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Theorem hhsscms 21856
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 2283 . 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 21807 . . 3  |-  H  e.  SH
62, 3, 5hhssmet 21854 . 2  |-  D  e.  ( Met `  H
)
7 simpl 443 . . . . . . . . . 10  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( Cau `  D ) )
82, 3, 5hhssims2 21853 . . . . . . . . . . 11  |-  D  =  ( ( normh  o.  -h  )  |`  ( H  X.  H ) )
98fveq2i 5528 . . . . . . . . . 10  |-  ( Cau `  D )  =  ( Cau `  ( (
normh  o.  -h  )  |`  ( H  X.  H
) ) )
107, 9syl6eleq 2373 . . . . . . . . 9  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( Cau `  ( ( normh  o.  -h  )  |`  ( H  X.  H ) ) ) )
11 eqid 2283 . . . . . . . . . . 11  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
1211hilxmet 21774 . . . . . . . . . 10  |-  ( normh  o. 
-h  )  e.  ( * Met `  ~H )
13 simpr 447 . . . . . . . . . 10  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f : NN --> H )
14 causs 18724 . . . . . . . . . 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 21811 . . . . . . . . . 10  |-  H  C_  ~H
18 fss 5397 . . . . . . . . . 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 21579 . . . . . . . . . 10  |-  ~H  e.  _V
21 nnex 9752 . . . . . . . . . 10  |-  NN  e.  _V
2220, 21elmap 6796 . . . . . . . . 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 2283 . . . . . . . . . 10  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
2524, 11hhims 21751 . . . . . . . . . 10  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
2624, 25hhcau 21777 . . . . . . . . 9  |-  Cauchy  =  ( ( Cau `  ( normh  o.  -h  ) )  i^i  ( ~H  ^m  NN ) )
2726elin2 3359 . . . . . . . 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 21781 . . . . . . 7  |-  ( f  e.  Cauchy  ->  E. x  e.  ~H  f  ~~>v  x )
30 vex 2791 . . . . . . . . 9  |-  f  e. 
_V
31 vex 2791 . . . . . . . . 9  |-  x  e. 
_V
3230, 31breldm 4883 . . . . . . . 8  |-  ( f 
~~>v  x  ->  f  e.  dom 
~~>v  )
3332rexlimivw 2663 . . . . . . 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 21817 . . . . . . 7  |-  ~~>v  : dom  ~~>v  --> ~H
36 ffun 5391 . . . . . . 7  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  Fun  ~~>v  )
37 funfvbrb 5638 . . . . . . 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 2283 . . . . . . . 8  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
4124, 25, 40hhlm 21778 . . . . . . 7  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
42 resss 4979 . . . . . . 7  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
4341, 42eqsstri 3208 . . . . . 6  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
4443ssbri 4065 . . . . 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 18070 . . . . . . 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 2287 . . . . 5  |-  ( MetOpen `  D )  =  ( ( MetOpen `  ( normh  o. 
-h  ) )t  H )
49 nnuz 10263 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
504a1i 10 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  ->  H  e.  CH )
5140mopntop 17986 . . . . . 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 5539 . . . . . . 7  |-  (  ~~>v  `  f )  e.  _V
5453chlimi 21814 . . . . . 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 10053 . . . . . 6  |-  1  e.  ZZ
5756a1i 10 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
1  e.  ZZ )
5848, 49, 50, 52, 55, 57, 13lmss 17026 . . . 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 4883 . . 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 18737 1  |-  D  e.  ( CMet `  H
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   <.cop 3643   class class class wbr 4023    X. cxp 4687   dom cdm 4689    |` cres 4691    o. ccom 4693   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   CCcc 8735   1c1 8738   NNcn 9746   ZZcz 10024   ↾t crest 13325   * Metcxmt 16369   MetOpencmopn 16372   Topctop 16631   ~~> tclm 16956   Caucca 18679   CMetcms 18680   IndMetcims 21147   ~Hchil 21499    +h cva 21500    .h csm 21501   normhcno 21503    -h cmv 21505   Cauchyccau 21506    ~~>v chli 21507   CHcch 21509
This theorem is referenced by:  hhssbn  21857
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-inf2 7342  ax-cc 8061  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  ax-addf 8816  ax-mulf 8817  ax-hilex 21579  ax-hfvadd 21580  ax-hvcom 21581  ax-hvass 21582  ax-hv0cl 21583  ax-hvaddid 21584  ax-hfvmul 21585  ax-hvmulid 21586  ax-hvmulass 21587  ax-hvdistr1 21588  ax-hvdistr2 21589  ax-hvmul0 21590  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663  ax-his4 21664  ax-hcompl 21781
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-se 4353  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-isom 5264  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-omul 6484  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  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-n0 9966  df-z 10025  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-fz 10783  df-fl 10925  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-rest 13327  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-ntr 16757  df-nei 16835  df-lm 16959  df-haus 17043  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-cfil 18681  df-cau 18682  df-cmet 18683  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-subgo 20969  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-vs 21155  df-nmcv 21156  df-ims 21157  df-ssp 21298  df-hnorm 21548  df-hba 21549  df-hvsub 21551  df-hlim 21552  df-hcau 21553  df-sh 21786  df-ch 21801  df-ch0 21832
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