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Theorem hhsscms 26922
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 2423 . 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 26872 . . 3  |-  H  e.  SH
62, 3, 5hhssmet 26920 . 2  |-  D  e.  ( Met `  H
)
7 simpl 459 . . . . . . . . . 10  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( Cau `  D ) )
82, 3, 5hhssims2 26919 . . . . . . . . . . 11  |-  D  =  ( ( normh  o.  -h  )  |`  ( H  X.  H ) )
98fveq2i 5882 . . . . . . . . . 10  |-  ( Cau `  D )  =  ( Cau `  ( (
normh  o.  -h  )  |`  ( H  X.  H
) ) )
107, 9syl6eleq 2521 . . . . . . . . 9  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( Cau `  ( ( normh  o.  -h  )  |`  ( H  X.  H ) ) ) )
11 eqid 2423 . . . . . . . . . . 11  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
1211hilxmet 26840 . . . . . . . . . 10  |-  ( normh  o. 
-h  )  e.  ( *Met `  ~H )
13 simpr 463 . . . . . . . . . 10  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f : NN --> H )
14 causs 22260 . . . . . . . . . 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 668 . . . . . . . . 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 236 . . . . . . . 8  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( Cau `  ( normh  o.  -h  ) ) )
174chssii 26876 . . . . . . . . . 10  |-  H  C_  ~H
18 fss 5752 . . . . . . . . . 10  |-  ( ( f : NN --> H  /\  H  C_  ~H )  -> 
f : NN --> ~H )
1913, 17, 18sylancl 667 . . . . . . . . 9  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f : NN --> ~H )
20 ax-hilex 26644 . . . . . . . . . 10  |-  ~H  e.  _V
21 nnex 10617 . . . . . . . . . 10  |-  NN  e.  _V
2220, 21elmap 7506 . . . . . . . . 9  |-  ( f  e.  ( ~H  ^m  NN )  <->  f : NN --> ~H )
2319, 22sylibr 216 . . . . . . . 8  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( ~H 
^m  NN ) )
24 eqid 2423 . . . . . . . . . 10  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
2524, 11hhims 26817 . . . . . . . . . 10  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
2624, 25hhcau 26843 . . . . . . . . 9  |-  Cauchy  =  ( ( Cau `  ( normh  o.  -h  ) )  i^i  ( ~H  ^m  NN ) )
2726elin2 3654 . . . . . . . 8  |-  ( f  e.  Cauchy 
<->  ( f  e.  ( Cau `  ( normh  o. 
-h  ) )  /\  f  e.  ( ~H  ^m  NN ) ) )
2816, 23, 27sylanbrc 669 . . . . . . 7  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  Cauchy )
29 ax-hcompl 26847 . . . . . . 7  |-  ( f  e.  Cauchy  ->  E. x  e.  ~H  f  ~~>v  x )
30 vex 3085 . . . . . . . . 9  |-  f  e. 
_V
31 vex 3085 . . . . . . . . 9  |-  x  e. 
_V
3230, 31breldm 5056 . . . . . . . 8  |-  ( f 
~~>v  x  ->  f  e.  dom 
~~>v  )
3332rexlimivw 2915 . . . . . . 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 26882 . . . . . . 7  |-  ~~>v  : dom  ~~>v  --> ~H
36 ffun 5746 . . . . . . 7  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  Fun  ~~>v  )
37 funfvbrb 6008 . . . . . . 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 200 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  ~~>v  (  ~~>v  `  f
) )
40 eqid 2423 . . . . . . . 8  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
4124, 25, 40hhlm 26844 . . . . . . 7  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
42 resss 5145 . . . . . . 7  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
4341, 42eqsstri 3495 . . . . . 6  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
4443ssbri 4464 . . . . 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 21531 . . . . . . 7  |-  ( ( ( normh  o.  -h  )  e.  ( *Met `  ~H )  /\  H  C_  ~H )  -> 
( ( MetOpen `  ( normh  o.  -h  ) )t  H )  =  ( MetOpen `  D ) )
4712, 17, 46mp2an 677 . . . . . 6  |-  ( (
MetOpen `  ( normh  o.  -h  ) )t  H )  =  (
MetOpen `  D )
4847eqcomi 2436 . . . . 5  |-  ( MetOpen `  D )  =  ( ( MetOpen `  ( normh  o. 
-h  ) )t  H )
49 nnuz 11196 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
504a1i 11 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  ->  H  e.  CH )
5140mopntop 21447 . . . . . 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 5889 . . . . . . 7  |-  (  ~~>v  `  f )  e.  _V
5453chlimi 26879 . . . . . 6  |-  ( ( H  e.  CH  /\  f : NN --> H  /\  f  ~~>v  (  ~~>v  `  f
) )  ->  (  ~~>v 
`  f )  e.  H )
5550, 13, 39, 54syl3anc 1265 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
(  ~~>v  `  f )  e.  H )
56 1zzd 10970 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
1  e.  ZZ )
5748, 49, 50, 52, 55, 56, 13lmss 20306 . . . 4  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
( f ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) (  ~~>v  `  f )  <->  f ( ~~> t `  ( MetOpen
`  D ) ) (  ~~>v  `  f )
) )
5845, 57mpbid 214 . . 3  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f ( ~~> t `  ( MetOpen `  D )
) (  ~~>v  `  f
) )
5930, 53breldm 5056 . . 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 22273 1  |-  D  e.  ( CMet `  H
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1438    e. wcel 1869   E.wrex 2777    C_ wss 3437   <.cop 4003   class class class wbr 4421    X. cxp 4849   dom cdm 4851    |` cres 4853    o. ccom 4855   Fun wfun 5593   -->wf 5595   ` cfv 5599  (class class class)co 6303    ^m cmap 7478   CCcc 9539   1c1 9542   NNcn 10611   ↾t crest 15312   *Metcxmt 18948   MetOpencmopn 18953   Topctop 19909   ~~> tclm 20234   Caucca 22215   CMetcms 22216   IndMetcims 26202   ~Hchil 26564    +h cva 26565    .h csm 26566   normhcno 26568    -h cmv 26570   Cauchyccau 26571    ~~>v chli 26572   CHcch 26574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cc 8867  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-pre-sup 9619  ax-addf 9620  ax-mulf 9621  ax-hilex 26644  ax-hfvadd 26645  ax-hvcom 26646  ax-hvass 26647  ax-hv0cl 26648  ax-hvaddid 26649  ax-hfvmul 26650  ax-hvmulid 26651  ax-hvmulass 26652  ax-hvdistr1 26653  ax-hvdistr2 26654  ax-hvmul0 26655  ax-hfi 26724  ax-his1 26727  ax-his2 26728  ax-his3 26729  ax-his4 26730  ax-hcompl 26847
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-oadd 7192  df-omul 7193  df-er 7369  df-map 7480  df-pm 7481  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fi 7929  df-sup 7960  df-inf 7961  df-oi 8029  df-card 8376  df-acn 8379  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-n0 10872  df-z 10940  df-uz 11162  df-q 11267  df-rp 11305  df-xneg 11411  df-xadd 11412  df-xmul 11413  df-ico 11643  df-icc 11644  df-fz 11787  df-fl 12029  df-seq 12215  df-exp 12274  df-cj 13156  df-re 13157  df-im 13158  df-sqrt 13292  df-abs 13293  df-clim 13545  df-rlim 13546  df-rest 15314  df-topgen 15335  df-psmet 18955  df-xmet 18956  df-met 18957  df-bl 18958  df-mopn 18959  df-fbas 18960  df-fg 18961  df-top 19913  df-bases 19914  df-topon 19915  df-ntr 20027  df-nei 20106  df-lm 20237  df-haus 20323  df-fil 20853  df-fm 20945  df-flim 20946  df-flf 20947  df-cfil 22217  df-cau 22218  df-cmet 22219  df-grpo 25911  df-gid 25912  df-ginv 25913  df-gdiv 25914  df-ablo 26002  df-subgo 26022  df-vc 26157  df-nv 26203  df-va 26206  df-ba 26207  df-sm 26208  df-0v 26209  df-vs 26210  df-nmcv 26211  df-ims 26212  df-ssp 26353  df-hnorm 26613  df-hba 26614  df-hvsub 26616  df-hlim 26617  df-hcau 26618  df-sh 26852  df-ch 26866  df-ch0 26898
This theorem is referenced by:  hhssbn  26923
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