( ph  ->  ( CMet `  ( Base `  K ) )  =  ( CMet `  ( Base `  L ) ) )
1411, 13eleq12d 2533 . . 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 710 . 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 2451 . . 3  |-  ( Base `  K )  =  (
Base `  K )
17 eqid 2451 . . 3  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
1816, 17iscms 20981 . 2  |-  ( K  e. CMetSp 
<->  ( K  e.  MetSp  /\  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( CMet `  ( Base `  K
) ) ) )
19 eqid 2451 . . 3  |-  ( Base `  L )  =  (
Base `  L )
20 eqid 2451 . . 3  |-  ( (
dist `  L )  |`  ( ( Base `  L
)  X.  ( Base `  L ) ) )  =  ( ( dist `  L )  |`  (
( Base `  L )  X.  ( Base `  L
) ) )
2119, 20iscms 20981 . 2  |-  ( L  e. CMetSp 
<->  ( L  e.  MetSp  /\  ( ( dist `  L
)  |`  ( ( Base `  L )  X.  ( Base `  L ) ) )  e.  ( CMet `  ( Base `  L
) ) ) )
2215, 18, 213bitr4g 288 1  |-  ( ph  ->  ( K  e. CMetSp  <->  L  e. CMetSp ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    X. cxp 4939    |` cres 4943   ` cfv 5519   Basecbs 14285   distcds 14358   TopOpenctopn 14471   MetSpcmt 20018   CMetcms 20890  CMetSpccms 20968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-res 4953  df-iota 5482  df-fun 5521  df-fv 5527  df-top 18628  df-topon 18631  df-topsp 18632  df-xms 20020  df-ms 20021  df-cms 20971
This theorem is referenced by:  srabn  20997
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