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Theorem cmsss 21958
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 459 . . . . . . 7  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  A  C_  X )
2 xpss12 5096 . . . . . . 7  |-  ( ( A  C_  X  /\  A  C_  X )  -> 
( A  X.  A
)  C_  ( X  X.  X ) )
31, 2sylancom 665 . . . . . 6  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( A  X.  A )  C_  ( X  X.  X
) )
43resabs1d 5291 . . . . 5  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( ( dist `  M
)  |`  ( X  X.  X ) )  |`  ( A  X.  A
) )  =  ( ( dist `  M
)  |`  ( A  X.  A ) ) )
5 cmsss.x . . . . . . . . . 10  |-  X  =  ( Base `  M
)
6 fvex 5858 . . . . . . . . . 10  |-  ( Base `  M )  e.  _V
75, 6eqeltri 2538 . . . . . . . . 9  |-  X  e. 
_V
87ssex 4581 . . . . . . . 8  |-  ( A 
C_  X  ->  A  e.  _V )
98adantl 464 . . . . . . 7  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  A  e.  _V )
10 cmsss.h . . . . . . . 8  |-  K  =  ( Ms  A )
11 eqid 2454 . . . . . . . 8  |-  ( dist `  M )  =  (
dist `  M )
1210, 11ressds 14905 . . . . . . 7  |-  ( A  e.  _V  ->  ( dist `  M )  =  ( dist `  K
) )
139, 12syl 16 . . . . . 6  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( dist `  M )  =  ( dist `  K
) )
1410, 5ressbas2 14777 . . . . . . . 8  |-  ( A 
C_  X  ->  A  =  ( Base `  K
) )
1514adantl 464 . . . . . . 7  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  A  =  ( Base `  K
) )
1615sqxpeqd 5014 . . . . . 6  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( A  X.  A )  =  ( ( Base `  K
)  X.  ( Base `  K ) ) )
1713, 16reseq12d 5263 . . . . 5  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( dist `  M )  |`  ( A  X.  A
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
184, 17eqtrd 2495 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( ( dist `  M
)  |`  ( X  X.  X ) )  |`  ( A  X.  A
) )  =  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) ) )
1915fveq2d 5852 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( CMet `  A )  =  ( CMet `  ( Base `  K ) ) )
2018, 19eleq12d 2536 . . 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
) ) ) )
21 eqid 2454 . . . . . 6  |-  ( (
dist `  M )  |`  ( X  X.  X
) )  =  ( ( dist `  M
)  |`  ( X  X.  X ) )
225, 21cmscmet 21954 . . . . 5  |-  ( M  e. CMetSp  ->  ( ( dist `  M )  |`  ( X  X.  X ) )  e.  ( CMet `  X
) )
2322adantr 463 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  (
( dist `  M )  |`  ( X  X.  X
) )  e.  (
CMet `  X )
)
24 eqid 2454 . . . . 5  |-  ( MetOpen `  ( ( dist `  M
)  |`  ( X  X.  X ) ) )  =  ( MetOpen `  (
( dist `  M )  |`  ( X  X.  X
) ) )
2524cmetss 21922 . . . 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 ) ) ) ) ) )
2623, 25syl 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 ) ) ) ) ) )
2720, 26bitr3d 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
) ) ) ) ) )
28 cmsms 21956 . . . 4  |-  ( M  e. CMetSp  ->  M  e.  MetSp )
29 ressms 21198 . . . . 5  |-  ( ( M  e.  MetSp  /\  A  e.  _V )  ->  ( Ms  A )  e.  MetSp )
3010, 29syl5eqel 2546 . . . 4  |-  ( ( M  e.  MetSp  /\  A  e.  _V )  ->  K  e.  MetSp )
3128, 8, 30syl2an 475 . . 3  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  K  e.  MetSp )
32 eqid 2454 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
33 eqid 2454 . . . . 5  |-  ( (
dist `  K )  |`  ( ( Base `  K
)  X.  ( Base `  K ) ) )  =  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )
3432, 33iscms 21953 . . . 4  |-  ( K  e. CMetSp 
<->  ( K  e.  MetSp  /\  ( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( CMet `  ( Base `  K
) ) ) )
3534baib 901 . . 3  |-  ( K  e.  MetSp  ->  ( K  e. CMetSp  <-> 
( ( dist `  K
)  |`  ( ( Base `  K )  X.  ( Base `  K ) ) )  e.  ( CMet `  ( Base `  K
) ) ) )
3631, 35syl 16 . 2  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( K  e. CMetSp  <->  ( ( dist `  K )  |`  (
( Base `  K )  X.  ( Base `  K
) ) )  e.  ( CMet `  ( Base `  K ) ) ) )
3728adantr 463 . . . . 5  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  M  e.  MetSp )
38 cmsss.j . . . . . 6  |-  J  =  ( TopOpen `  M )
3938, 5, 21mstopn 21124 . . . . 5  |-  ( M  e.  MetSp  ->  J  =  ( MetOpen `  ( ( dist `  M )  |`  ( X  X.  X
) ) ) )
4037, 39syl 16 . . . 4  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  J  =  ( MetOpen `  (
( dist `  M )  |`  ( X  X.  X
) ) ) )
4140fveq2d 5852 . . 3  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( Clsd `  J )  =  ( Clsd `  ( MetOpen
`  ( ( dist `  M )  |`  ( X  X.  X ) ) ) ) )
4241eleq2d 2524 . 2  |-  ( ( M  e. CMetSp  /\  A  C_  X )  ->  ( A  e.  ( Clsd `  J )  <->  A  e.  ( Clsd `  ( MetOpen `  (
( dist `  M )  |`  ( X  X.  X
) ) ) ) ) )
4327, 36, 423bitr4d 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 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461    X. cxp 4986    |` cres 4990   ` cfv 5570  (class class class)co 6270   Basecbs 14719   ↾s cress 14720   distcds 14796   TopOpenctopn 14914   MetOpencmopn 18606   Clsdccld 19687   MetSpcmt 20990   CMetcms 21862  CMetSpccms 21940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fi 7863  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ico 11538  df-icc 11539  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-tset 14806  df-ds 14809  df-rest 14915  df-topn 14916  df-topgen 14936  df-psmet 18609  df-xmet 18610  df-met 18611  df-bl 18612  df-mopn 18613  df-fbas 18614  df-fg 18615  df-top 19569  df-bases 19571  df-topon 19572  df-topsp 19573  df-cld 19690  df-ntr 19691  df-cls 19692  df-nei 19769  df-haus 19986  df-fil 20516  df-flim 20609  df-xms 20992  df-ms 20993  df-cfil 21863  df-cmet 21865  df-cms 21943
This theorem is referenced by:  lssbn  21959  resscdrg  21967  srabn  21969  ishl2  21979  recms  21981  pjthlem2  22022
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