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Theorem cmsss 18772
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 447 . . . . . . 7  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  A  C_  X )
2 xpss12 4792 . . . . . . 7  |-  ( ( A  C_  X  /\  A  C_  X )  -> 
( A  X.  A
)  C_  ( X  X.  X ) )
31, 2sylancom 648 . . . . . 6  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( A  X.  A )  C_  ( X  X.  X
) )
4 resabs1 4984 . . . . . 6  |-  ( ( A  X.  A ) 
C_  ( X  X.  X )  ->  (
( ( dist `  M
)  |`  ( X  X.  X ) )  |`  ( A  X.  A
) )  =  ( ( dist `  M
)  |`  ( A  X.  A ) ) )
53, 4syl 15 . . . . 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 5539 . . . . . . . . . 10  |-  ( Base `  M )  e.  _V
86, 7eqeltri 2353 . . . . . . . . 9  |-  X  e. 
_V
98ssex 4158 . . . . . . . 8  |-  ( A 
C_  X  ->  A  e.  _V )
109adantl 452 . . . . . . 7  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  A  e.  _V )
11 cmsss.h . . . . . . . 8  |-  K  =  ( Ms  A )
12 eqid 2283 . . . . . . . 8  |-  ( dist `  M )  =  (
dist `  M )
1311, 12ressds 13318 . . . . . . 7  |-  ( A  e.  _V  ->  ( dist `  M )  =  ( dist `  K
) )
1410, 13syl 15 . . . . . 6  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( dist `  M )  =  ( dist `  K
) )
1511, 6ressbas2 13199 . . . . . . . 8  |-  ( A 
C_  X  ->  A  =  ( Base `  K
) )
1615adantl 452 . . . . . . 7  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  A  =  ( Base `  K
) )
1716, 16xpeq12d 4714 . . . . . 6  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( A  X.  A )  =  ( ( Base `  K
)  X.  ( Base `  K ) ) )
1814, 17reseq12d 4956 . . . . 5  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( dist `  M )  |`  ( A  X.  A
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
195, 18eqtrd 2315 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( ( dist `  M
)  |`  ( X  X.  X ) )  |`  ( A  X.  A
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
2016fveq2d 5529 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( CMet `  A )  =  ( CMet `  ( Base `  K ) ) )
2119, 20eleq12d 2351 . . 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 2283 . . . . . 6  |-  ( (
dist `  M )  |`  ( X  X.  X
) )  =  ( ( dist `  M
)  |`  ( X  X.  X ) )
236, 22cmscmet 18768 . . . . 5  |-  ( M  e. CMetSp  ->  ( ( dist `  M )  |`  ( X  X.  X ) )  e.  ( CMet `  X
) )
2423adantr 451 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( dist `  M )  |`  ( X  X.  X
) )  e.  (
CMet `  X )
)
25 eqid 2283 . . . . 5  |-  ( MetOpen `  ( ( dist `  M
)  |`  ( X  X.  X ) ) )  =  ( MetOpen `  (
( dist `  M )  |`  ( X  X.  X
) ) )
2625cmetss 18740 . . . 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 15 . . 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 246 . 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 18770 . . . 4  |-  ( M  e. CMetSp  ->  M  e.  MetSp )
30 ressms 18072 . . . . 5  |-  ( ( M  e.  MetSp  /\  A  e.  _V )  ->  ( Ms  A )  e.  MetSp )
3111, 30syl5eqel 2367 . . . 4  |-  ( ( M  e.  MetSp  /\  A  e.  _V )  ->  K  e.  MetSp )
3229, 9, 31syl2an 463 . . 3  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  K  e.  MetSp )
33 eqid 2283 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
34 eqid 2283 . . . . 5  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
3533, 34iscms 18767 . . . 4  |-  ( K  e. CMetSp 
<->  ( K  e.  MetSp  /\  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( CMet `  ( Base `  K
) ) ) )
3635baib 871 . . 3  |-  ( K  e.  MetSp  ->  ( K  e. CMetSp  <-> 
( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( CMet `  ( Base `  K
) ) ) )
3732, 36syl 15 . 2  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( K  e. CMetSp  <->  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  e.  ( CMet `  ( Base `  K ) ) ) )
3829adantr 451 . . . . 5  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  M  e.  MetSp )
39 cmsss.j . . . . . 6  |-  J  =  ( TopOpen `  M )
4039, 6, 22mstopn 17998 . . . . 5  |-  ( M  e.  MetSp  ->  J  =  ( MetOpen `  ( ( dist `  M )  |`  ( X  X.  X
) ) ) )
4138, 40syl 15 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  J  =  ( MetOpen `  (
( dist `  M )  |`  ( X  X.  X
) ) ) )
4241fveq2d 5529 . . 3  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( Clsd `  J )  =  ( Clsd `  ( MetOpen
`  ( ( dist `  M )  |`  ( X  X.  X ) ) ) ) )
4342eleq2d 2350 . 2  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( A  e.  ( Clsd `  J )  <->  A  e.  ( Clsd `  ( MetOpen `  (
( dist `  M )  |`  ( X  X.  X
) ) ) ) ) )
4428, 37, 433bitr4d 276 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152    X. cxp 4687    |` cres 4691   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   distcds 13217   TopOpenctopn 13326   MetOpencmopn 16372   Clsdccld 16753   MetSpcmt 17883   CMetcms 18680  CMetSpccms 18754
This theorem is referenced by:  lssbn  18773  resscdrg  18775  srabn  18777  ishl2  18787  pjthlem2  18802
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ico 10662  df-icc 10663  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-tset 13227  df-ds 13230  df-rest 13327  df-topn 13328  df-topgen 13344  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-haus 17043  df-fbas 17520  df-fg 17521  df-fil 17541  df-flim 17634  df-xms 17885  df-ms 17886  df-cfil 18681  df-cmet 18683  df-cms 18757
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