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Theorem nlelchi 22637
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 22636 . 2  |-  ( null `  T )  e.  SH
3 vex 2792 . . . . . 6  |-  x  e. 
_V
43hlimveci 21765 . . . . 5  |-  ( f 
~~>v  x  ->  x  e.  ~H )
54adantl 452 . . . 4  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  x  e.  ~H )
6 eqid 2284 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
76cnfldhaus 18290 . . . . . 6  |-  ( TopOpen ` fld )  e.  Haus
87a1i 10 . . . . 5  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( TopOpen
` fld
)  e.  Haus )
9 eqid 2284 . . . . . . . . . 10  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
10 eqid 2284 . . . . . . . . . . 11  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
119, 10hhims 21747 . . . . . . . . . 10  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
12 eqid 2284 . . . . . . . . . 10  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
139, 11, 12hhlm 21774 . . . . . . . . 9  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
14 resss 4978 . . . . . . . . 9  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
1513, 14eqsstri 3209 . . . . . . . 8  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
1615ssbri 4066 . . . . . . 7  |-  ( f 
~~>v  x  ->  f ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) ) x )
1716adantl 452 . . . . . 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 22481 . . . . . . . 8  |-  ConFn  =  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( TopOpen ` fld ) )
2018, 19eleqtri 2356 . . . . . . 7  |-  T  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( TopOpen ` fld ) )
2120a1i 10 . . . . . 6  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  T  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( TopOpen ` fld ) ) )
2217, 21lmcn 17029 . . . . 5  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T  o.  f )
( ~~> t `  ( TopOpen
` fld
) ) ( T `
 x ) )
231lnfnfi 22617 . . . . . . . . . 10  |-  T : ~H
--> CC
24 ffvelrn 5625 . . . . . . . . . . 11  |-  ( ( f : NN --> ( null `  T )  /\  n  e.  NN )  ->  (
f `  n )  e.  ( null `  T
) )
2524adantlr 695 . . . . . . . . . 10  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( f `  n )  e.  (
null `  T )
)
26 elnlfn2 22505 . . . . . . . . . 10  |-  ( ( T : ~H --> CC  /\  ( f `  n
)  e.  ( null `  T ) )  -> 
( T `  (
f `  n )
)  =  0 )
2723, 25, 26sylancr 644 . . . . . . . . 9  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( T `  ( f `  n
) )  =  0 )
28 fvco3 5558 . . . . . . . . . 10  |-  ( ( f : NN --> ( null `  T )  /\  n  e.  NN )  ->  (
( T  o.  f
) `  n )  =  ( T `  ( f `  n
) ) )
2928adantlr 695 . . . . . . . . 9  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( ( T  o.  f ) `  n )  =  ( T `  ( f `
 n ) ) )
30 c0ex 8828 . . . . . . . . . . 11  |-  0  e.  _V
3130fvconst2 5691 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
( NN  X.  {
0 } ) `  n )  =  0 )
3231adantl 452 . . . . . . . . 9  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( ( NN 
X.  { 0 } ) `  n )  =  0 )
3327, 29, 323eqtr4d 2326 . . . . . . . 8  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( ( T  o.  f ) `  n )  =  ( ( NN  X.  {
0 } ) `  n ) )
3433ralrimiva 2627 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  A. n  e.  NN  ( ( T  o.  f ) `  n )  =  ( ( NN  X.  {
0 } ) `  n ) )
35 ffn 5355 . . . . . . . . . 10  |-  ( T : ~H --> CC  ->  T  Fn  ~H )
3623, 35ax-mp 8 . . . . . . . . 9  |-  T  Fn  ~H
37 simpl 443 . . . . . . . . . 10  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  f : NN --> ( null `  T
) )
382shssii 21788 . . . . . . . . . 10  |-  ( null `  T )  C_  ~H
39 fss 5363 . . . . . . . . . 10  |-  ( ( f : NN --> ( null `  T )  /\  ( null `  T )  C_  ~H )  ->  f : NN --> ~H )
4037, 38, 39sylancl 643 . . . . . . . . 9  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  f : NN --> ~H )
41 fnfco 5373 . . . . . . . . 9  |-  ( ( T  Fn  ~H  /\  f : NN --> ~H )  ->  ( T  o.  f
)  Fn  NN )
4236, 40, 41sylancr 644 . . . . . . . 8  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T  o.  f )  Fn  NN )
4330fconst 5393 . . . . . . . . 9  |-  ( NN 
X.  { 0 } ) : NN --> { 0 }
44 ffn 5355 . . . . . . . . 9  |-  ( ( NN  X.  { 0 } ) : NN --> { 0 }  ->  ( NN  X.  { 0 } )  Fn  NN )
4543, 44ax-mp 8 . . . . . . . 8  |-  ( NN 
X.  { 0 } )  Fn  NN
46 eqfnfv 5584 . . . . . . . 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 643 . . . . . . 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 223 . . . . . 6  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T  o.  f )  =  ( NN  X.  { 0 } ) )
496cnfldtopon 18288 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
5049a1i 10 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )
51 0cn 8827 . . . . . . . 8  |-  0  e.  CC
5251a1i 10 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  0  e.  CC )
53 1z 10049 . . . . . . . 8  |-  1  e.  ZZ
5453a1i 10 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  1  e.  ZZ )
55 nnuz 10259 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
5655lmconst 16987 . . . . . . 7  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  0  e.  CC  /\  1  e.  ZZ )  ->  ( NN  X.  { 0 } ) ( ~~> t `  ( TopOpen ` fld ) ) 0 )
5750, 52, 54, 56syl3anc 1182 . . . . . 6  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( NN  X.  { 0 } ) ( ~~> t `  ( TopOpen ` fld ) ) 0 )
5848, 57eqbrtrd 4044 . . . . 5  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T  o.  f )
( ~~> t `  ( TopOpen
` fld
) ) 0 )
598, 22, 58lmmo 17104 . . . 4  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T `  x )  =  0 )
60 elnlfn 22504 . . . . 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 645 . . 3  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  x  e.  ( null `  T
) )
6362gen2 1534 . 2  |-  A. f A. x ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  x  e.  ( null `  T
) )
64 isch2 21799 . 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 886 1  |-  ( null `  T )  e.  CH
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1685   A.wral 2544    C_ wss 3153   {csn 3641   <.cop 3644   class class class wbr 4024    X. cxp 4686    |` cres 4690    o. ccom 4692    Fn wfn 5216   -->wf 5217   ` cfv 5221  (class class class)co 5820    ^m cmap 6768   CCcc 8731   0cc0 8733   1c1 8734   NNcn 9742   ZZcz 10020   TopOpenctopn 13322   MetOpencmopn 16368  ℂfldccnfld 16373  TopOnctopon 16628    Cn ccn 16950   ~~> tclm 16952   Hauscha 17032   ~Hchil 21495    +h cva 21496    .h csm 21497   normhcno 21499    -h cmv 21501    ~~>v chli 21503   SHcsh 21504   CHcch 21505   nullcnl 21528   ConFnccnfn 21529   LinFnclf 21530
This theorem is referenced by:  riesz3i  22638
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-addf 8812  ax-mulf 8813  ax-hilex 21575  ax-hfvadd 21576  ax-hvcom 21577  ax-hvass 21578  ax-hv0cl 21579  ax-hvaddid 21580  ax-hfvmul 21581  ax-hvmulid 21582  ax-hvmulass 21583  ax-hvdistr1 21584  ax-hvdistr2 21585  ax-hvmul0 21586  ax-hfi 21654  ax-his1 21657  ax-his2 21658  ax-his3 21659  ax-his4 21660
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-map 6770  df-pm 6771  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-sup 7190  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-7 9805  df-8 9806  df-9 9807  df-10 9808  df-n0 9962  df-z 10021  df-dec 10121  df-uz 10227  df-q 10313  df-rp 10351  df-xneg 10448  df-xadd 10449  df-xmul 10450  df-icc 10659  df-fz 10779  df-seq 11043  df-exp 11101  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-plusg 13217  df-mulr 13218  df-starv 13219  df-tset 13223  df-ple 13224  df-ds 13226  df-rest 13323  df-topn 13324  df-topgen 13340  df-xmet 16369  df-met 16370  df-bl 16371  df-mopn 16372  df-cnfld 16374  df-top 16632  df-bases 16634  df-topon 16635  df-topsp 16636  df-cn 16953  df-cnp 16954  df-lm 16955  df-haus 17039  df-xms 17881  df-ms 17882  df-grpo 20852  df-gid 20853  df-ginv 20854  df-gdiv 20855  df-ablo 20943  df-vc 21096  df-nv 21142  df-va 21145  df-ba 21146  df-sm 21147  df-0v 21148  df-vs 21149  df-nmcv 21150  df-ims 21151  df-hnorm 21544  df-hvsub 21547  df-hlim 21548  df-sh 21782  df-ch 21797  df-nlfn 22422  df-cnfn 22423  df-lnfn 22424
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