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Theorem hhsscms 22780
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 2437 . 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 22731 . . 3  |-  H  e.  SH
62, 3, 5hhssmet 22778 . 2  |-  D  e.  ( Met `  H
)
7 simpl 445 . . . . . . . . . 10  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( Cau `  D ) )
82, 3, 5hhssims2 22777 . . . . . . . . . . 11  |-  D  =  ( ( normh  o.  -h  )  |`  ( H  X.  H ) )
98fveq2i 5732 . . . . . . . . . 10  |-  ( Cau `  D )  =  ( Cau `  ( (
normh  o.  -h  )  |`  ( H  X.  H
) ) )
107, 9syl6eleq 2527 . . . . . . . . 9  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( Cau `  ( ( normh  o.  -h  )  |`  ( H  X.  H ) ) ) )
11 eqid 2437 . . . . . . . . . . 11  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
1211hilxmet 22698 . . . . . . . . . 10  |-  ( normh  o. 
-h  )  e.  ( * Met `  ~H )
13 simpr 449 . . . . . . . . . 10  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f : NN --> H )
14 causs 19252 . . . . . . . . . 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 646 . . . . . . . . 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 225 . . . . . . . 8  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( Cau `  ( normh  o.  -h  ) ) )
174chssii 22735 . . . . . . . . . 10  |-  H  C_  ~H
18 fss 5600 . . . . . . . . . 10  |-  ( ( f : NN --> H  /\  H  C_  ~H )  -> 
f : NN --> ~H )
1913, 17, 18sylancl 645 . . . . . . . . 9  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f : NN --> ~H )
20 ax-hilex 22503 . . . . . . . . . 10  |-  ~H  e.  _V
21 nnex 10007 . . . . . . . . . 10  |-  NN  e.  _V
2220, 21elmap 7043 . . . . . . . . 9  |-  ( f  e.  ( ~H  ^m  NN )  <->  f : NN --> ~H )
2319, 22sylibr 205 . . . . . . . 8  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  ( ~H 
^m  NN ) )
24 eqid 2437 . . . . . . . . . 10  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
2524, 11hhims 22675 . . . . . . . . . 10  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
2624, 25hhcau 22701 . . . . . . . . 9  |-  Cauchy  =  ( ( Cau `  ( normh  o.  -h  ) )  i^i  ( ~H  ^m  NN ) )
2726elin2 3532 . . . . . . . 8  |-  ( f  e.  Cauchy 
<->  ( f  e.  ( Cau `  ( normh  o. 
-h  ) )  /\  f  e.  ( ~H  ^m  NN ) ) )
2816, 23, 27sylanbrc 647 . . . . . . 7  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  Cauchy )
29 ax-hcompl 22705 . . . . . . 7  |-  ( f  e.  Cauchy  ->  E. x  e.  ~H  f  ~~>v  x )
30 vex 2960 . . . . . . . . 9  |-  f  e. 
_V
31 vex 2960 . . . . . . . . 9  |-  x  e. 
_V
3230, 31breldm 5075 . . . . . . . 8  |-  ( f 
~~>v  x  ->  f  e.  dom 
~~>v  )
3332rexlimivw 2827 . . . . . . 7  |-  ( E. x  e.  ~H  f  ~~>v  x  ->  f  e.  dom 
~~>v  )
3428, 29, 333syl 19 . . . . . 6  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  dom  ~~>v  )
35 hlimf 22741 . . . . . . 7  |-  ~~>v  : dom  ~~>v  --> ~H
36 ffun 5594 . . . . . . 7  |-  (  ~~>v  : dom  ~~>v  --> ~H  ->  Fun  ~~>v  )
37 funfvbrb 5844 . . . . . . 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 190 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  ~~>v  (  ~~>v  `  f
) )
40 eqid 2437 . . . . . . . 8  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
4124, 25, 40hhlm 22702 . . . . . . 7  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
42 resss 5171 . . . . . . 7  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
4341, 42eqsstri 3379 . . . . . 6  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
4443ssbri 4255 . . . . 5  |-  ( f 
~~>v  (  ~~>v  `  f )  ->  f ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) (  ~~>v  `  f )
)
4539, 44syl 16 . . . 4  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) (  ~~>v  `  f )
)
468, 40, 1metrest 18555 . . . . . . 7  |-  ( ( ( normh  o.  -h  )  e.  ( * Met `  ~H )  /\  H  C_  ~H )  -> 
( ( MetOpen `  ( normh  o.  -h  ) )t  H )  =  ( MetOpen `  D ) )
4712, 17, 46mp2an 655 . . . . . 6  |-  ( (
MetOpen `  ( normh  o.  -h  ) )t  H )  =  (
MetOpen `  D )
4847eqcomi 2441 . . . . 5  |-  ( MetOpen `  D )  =  ( ( MetOpen `  ( normh  o. 
-h  ) )t  H )
49 nnuz 10522 . . . . 5  |-  NN  =  ( ZZ>= `  1 )
504a1i 11 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  ->  H  e.  CH )
5140mopntop 18471 . . . . . 6  |-  ( (
normh  o.  -h  )  e.  ( * Met `  ~H )  ->  ( MetOpen `  ( normh  o.  -h  ) )  e.  Top )
5212, 51mp1i 12 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
( MetOpen `  ( normh  o. 
-h  ) )  e. 
Top )
53 fvex 5743 . . . . . . 7  |-  (  ~~>v  `  f )  e.  _V
5453chlimi 22738 . . . . . 6  |-  ( ( H  e.  CH  /\  f : NN --> H  /\  f  ~~>v  (  ~~>v  `  f
) )  ->  (  ~~>v 
`  f )  e.  H )
5550, 13, 39, 54syl3anc 1185 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
(  ~~>v  `  f )  e.  H )
56 1z 10312 . . . . . 6  |-  1  e.  ZZ
5756a1i 11 . . . . 5  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
1  e.  ZZ )
5848, 49, 50, 52, 55, 57, 13lmss 17363 . . . 4  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
( f ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) ) (  ~~>v  `  f )  <->  f ( ~~> t `  ( MetOpen
`  D ) ) (  ~~>v  `  f )
) )
5945, 58mpbid 203 . . 3  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f ( ~~> t `  ( MetOpen `  D )
) (  ~~>v  `  f
) )
6030, 53breldm 5075 . . 3  |-  ( f ( ~~> t `  ( MetOpen
`  D ) ) (  ~~>v  `  f )  ->  f  e.  dom  ( ~~> t `  ( MetOpen `  D
) ) )
6159, 60syl 16 . 2  |-  ( ( f  e.  ( Cau `  D )  /\  f : NN --> H )  -> 
f  e.  dom  ( ~~> t `  ( MetOpen `  D
) ) )
621, 6, 61iscmet3i 19265 1  |-  D  e.  ( CMet `  H
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2707    C_ wss 3321   <.cop 3818   class class class wbr 4213    X. cxp 4877   dom cdm 4879    |` cres 4881    o. ccom 4883   Fun wfun 5449   -->wf 5451   ` cfv 5455  (class class class)co 6082    ^m cmap 7019   CCcc 8989   1c1 8992   NNcn 10001   ZZcz 10283   ↾t crest 13649   * Metcxmt 16687   MetOpencmopn 16692   Topctop 16959   ~~> tclm 17291   Caucca 19207   CMetcms 19208   IndMetcims 22071   ~Hchil 22423    +h cva 22424    .h csm 22425   normhcno 22427    -h cmv 22429   Cauchyccau 22430    ~~>v chli 22431   CHcch 22433
This theorem is referenced by:  hhssbn  22781
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cc 8316  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069  ax-addf 9070  ax-mulf 9071  ax-hilex 22503  ax-hfvadd 22504  ax-hvcom 22505  ax-hvass 22506  ax-hv0cl 22507  ax-hvaddid 22508  ax-hfvmul 22509  ax-hvmulid 22510  ax-hvmulass 22511  ax-hvdistr1 22512  ax-hvdistr2 22513  ax-hvmul0 22514  ax-hfi 22582  ax-his1 22585  ax-his2 22586  ax-his3 22587  ax-his4 22588  ax-hcompl 22705
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-oadd 6729  df-omul 6730  df-er 6906  df-map 7021  df-pm 7022  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-fi 7417  df-sup 7447  df-oi 7480  df-card 7827  df-acn 7830  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-n0 10223  df-z 10284  df-uz 10490  df-q 10576  df-rp 10614  df-xneg 10711  df-xadd 10712  df-xmul 10713  df-ico 10923  df-icc 10924  df-fz 11045  df-fl 11203  df-seq 11325  df-exp 11384  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-clim 12283  df-rlim 12284  df-rest 13651  df-topgen 13668  df-psmet 16695  df-xmet 16696  df-met 16697  df-bl 16698  df-mopn 16699  df-fbas 16700  df-fg 16701  df-top 16964  df-bases 16966  df-topon 16967  df-ntr 17085  df-nei 17163  df-lm 17294  df-haus 17380  df-fil 17879  df-fm 17971  df-flim 17972  df-flf 17973  df-cfil 19209  df-cau 19210  df-cmet 19211  df-grpo 21780  df-gid 21781  df-ginv 21782  df-gdiv 21783  df-ablo 21871  df-subgo 21891  df-vc 22026  df-nv 22072  df-va 22075  df-ba 22076  df-sm 22077  df-0v 22078  df-vs 22079  df-nmcv 22080  df-ims 22081  df-ssp 22222  df-hnorm 22472  df-hba 22473  df-hvsub 22475  df-hlim 22476  df-hcau 22477  df-sh 22710  df-ch 22725  df-ch0 22756
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