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Theorem hhsscms 26609
Description: The induced metric of a closed subspace is complete. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 14-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
hhssims2.1  |-  W  = 
<. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.
hhssims2.3  |-  D  =  ( IndMet `  W )
hhsscms.3  |-  H  e. 
CH
Assertion
Ref Expression
hhsscms  |-  D  e.  ( CMet `  H
)

Proof of Theorem hhsscms
Dummy variables  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2402 . 2  |-  ( MetOpen `  D )  =  (
MetOpen `  D )
2 hhssims2.1 . . 3  |-  W  = 
<. <. (  +h  |`  ( H  X.  H ) ) ,  (  .h  |`  ( CC  X.  H ) )
>. ,  ( normh  |`  H ) >.
3 hhssims2.3 . . 3  |-  D  =  ( IndMet `  W )
4 hhsscms.3 . . . 4  |-  H  e. 
CH
54chshii 26559 . . 3  |-  H  e.  SH
62, 3, 5hhssmet 26607 . 2  |-  D  e.  ( Met `  H
)
7 simpl 455 . . . . . . . . . 10  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( Cau `  D ) )
82, 3, 5hhssims2 26606 . . . . . . . . . . 11  |-  D  =  ( ( normh  o.  -h  )  |`  ( H  X.  H ) )
98fveq2i 5852 . . . . . . . . . 10  |-  ( Cau `  D )  =  ( Cau `  ( (
normh  o.  -h  )  |`  ( H  X.  H
) ) )
107, 9syl6eleq 2500 . . . . . . . . 9  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( Cau `  ( ( normh  o.  -h  )  |`  ( H  X.  H ) ) ) )
11 eqid 2402 . . . . . . . . . . 11  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
1211hilxmet 26526 . . . . . . . . . 10  |-  ( normh  o. 
-h  )  e.  ( *Met `  ~H )
13 simpr 459 . . . . . . . . . 10  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f : NN --> H )
14 causs 22029 . . . . . . . . . 10  |-  ( ( ( normh  o.  -h  )  e.  ( *Met `  ~H )  /\  f : NN --> H )  ->  ( f  e.  ( Cau `  ( normh  o.  -h  ) )  <-> 
f  e.  ( Cau `  ( ( normh  o.  -h  )  |`  ( H  X.  H ) ) ) ) )
1512, 13, 14sylancr 661 . . . . . . . . 9  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
( f  e.  ( Cau `  ( normh  o. 
-h  ) )  <->  f  e.  ( Cau `  ( (
normh  o.  -h  )  |`  ( H  X.  H
) ) ) ) )
1610, 15mpbird 232 . . . . . . . 8  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( Cau `  ( normh  o.  -h  ) ) )
174chssii 26563 . . . . . . . . . 10  |-  H  C_  ~H
18 fss 5722 . . . . . . . . . 10  |-  ( ( f : NN --> H  /\  H  C_  ~H )  -> 
f : NN --> ~H )
1913, 17, 18sylancl 660 . . . . . . . . 9  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f : NN --> ~H )
20 ax-hilex 26330 . . . . . . . . . 10  |-  ~H  e.  _V
21 nnex 10582 . . . . . . . . . 10  |-  NN  e.  _V
2220, 21elmap 7485 . . . . . . . . 9  |-  ( f  e.  ( ~H  ^m  NN )  <->  f : NN --> ~H )
2319, 22sylibr 212 . . . . . . . 8  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( ~H 
^m  NN ) )
24 eqid 2402 . . . . . . . . . 10  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
2524, 11hhims 26503 . . . . . . . . . 10  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
2624, 25hhcau 26529 . . . . . . . . 9  |-  Cauchy  =  ( ( Cau `  ( normh  o.  -h  ) )  i^i  ( ~H  ^m  NN ) )
2726elin2 3630 . . . . . . . 8  |-  ( f  e.  Cauchy 
<->  ( f  e.  ( Cau `  ( normh  o. 
-h  ) )  /\  f  e.  ( ~H  ^m  NN ) ) )
2816, 23, 27sylanbrc 662 . . . . . . 7  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  Cauchy )
29 ax-hcompl 26533 . . . . . . 7  |-  ( f  e.  Cauchy  ->  E. x  e.  ~H  f  ~~>v  x )
30 vex 3062 . . . . . . . . 9  |-  f  e. 
_V
31 vex 3062 . . . . . . . . 9  |-  x  e. 
_V
3230, 31breldm 5028 . . . . . . . 8  |-  ( f 
~~>v  x  ->  f  e.  dom 
~~>v  )
3332rexlimivw 2893 . . . . . . 7  |-  ( E. x  e.  ~H  f  ~~>v  x  ->  f  e.  dom 
~~>v  )
3428, 29, 333syl 18 . . . . . 6  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  dom  ~~>v  )
35 hlimf 26569 . . . . . . 7  |-  ~~>v  : dom  ~~>v  --> ~H
36 ffun 5716 . . . . . . 7  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  Fun  ~~>v  )
37 funfvbrb 5978 . . . . . . 7  |-  ( Fun  ~~>v 
->  ( f  e.  dom  ~~>v  <->  f  ~~>v  (  ~~>v  `  f )
) )
3835, 36, 37mp2b 10 . . . . . 6  |-  ( f  e.  dom  ~~>v  <->  f  ~~>v  ( 
~~>v  `  f ) )
3934, 38sylib 196 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  ~~>v  (  ~~>v  `  f
) )
40 eqid 2402 . . . . . . . 8  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
4124, 25, 40hhlm 26530 . . . . . . 7  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
42 resss 5117 . . . . . . 7  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
4341, 42eqsstri 3472 . . . . . 6  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
4443ssbri 4437 . . . . 5  |-  ( f 
~~>v  (  ~~>v  `  f )  ->  f ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) (  ~~>v  `  f )
)
4539, 44syl 17 . . . 4  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) (  ~~>v  `  f )
)
468, 40, 1metrest 21319 . . . . . . 7  |-  ( ( ( normh  o.  -h  )  e.  ( *Met `  ~H )  /\  H  C_  ~H )  -> 
( ( MetOpen `  ( normh  o.  -h  ) )t  H )  =  ( MetOpen `  D ) )
4712, 17, 46mp2an 670 . . . . . 6  |-  ( (
MetOpen `  ( normh  o.  -h  ) )t  H )  =  (
MetOpen `  D )
4847eqcomi 2415 . . . . 5  |-  ( MetOpen `  D )  =  ( ( MetOpen `  ( normh  o. 
-h  ) )t  H )
49 nnuz 11162 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
504a1i 11 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  ->  H  e.  CH )
5140mopntop 21235 . . . . . 6  |-  ( (
normh  o.  -h  )  e.  ( *Met `  ~H )  ->  ( MetOpen `  ( normh  o.  -h  )
)  e.  Top )
5212, 51mp1i 13 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
( MetOpen `  ( normh  o. 
-h  ) )  e. 
Top )
53 fvex 5859 . . . . . . 7  |-  (  ~~>v  `  f )  e.  _V
5453chlimi 26566 . . . . . 6  |-  ( ( H  e.  CH  /\  f : NN --> H  /\  f  ~~>v  (  ~~>v  `  f
) )  ->  (  ~~>v 
`  f )  e.  H )
5550, 13, 39, 54syl3anc 1230 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
(  ~~>v  `  f )  e.  H )
56 1zzd 10936 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
1  e.  ZZ )
5748, 49, 50, 52, 55, 56, 13lmss 20092 . . . 4  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
( f ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) (  ~~>v  `  f )  <->  f ( ~~> t `  ( MetOpen
`  D ) ) (  ~~>v  `  f )
) )
5845, 57mpbid 210 . . 3  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f ( ~~> t `  ( MetOpen `  D )
) (  ~~>v  `  f
) )
5930, 53breldm 5028 . . 3  |-  ( f ( ~~> t `  ( MetOpen
`  D ) ) (  ~~>v  `  f )  ->  f  e.  dom  ( ~~> t `  ( MetOpen `  D
) ) )
6058, 59syl 17 . 2  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  dom  ( ~~> t `  ( MetOpen `  D
) ) )
611, 6, 60iscmet3i 22042 1  |-  D  e.  ( CMet `  H
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   E.wrex 2755    C_ wss 3414   <.cop 3978   class class class wbr 4395    X. cxp 4821   dom cdm 4823    |` cres 4825    o. ccom 4827   Fun wfun 5563   -->wf 5565   ` cfv 5569  (class class class)co 6278    ^m cmap 7457   CCcc 9520   1c1 9523   NNcn 10576   ↾t crest 15035   *Metcxmt 18723   MetOpencmopn 18728   Topctop 19686   ~~> tclm 20020   Caucca 21984   CMetcms 21985   IndMetcims 25898   ~Hchil 26250    +h cva 26251    .h csm 26252   normhcno 26254    -h cmv 26256   Cauchyccau 26257    ~~>v chli 26258   CHcch 26260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cc 8847  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602  ax-hilex 26330  ax-hfvadd 26331  ax-hvcom 26332  ax-hvass 26333  ax-hv0cl 26334  ax-hvaddid 26335  ax-hfvmul 26336  ax-hvmulid 26337  ax-hvmulass 26338  ax-hvdistr1 26339  ax-hvdistr2 26340  ax-hvmul0 26341  ax-hfi 26410  ax-his1 26413  ax-his2 26414  ax-his3 26415  ax-his4 26416  ax-hcompl 26533
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-omul 7172  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-acn 8355  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-n0 10837  df-z 10906  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ico 11588  df-icc 11589  df-fz 11727  df-fl 11966  df-seq 12152  df-exp 12211  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-rlim 13461  df-rest 15037  df-topgen 15058  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-fbas 18736  df-fg 18737  df-top 19691  df-bases 19693  df-topon 19694  df-ntr 19813  df-nei 19892  df-lm 20023  df-haus 20109  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-cfil 21986  df-cau 21987  df-cmet 21988  df-grpo 25607  df-gid 25608  df-ginv 25609  df-gdiv 25610  df-ablo 25698  df-subgo 25718  df-vc 25853  df-nv 25899  df-va 25902  df-ba 25903  df-sm 25904  df-0v 25905  df-vs 25906  df-nmcv 25907  df-ims 25908  df-ssp 26049  df-hnorm 26299  df-hba 26300  df-hvsub 26302  df-hlim 26303  df-hcau 26304  df-sh 26538  df-ch 26553  df-ch0 26585
This theorem is referenced by:  hhssbn  26610
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