HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  nlelchi Unicode version

Theorem nlelchi 23517
Description: The null space of a continuous linear functional is a closed subspace. Remark 3.8 of [Beran] p. 103. (Contributed by NM, 11-Feb-2006.) (Proof shortened by Mario Carneiro, 19-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
nlelch.1  |-  T  e. 
LinFn
nlelch.2  |-  T  e. 
ConFn
Assertion
Ref Expression
nlelchi  |-  ( null `  T )  e.  CH

Proof of Theorem nlelchi
Dummy variables  f  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nlelch.1 . . 3  |-  T  e. 
LinFn
21nlelshi 23516 . 2  |-  ( null `  T )  e.  SH
3 vex 2919 . . . . . 6  |-  x  e. 
_V
43hlimveci 22645 . . . . 5  |-  ( f 
~~>v  x  ->  x  e.  ~H )
54adantl 453 . . . 4  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  x  e.  ~H )
6 eqid 2404 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
76cnfldhaus 18772 . . . . . 6  |-  ( TopOpen ` fld )  e.  Haus
87a1i 11 . . . . 5  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( TopOpen
` fld
)  e.  Haus )
9 eqid 2404 . . . . . . . . . 10  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
10 eqid 2404 . . . . . . . . . . 11  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
119, 10hhims 22627 . . . . . . . . . 10  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
12 eqid 2404 . . . . . . . . . 10  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
139, 11, 12hhlm 22654 . . . . . . . . 9  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
14 resss 5129 . . . . . . . . 9  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
1513, 14eqsstri 3338 . . . . . . . 8  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
1615ssbri 4214 . . . . . . 7  |-  ( f 
~~>v  x  ->  f ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) ) x )
1716adantl 453 . . . . . 6  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  f
( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) ) x )
18 nlelch.2 . . . . . . . 8  |-  T  e. 
ConFn
1910, 12, 6hhcnf 23361 . . . . . . . 8  |-  ConFn  =  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( TopOpen ` fld ) )
2018, 19eleqtri 2476 . . . . . . 7  |-  T  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( TopOpen ` fld ) )
2120a1i 11 . . . . . 6  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  T  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( TopOpen ` fld ) ) )
2217, 21lmcn 17323 . . . . 5  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T  o.  f )
( ~~> t `  ( TopOpen
` fld
) ) ( T `
 x ) )
231lnfnfi 23497 . . . . . . . . . 10  |-  T : ~H
--> CC
24 ffvelrn 5827 . . . . . . . . . . 11  |-  ( ( f : NN --> ( null `  T )  /\  n  e.  NN )  ->  (
f `  n )  e.  ( null `  T
) )
2524adantlr 696 . . . . . . . . . 10  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( f `  n )  e.  (
null `  T )
)
26 elnlfn2 23385 . . . . . . . . . 10  |-  ( ( T : ~H --> CC  /\  ( f `  n
)  e.  ( null `  T ) )  -> 
( T `  (
f `  n )
)  =  0 )
2723, 25, 26sylancr 645 . . . . . . . . 9  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( T `  ( f `  n
) )  =  0 )
28 fvco3 5759 . . . . . . . . . 10  |-  ( ( f : NN --> ( null `  T )  /\  n  e.  NN )  ->  (
( T  o.  f
) `  n )  =  ( T `  ( f `  n
) ) )
2928adantlr 696 . . . . . . . . 9  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( ( T  o.  f ) `  n )  =  ( T `  ( f `
 n ) ) )
30 c0ex 9041 . . . . . . . . . . 11  |-  0  e.  _V
3130fvconst2 5906 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
( NN  X.  {
0 } ) `  n )  =  0 )
3231adantl 453 . . . . . . . . 9  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( ( NN 
X.  { 0 } ) `  n )  =  0 )
3327, 29, 323eqtr4d 2446 . . . . . . . 8  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( ( T  o.  f ) `  n )  =  ( ( NN  X.  {
0 } ) `  n ) )
3433ralrimiva 2749 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  A. n  e.  NN  ( ( T  o.  f ) `  n )  =  ( ( NN  X.  {
0 } ) `  n ) )
35 ffn 5550 . . . . . . . . . 10  |-  ( T : ~H --> CC  ->  T  Fn  ~H )
3623, 35ax-mp 8 . . . . . . . . 9  |-  T  Fn  ~H
37 simpl 444 . . . . . . . . . 10  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  f : NN --> ( null `  T
) )
382shssii 22668 . . . . . . . . . 10  |-  ( null `  T )  C_  ~H
39 fss 5558 . . . . . . . . . 10  |-  ( ( f : NN --> ( null `  T )  /\  ( null `  T )  C_  ~H )  ->  f : NN --> ~H )
4037, 38, 39sylancl 644 . . . . . . . . 9  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  f : NN --> ~H )
41 fnfco 5568 . . . . . . . . 9  |-  ( ( T  Fn  ~H  /\  f : NN --> ~H )  ->  ( T  o.  f
)  Fn  NN )
4236, 40, 41sylancr 645 . . . . . . . 8  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T  o.  f )  Fn  NN )
4330fconst 5588 . . . . . . . . 9  |-  ( NN 
X.  { 0 } ) : NN --> { 0 }
44 ffn 5550 . . . . . . . . 9  |-  ( ( NN  X.  { 0 } ) : NN --> { 0 }  ->  ( NN  X.  { 0 } )  Fn  NN )
4543, 44ax-mp 8 . . . . . . . 8  |-  ( NN 
X.  { 0 } )  Fn  NN
46 eqfnfv 5786 . . . . . . . 8  |-  ( ( ( T  o.  f
)  Fn  NN  /\  ( NN  X.  { 0 } )  Fn  NN )  ->  ( ( T  o.  f )  =  ( NN  X.  {
0 } )  <->  A. n  e.  NN  ( ( T  o.  f ) `  n )  =  ( ( NN  X.  {
0 } ) `  n ) ) )
4742, 45, 46sylancl 644 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  (
( T  o.  f
)  =  ( NN 
X.  { 0 } )  <->  A. n  e.  NN  ( ( T  o.  f ) `  n
)  =  ( ( NN  X.  { 0 } ) `  n
) ) )
4834, 47mpbird 224 . . . . . 6  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T  o.  f )  =  ( NN  X.  { 0 } ) )
496cnfldtopon 18770 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
5049a1i 11 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )
51 0cn 9040 . . . . . . . 8  |-  0  e.  CC
5251a1i 11 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  0  e.  CC )
53 1z 10267 . . . . . . . 8  |-  1  e.  ZZ
5453a1i 11 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  1  e.  ZZ )
55 nnuz 10477 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
5655lmconst 17279 . . . . . . 7  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  0  e.  CC  /\  1  e.  ZZ )  ->  ( NN  X.  { 0 } ) ( ~~> t `  ( TopOpen ` fld ) ) 0 )
5750, 52, 54, 56syl3anc 1184 . . . . . 6  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( NN  X.  { 0 } ) ( ~~> t `  ( TopOpen ` fld ) ) 0 )
5848, 57eqbrtrd 4192 . . . . 5  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T  o.  f )
( ~~> t `  ( TopOpen
` fld
) ) 0 )
598, 22, 58lmmo 17398 . . . 4  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T `  x )  =  0 )
60 elnlfn 23384 . . . . 5  |-  ( T : ~H --> CC  ->  ( x  e.  ( null `  T )  <->  ( x  e.  ~H  /\  ( T `
 x )  =  0 ) ) )
6123, 60ax-mp 8 . . . 4  |-  ( x  e.  ( null `  T
)  <->  ( x  e. 
~H  /\  ( T `  x )  =  0 ) )
625, 59, 61sylanbrc 646 . . 3  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  x  e.  ( null `  T
) )
6362gen2 1553 . 2  |-  A. f A. x ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  x  e.  ( null `  T
) )
64 isch2 22679 . 2  |-  ( (
null `  T )  e.  CH  <->  ( ( null `  T )  e.  SH  /\ 
A. f A. x
( ( f : NN --> ( null `  T
)  /\  f  ~~>v  x )  ->  x  e.  ( null `  T )
) ) )
652, 63, 64mpbir2an 887 1  |-  ( null `  T )  e.  CH
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546    = wceq 1649    e. wcel 1721   A.wral 2666    C_ wss 3280   {csn 3774   <.cop 3777   class class class wbr 4172    X. cxp 4835    |` cres 4839    o. ccom 4841    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    ^m cmap 6977   CCcc 8944   0cc0 8946   1c1 8947   NNcn 9956   ZZcz 10238   TopOpenctopn 13604   MetOpencmopn 16646  ℂfldccnfld 16658  TopOnctopon 16914    Cn ccn 17242   ~~> tclm 17244   Hauscha 17326   ~Hchil 22375    +h cva 22376    .h csm 22377   normhcno 22379    -h cmv 22381    ~~>v chli 22383   SHcsh 22384   CHcch 22385   nullcnl 22408   ConFnccnfn 22409   LinFnclf 22410
This theorem is referenced by:  riesz3i  23518
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026  ax-hilex 22455  ax-hfvadd 22456  ax-hvcom 22457  ax-hvass 22458  ax-hv0cl 22459  ax-hvaddid 22460  ax-hfvmul 22461  ax-hvmulid 22462  ax-hvmulass 22463  ax-hvdistr1 22464  ax-hvdistr2 22465  ax-hvmul0 22466  ax-hfi 22534  ax-his1 22537  ax-his2 22538  ax-his3 22539  ax-his4 22540
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-icc 10879  df-fz 11000  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-rest 13605  df-topn 13606  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cn 17245  df-cnp 17246  df-lm 17247  df-haus 17333  df-xms 18303  df-ms 18304  df-grpo 21732  df-gid 21733  df-ginv 21734  df-gdiv 21735  df-ablo 21823  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-vs 22031  df-nmcv 22032  df-ims 22033  df-hnorm 22424  df-hvsub 22427  df-hlim 22428  df-sh 22662  df-ch 22677  df-nlfn 23302  df-cnfn 23303  df-lnfn 23304
  Copyright terms: Public domain W3C validator