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Theorem hhsscms 21872
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 2296 . 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 21823 . . 3  |-  H  e.  SH
62, 3, 5hhssmet 21870 . 2  |-  D  e.  ( Met `  H
)
7 simpl 443 . . . . . . . . . 10  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( Cau `  D ) )
82, 3, 5hhssims2 21869 . . . . . . . . . . 11  |-  D  =  ( ( normh  o.  -h  )  |`  ( H  X.  H ) )
98fveq2i 5544 . . . . . . . . . 10  |-  ( Cau `  D )  =  ( Cau `  ( (
normh  o.  -h  )  |`  ( H  X.  H
) ) )
107, 9syl6eleq 2386 . . . . . . . . 9  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( Cau `  ( ( normh  o.  -h  )  |`  ( H  X.  H ) ) ) )
11 eqid 2296 . . . . . . . . . . 11  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
1211hilxmet 21790 . . . . . . . . . 10  |-  ( normh  o. 
-h  )  e.  ( * Met `  ~H )
13 simpr 447 . . . . . . . . . 10  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f : NN --> H )
14 causs 18740 . . . . . . . . . 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 644 . . . . . . . . 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 223 . . . . . . . 8  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( Cau `  ( normh  o.  -h  ) ) )
174chssii 21827 . . . . . . . . . 10  |-  H  C_  ~H
18 fss 5413 . . . . . . . . . 10  |-  ( ( f : NN --> H  /\  H  C_  ~H )  -> 
f : NN --> ~H )
1913, 17, 18sylancl 643 . . . . . . . . 9  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f : NN --> ~H )
20 ax-hilex 21595 . . . . . . . . . 10  |-  ~H  e.  _V
21 nnex 9768 . . . . . . . . . 10  |-  NN  e.  _V
2220, 21elmap 6812 . . . . . . . . 9  |-  ( f  e.  ( ~H  ^m  NN )  <->  f : NN --> ~H )
2319, 22sylibr 203 . . . . . . . 8  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( ~H 
^m  NN ) )
24 eqid 2296 . . . . . . . . . 10  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
2524, 11hhims 21767 . . . . . . . . . 10  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
2624, 25hhcau 21793 . . . . . . . . 9  |-  Cauchy  =  ( ( Cau `  ( normh  o.  -h  ) )  i^i  ( ~H  ^m  NN ) )
2726elin2 3372 . . . . . . . 8  |-  ( f  e.  Cauchy 
<->  ( f  e.  ( Cau `  ( normh  o. 
-h  ) )  /\  f  e.  ( ~H  ^m  NN ) ) )
2816, 23, 27sylanbrc 645 . . . . . . 7  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  Cauchy )
29 ax-hcompl 21797 . . . . . . 7  |-  ( f  e.  Cauchy  ->  E. x  e.  ~H  f  ~~>v  x )
30 vex 2804 . . . . . . . . 9  |-  f  e. 
_V
31 vex 2804 . . . . . . . . 9  |-  x  e. 
_V
3230, 31breldm 4899 . . . . . . . 8  |-  ( f 
~~>v  x  ->  f  e.  dom 
~~>v  )
3332rexlimivw 2676 . . . . . . 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 21833 . . . . . . 7  |-  ~~>v  : dom  ~~>v  --> ~H
36 ffun 5407 . . . . . . 7  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  Fun  ~~>v  )
37 funfvbrb 5654 . . . . . . 7  |-  ( Fun  ~~>v 
->  ( f  e.  dom  ~~>v  <->  f  ~~>v  (  ~~>v  `  f )
) )
3835, 36, 37mp2b 9 . . . . . 6  |-  ( f  e.  dom  ~~>v  <->  f  ~~>v  ( 
~~>v  `  f ) )
3934, 38sylib 188 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  ~~>v  (  ~~>v  `  f
) )
40 eqid 2296 . . . . . . . 8  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
4124, 25, 40hhlm 21794 . . . . . . 7  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
42 resss 4995 . . . . . . 7  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
4341, 42eqsstri 3221 . . . . . 6  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
4443ssbri 4081 . . . . 5  |-  ( f 
~~>v  (  ~~>v  `  f )  ->  f ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) (  ~~>v  `  f )
)
4539, 44syl 15 . . . 4  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) (  ~~>v  `  f )
)
468, 40, 1metrest 18086 . . . . . . 7  |-  ( ( ( normh  o.  -h  )  e.  ( * Met `  ~H )  /\  H  C_  ~H )  -> 
( ( MetOpen `  ( normh  o.  -h  ) )t  H )  =  ( MetOpen `  D ) )
4712, 17, 46mp2an 653 . . . . . 6  |-  ( (
MetOpen `  ( normh  o.  -h  ) )t  H )  =  (
MetOpen `  D )
4847eqcomi 2300 . . . . 5  |-  ( MetOpen `  D )  =  ( ( MetOpen `  ( normh  o. 
-h  ) )t  H )
49 nnuz 10279 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
504a1i 10 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  ->  H  e.  CH )
5140mopntop 18002 . . . . . 6  |-  ( (
normh  o.  -h  )  e.  ( * Met `  ~H )  ->  ( MetOpen `  ( normh  o.  -h  ) )  e.  Top )
5212, 51mp1i 11 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
( MetOpen `  ( normh  o. 
-h  ) )  e. 
Top )
53 fvex 5555 . . . . . . 7  |-  (  ~~>v  `  f )  e.  _V
5453chlimi 21830 . . . . . 6  |-  ( ( H  e.  CH  /\  f : NN --> H  /\  f  ~~>v  (  ~~>v  `  f
) )  ->  (  ~~>v 
`  f )  e.  H )
5550, 13, 39, 54syl3anc 1182 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
(  ~~>v  `  f )  e.  H )
56 1z 10069 . . . . . 6  |-  1  e.  ZZ
5756a1i 10 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
1  e.  ZZ )
5848, 49, 50, 52, 55, 57, 13lmss 17042 . . . 4  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
( f ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) (  ~~>v  `  f )  <->  f ( ~~> t `  ( MetOpen
`  D ) ) (  ~~>v  `  f )
) )
5945, 58mpbid 201 . . 3  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f ( ~~> t `  ( MetOpen `  D )
) (  ~~>v  `  f
) )
6030, 53breldm 4899 . . 3  |-  ( f ( ~~> t `  ( MetOpen
`  D ) ) (  ~~>v  `  f )  ->  f  e.  dom  ( ~~> t `  ( MetOpen `  D
) ) )
6159, 60syl 15 . 2  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  dom  ( ~~> t `  ( MetOpen `  D
) ) )
621, 6, 61iscmet3i 18753 1  |-  D  e.  ( CMet `  H
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   <.cop 3656   class class class wbr 4039    X. cxp 4703   dom cdm 4705    |` cres 4707    o. ccom 4709   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   CCcc 8751   1c1 8754   NNcn 9762   ZZcz 10040   ↾t crest 13341   * Metcxmt 16385   MetOpencmopn 16388   Topctop 16647   ~~> tclm 16972   Caucca 18695   CMetcms 18696   IndMetcims 21163   ~Hchil 21515    +h cva 21516    .h csm 21517   normhcno 21519    -h cmv 21521   Cauchyccau 21522    ~~>v chli 21523   CHcch 21525
This theorem is referenced by:  hhssbn  21873
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cc 8077  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833  ax-hilex 21595  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvmulass 21603  ax-hvdistr1 21604  ax-hvdistr2 21605  ax-hvmul0 21606  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679  ax-his4 21680  ax-hcompl 21797
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ico 10678  df-icc 10679  df-fz 10799  df-fl 10941  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-rest 13343  df-topgen 13360  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-top 16652  df-bases 16654  df-topon 16655  df-ntr 16773  df-nei 16851  df-lm 16975  df-haus 17059  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-cfil 18697  df-cau 18698  df-cmet 18699  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877  df-ablo 20965  df-subgo 20985  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-vs 21171  df-nmcv 21172  df-ims 21173  df-ssp 21314  df-hnorm 21564  df-hba 21565  df-hvsub 21567  df-hlim 21568  df-hcau 21569  df-sh 21802  df-ch 21817  df-ch0 21848
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