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Theorem cmsss 20866
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 461 . . . . . . 7  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  A  C_  X )
2 xpss12 4950 . . . . . . 7  |-  ( ( A  C_  X  /\  A  C_  X )  -> 
( A  X.  A
)  C_  ( X  X.  X ) )
31, 2sylancom 667 . . . . . 6  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( A  X.  A )  C_  ( X  X.  X
) )
4 resabs1 5144 . . . . . 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 5706 . . . . . . . . . 10  |-  ( Base `  M )  e.  _V
86, 7eqeltri 2513 . . . . . . . . 9  |-  X  e. 
_V
98ssex 4441 . . . . . . . 8  |-  ( A 
C_  X  ->  A  e.  _V )
109adantl 466 . . . . . . 7  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  A  e.  _V )
11 cmsss.h . . . . . . . 8  |-  K  =  ( Ms  A )
12 eqid 2443 . . . . . . . 8  |-  ( dist `  M )  =  (
dist `  M )
1311, 12ressds 14357 . . . . . . 7  |-  ( A  e.  _V  ->  ( dist `  M )  =  ( dist `  K
) )
1410, 13syl 16 . . . . . 6  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( dist `  M )  =  ( dist `  K
) )
1511, 6ressbas2 14234 . . . . . . . 8  |-  ( A 
C_  X  ->  A  =  ( Base `  K
) )
1615adantl 466 . . . . . . 7  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  A  =  ( Base `  K
) )
1716, 16xpeq12d 4870 . . . . . 6  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( A  X.  A )  =  ( ( Base `  K
)  X.  ( Base `  K ) ) )
1814, 17reseq12d 5116 . . . . 5  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( dist `  M )  |`  ( A  X.  A
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
195, 18eqtrd 2475 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( ( dist `  M
)  |`  ( X  X.  X ) )  |`  ( A  X.  A
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
2016fveq2d 5700 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( CMet `  A )  =  ( CMet `  ( Base `  K ) ) )
2119, 20eleq12d 2511 . . 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 2443 . . . . . 6  |-  ( (
dist `  M )  |`  ( X  X.  X
) )  =  ( ( dist `  M
)  |`  ( X  X.  X ) )
236, 22cmscmet 20862 . . . . 5  |-  ( M  e. CMetSp  ->  ( ( dist `  M )  |`  ( X  X.  X ) )  e.  ( CMet `  X
) )
2423adantr 465 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( dist `  M )  |`  ( X  X.  X
) )  e.  (
CMet `  X )
)
25 eqid 2443 . . . . 5  |-  ( MetOpen `  ( ( dist `  M
)  |`  ( X  X.  X ) ) )  =  ( MetOpen `  (
( dist `  M )  |`  ( X  X.  X
) ) )
2625cmetss 20830 . . . 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 255 . 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 20864 . . . 4  |-  ( M  e. CMetSp  ->  M  e.  MetSp )
30 ressms 20106 . . . . 5  |-  ( ( M  e.  MetSp  /\  A  e.  _V )  ->  ( Ms  A )  e.  MetSp )
3111, 30syl5eqel 2527 . . . 4  |-  ( ( M  e.  MetSp  /\  A  e.  _V )  ->  K  e.  MetSp )
3229, 9, 31syl2an 477 . . 3  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  K  e.  MetSp )
33 eqid 2443 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
34 eqid 2443 . . . . 5  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
3533, 34iscms 20861 . . . 4  |-  ( K  e. CMetSp 
<->  ( K  e.  MetSp  /\  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( CMet `  ( Base `  K
) ) ) )
3635baib 896 . . 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 465 . . . . 5  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  M  e.  MetSp )
39 cmsss.j . . . . . 6  |-  J  =  ( TopOpen `  M )
4039, 6, 22mstopn 20032 . . . . 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 5700 . . 3  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( Clsd `  J )  =  ( Clsd `  ( MetOpen
`  ( ( dist `  M )  |`  ( X  X.  X ) ) ) ) )
4342eleq2d 2510 . 2  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( A  e.  ( Clsd `  J )  <->  A  e.  ( Clsd `  ( MetOpen `  (
( dist `  M )  |`  ( X  X.  X
) ) ) ) ) )
4428, 37, 433bitr4d 285 1  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( K  e. CMetSp  <->  A  e.  ( Clsd `  J ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2977    C_ wss 3333    X. cxp 4843    |` cres 4847   ` cfv 5423  (class class class)co 6096   Basecbs 14179   ↾s cress 14180   distcds 14252   TopOpenctopn 14365   MetOpencmopn 17811   Clsdccld 18625   MetSpcmt 19898   CMetcms 20770  CMetSpccms 20848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fi 7666  df-sup 7696  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-q 10959  df-rp 10997  df-xneg 11094  df-xadd 11095  df-xmul 11096  df-ico 11311  df-icc 11312  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-tset 14262  df-ds 14265  df-rest 14366  df-topn 14367  df-topgen 14387  df-psmet 17814  df-xmet 17815  df-met 17816  df-bl 17817  df-mopn 17818  df-fbas 17819  df-fg 17820  df-top 18508  df-bases 18510  df-topon 18511  df-topsp 18512  df-cld 18628  df-ntr 18629  df-cls 18630  df-nei 18707  df-haus 18924  df-fil 19424  df-flim 19517  df-xms 19900  df-ms 19901  df-cfil 20771  df-cmet 20773  df-cms 20851
This theorem is referenced by:  lssbn  20867  resscdrg  20875  srabn  20877  ishl2  20887  recms  20889  pjthlem2  20930
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