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Theorem cmsss 19308
Description: The restriction of a complete metric space is complete iff it is closed. (Contributed by Mario Carneiro, 15-Oct-2015.)
Hypotheses
Ref Expression
cmsss.h  |-  K  =  ( Ms  A )
cmsss.x  |-  X  =  ( Base `  M
)
cmsss.j  |-  J  =  ( TopOpen `  M )
Assertion
Ref Expression
cmsss  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( K  e. CMetSp  <->  A  e.  ( Clsd `  J ) ) )

Proof of Theorem cmsss
StepHypRef Expression
1 simpr 449 . . . . . . 7  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  A  C_  X )
2 xpss12 4984 . . . . . . 7  |-  ( ( A  C_  X  /\  A  C_  X )  -> 
( A  X.  A
)  C_  ( X  X.  X ) )
31, 2sylancom 650 . . . . . 6  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( A  X.  A )  C_  ( X  X.  X
) )
4 resabs1 5178 . . . . . 6  |-  ( ( A  X.  A ) 
C_  ( X  X.  X )  ->  (
( ( dist `  M
)  |`  ( X  X.  X ) )  |`  ( A  X.  A
) )  =  ( ( dist `  M
)  |`  ( A  X.  A ) ) )
53, 4syl 16 . . . . 5  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( ( dist `  M
)  |`  ( X  X.  X ) )  |`  ( A  X.  A
) )  =  ( ( dist `  M
)  |`  ( A  X.  A ) ) )
6 cmsss.x . . . . . . . . . 10  |-  X  =  ( Base `  M
)
7 fvex 5745 . . . . . . . . . 10  |-  ( Base `  M )  e.  _V
86, 7eqeltri 2508 . . . . . . . . 9  |-  X  e. 
_V
98ssex 4350 . . . . . . . 8  |-  ( A 
C_  X  ->  A  e.  _V )
109adantl 454 . . . . . . 7  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  A  e.  _V )
11 cmsss.h . . . . . . . 8  |-  K  =  ( Ms  A )
12 eqid 2438 . . . . . . . 8  |-  ( dist `  M )  =  (
dist `  M )
1311, 12ressds 13646 . . . . . . 7  |-  ( A  e.  _V  ->  ( dist `  M )  =  ( dist `  K
) )
1410, 13syl 16 . . . . . 6  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( dist `  M )  =  ( dist `  K
) )
1511, 6ressbas2 13525 . . . . . . . 8  |-  ( A 
C_  X  ->  A  =  ( Base `  K
) )
1615adantl 454 . . . . . . 7  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  A  =  ( Base `  K
) )
1716, 16xpeq12d 4906 . . . . . 6  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( A  X.  A )  =  ( ( Base `  K
)  X.  ( Base `  K ) ) )
1814, 17reseq12d 5150 . . . . 5  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( dist `  M )  |`  ( A  X.  A
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
195, 18eqtrd 2470 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( ( dist `  M
)  |`  ( X  X.  X ) )  |`  ( A  X.  A
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
2016fveq2d 5735 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( CMet `  A )  =  ( CMet `  ( Base `  K ) ) )
2119, 20eleq12d 2506 . . 3  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( ( ( dist `  M )  |`  ( X  X.  X ) )  |`  ( A  X.  A
) )  e.  (
CMet `  A )  <->  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( CMet `  ( Base `  K
) ) ) )
22 eqid 2438 . . . . . 6  |-  ( (
dist `  M )  |`  ( X  X.  X
) )  =  ( ( dist `  M
)  |`  ( X  X.  X ) )
236, 22cmscmet 19304 . . . . 5  |-  ( M  e. CMetSp  ->  ( ( dist `  M )  |`  ( X  X.  X ) )  e.  ( CMet `  X
) )
2423adantr 453 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( dist `  M )  |`  ( X  X.  X
) )  e.  (
CMet `  X )
)
25 eqid 2438 . . . . 5  |-  ( MetOpen `  ( ( dist `  M
)  |`  ( X  X.  X ) ) )  =  ( MetOpen `  (
( dist `  M )  |`  ( X  X.  X
) ) )
2625cmetss 19272 . . . 4  |-  ( ( ( dist `  M
)  |`  ( X  X.  X ) )  e.  ( CMet `  X
)  ->  ( (
( ( dist `  M
)  |`  ( X  X.  X ) )  |`  ( A  X.  A
) )  e.  (
CMet `  A )  <->  A  e.  ( Clsd `  ( MetOpen
`  ( ( dist `  M )  |`  ( X  X.  X ) ) ) ) ) )
2724, 26syl 16 . . 3  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( ( ( dist `  M )  |`  ( X  X.  X ) )  |`  ( A  X.  A
) )  e.  (
CMet `  A )  <->  A  e.  ( Clsd `  ( MetOpen
`  ( ( dist `  M )  |`  ( X  X.  X ) ) ) ) ) )
2821, 27bitr3d 248 . 2  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( CMet `  ( Base `  K
) )  <->  A  e.  ( Clsd `  ( MetOpen `  (
( dist `  M )  |`  ( X  X.  X
) ) ) ) ) )
29 cmsms 19306 . . . 4  |-  ( M  e. CMetSp  ->  M  e.  MetSp )
30 ressms 18561 . . . . 5  |-  ( ( M  e.  MetSp  /\  A  e.  _V )  ->  ( Ms  A )  e.  MetSp )
3111, 30syl5eqel 2522 . . . 4  |-  ( ( M  e.  MetSp  /\  A  e.  _V )  ->  K  e.  MetSp )
3229, 9, 31syl2an 465 . . 3  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  K  e.  MetSp )
33 eqid 2438 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
34 eqid 2438 . . . . 5  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
3533, 34iscms 19303 . . . 4  |-  ( K  e. CMetSp 
<->  ( K  e.  MetSp  /\  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( CMet `  ( Base `  K
) ) ) )
3635baib 873 . . 3  |-  ( K  e.  MetSp  ->  ( K  e. CMetSp  <-> 
( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( CMet `  ( Base `  K
) ) ) )
3732, 36syl 16 . 2  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( K  e. CMetSp  <->  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  e.  ( CMet `  ( Base `  K ) ) ) )
3829adantr 453 . . . . 5  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  M  e.  MetSp )
39 cmsss.j . . . . . 6  |-  J  =  ( TopOpen `  M )
4039, 6, 22mstopn 18487 . . . . 5  |-  ( M  e.  MetSp  ->  J  =  ( MetOpen `  ( ( dist `  M )  |`  ( X  X.  X
) ) ) )
4138, 40syl 16 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  J  =  ( MetOpen `  (
( dist `  M )  |`  ( X  X.  X
) ) ) )
4241fveq2d 5735 . . 3  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( Clsd `  J )  =  ( Clsd `  ( MetOpen
`  ( ( dist `  M )  |`  ( X  X.  X ) ) ) ) )
4342eleq2d 2505 . 2  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( A  e.  ( Clsd `  J )  <->  A  e.  ( Clsd `  ( MetOpen `  (
( dist `  M )  |`  ( X  X.  X
) ) ) ) ) )
4428, 37, 433bitr4d 278 1  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( K  e. CMetSp  <->  A  e.  ( Clsd `  J ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    C_ wss 3322    X. cxp 4879    |` cres 4883   ` cfv 5457  (class class class)co 6084   Basecbs 13474   ↾s cress 13475   distcds 13543   TopOpenctopn 13654   MetOpencmopn 16696   Clsdccld 17085   MetSpcmt 18353   CMetcms 19212  CMetSpccms 19290
This theorem is referenced by:  lssbn  19309  resscdrg  19317  srabn  19319  ishl2  19329  pjthlem2  19344  recms  24348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-fi 7419  df-sup 7449  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-10 10071  df-n0 10227  df-z 10288  df-dec 10388  df-uz 10494  df-q 10580  df-rp 10618  df-xneg 10715  df-xadd 10716  df-xmul 10717  df-ico 10927  df-icc 10928  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-tset 13553  df-ds 13556  df-rest 13655  df-topn 13656  df-topgen 13672  df-psmet 16699  df-xmet 16700  df-met 16701  df-bl 16702  df-mopn 16703  df-fbas 16704  df-fg 16705  df-top 16968  df-bases 16970  df-topon 16971  df-topsp 16972  df-cld 17088  df-ntr 17089  df-cls 17090  df-nei 17167  df-haus 17384  df-fil 17883  df-flim 17976  df-xms 18355  df-ms 18356  df-cfil 19213  df-cmet 19215  df-cms 19293
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