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Theorem cmsss 19260
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 448 . . . . . . 7  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  A  C_  X )
2 xpss12 4944 . . . . . . 7  |-  ( ( A  C_  X  /\  A  C_  X )  -> 
( A  X.  A
)  C_  ( X  X.  X ) )
31, 2sylancom 649 . . . . . 6  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( A  X.  A )  C_  ( X  X.  X
) )
4 resabs1 5138 . . . . . 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 5705 . . . . . . . . . 10  |-  ( Base `  M )  e.  _V
86, 7eqeltri 2478 . . . . . . . . 9  |-  X  e. 
_V
98ssex 4311 . . . . . . . 8  |-  ( A 
C_  X  ->  A  e.  _V )
109adantl 453 . . . . . . 7  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  A  e.  _V )
11 cmsss.h . . . . . . . 8  |-  K  =  ( Ms  A )
12 eqid 2408 . . . . . . . 8  |-  ( dist `  M )  =  (
dist `  M )
1311, 12ressds 13600 . . . . . . 7  |-  ( A  e.  _V  ->  ( dist `  M )  =  ( dist `  K
) )
1410, 13syl 16 . . . . . 6  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( dist `  M )  =  ( dist `  K
) )
1511, 6ressbas2 13479 . . . . . . . 8  |-  ( A 
C_  X  ->  A  =  ( Base `  K
) )
1615adantl 453 . . . . . . 7  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  A  =  ( Base `  K
) )
1716, 16xpeq12d 4866 . . . . . 6  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( A  X.  A )  =  ( ( Base `  K
)  X.  ( Base `  K ) ) )
1814, 17reseq12d 5110 . . . . 5  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( dist `  M )  |`  ( A  X.  A
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
195, 18eqtrd 2440 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( ( dist `  M
)  |`  ( X  X.  X ) )  |`  ( A  X.  A
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
2016fveq2d 5695 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( CMet `  A )  =  ( CMet `  ( Base `  K ) ) )
2119, 20eleq12d 2476 . . 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 2408 . . . . . 6  |-  ( (
dist `  M )  |`  ( X  X.  X
) )  =  ( ( dist `  M
)  |`  ( X  X.  X ) )
236, 22cmscmet 19256 . . . . 5  |-  ( M  e. CMetSp  ->  ( ( dist `  M )  |`  ( X  X.  X ) )  e.  ( CMet `  X
) )
2423adantr 452 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( dist `  M )  |`  ( X  X.  X
) )  e.  (
CMet `  X )
)
25 eqid 2408 . . . . 5  |-  ( MetOpen `  ( ( dist `  M
)  |`  ( X  X.  X ) ) )  =  ( MetOpen `  (
( dist `  M )  |`  ( X  X.  X
) ) )
2625cmetss 19224 . . . 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 247 . 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 19258 . . . 4  |-  ( M  e. CMetSp  ->  M  e.  MetSp )
30 ressms 18513 . . . . 5  |-  ( ( M  e.  MetSp  /\  A  e.  _V )  ->  ( Ms  A )  e.  MetSp )
3111, 30syl5eqel 2492 . . . 4  |-  ( ( M  e.  MetSp  /\  A  e.  _V )  ->  K  e.  MetSp )
3229, 9, 31syl2an 464 . . 3  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  K  e.  MetSp )
33 eqid 2408 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
34 eqid 2408 . . . . 5  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
3533, 34iscms 19255 . . . 4  |-  ( K  e. CMetSp 
<->  ( K  e.  MetSp  /\  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( CMet `  ( Base `  K
) ) ) )
3635baib 872 . . 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 452 . . . . 5  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  M  e.  MetSp )
39 cmsss.j . . . . . 6  |-  J  =  ( TopOpen `  M )
4039, 6, 22mstopn 18439 . . . . 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 5695 . . 3  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( Clsd `  J )  =  ( Clsd `  ( MetOpen
`  ( ( dist `  M )  |`  ( X  X.  X ) ) ) ) )
4342eleq2d 2475 . 2  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( A  e.  ( Clsd `  J )  <->  A  e.  ( Clsd `  ( MetOpen `  (
( dist `  M )  |`  ( X  X.  X
) ) ) ) ) )
4428, 37, 433bitr4d 277 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2920    C_ wss 3284    X. cxp 4839    |` cres 4843   ` cfv 5417  (class class class)co 6044   Basecbs 13428   ↾s cress 13429   distcds 13497   TopOpenctopn 13608   MetOpencmopn 16650   Clsdccld 17039   MetSpcmt 18305   CMetcms 19164  CMetSpccms 19242
This theorem is referenced by:  lssbn  19261  resscdrg  19269  srabn  19271  ishl2  19281  pjthlem2  19296  recms  24300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-fi 7378  df-sup 7408  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-7 10023  df-8 10024  df-9 10025  df-10 10026  df-n0 10182  df-z 10243  df-dec 10343  df-uz 10449  df-q 10535  df-rp 10573  df-xneg 10670  df-xadd 10671  df-xmul 10672  df-ico 10882  df-icc 10883  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-tset 13507  df-ds 13510  df-rest 13609  df-topn 13610  df-topgen 13626  df-psmet 16653  df-xmet 16654  df-met 16655  df-bl 16656  df-mopn 16657  df-fbas 16658  df-fg 16659  df-top 16922  df-bases 16924  df-topon 16925  df-topsp 16926  df-cld 17042  df-ntr 17043  df-cls 17044  df-nei 17121  df-haus 17337  df-fil 17835  df-flim 17928  df-xms 18307  df-ms 18308  df-cfil 19165  df-cmet 19167  df-cms 19245
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