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Theorem nlelchi 22471
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
StepHypRef Expression
1 nlelch.1 . . 3  |-  T  e. 
LinFn
21nlelshi 22470 . 2  |-  ( null `  T )  e.  SH
3 vex 2730 . . . . . 6  |-  x  e. 
_V
43hlimveci 21599 . . . . 5  |-  ( f 
~~>v  x  ->  x  e.  ~H )
54adantl 454 . . . 4  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  x  e.  ~H )
6 eqid 2253 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
76cnfldhaus 18126 . . . . . 6  |-  ( TopOpen ` fld )  e.  Haus
87a1i 12 . . . . 5  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( TopOpen
` fld
)  e.  Haus )
9 eqid 2253 . . . . . . . . . 10  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
10 eqid 2253 . . . . . . . . . . 11  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
119, 10hhims 21581 . . . . . . . . . 10  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
12 eqid 2253 . . . . . . . . . 10  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
139, 11, 12hhlm 21608 . . . . . . . . 9  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
14 resss 4886 . . . . . . . . 9  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
1513, 14eqsstri 3129 . . . . . . . 8  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
1615ssbri 3962 . . . . . . 7  |-  ( f 
~~>v  x  ->  f ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) ) x )
1716adantl 454 . . . . . 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 22315 . . . . . . . 8  |-  ConFn  =  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( TopOpen ` fld ) )
2018, 19eleqtri 2325 . . . . . . 7  |-  T  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( TopOpen ` fld ) )
2120a1i 12 . . . . . 6  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  T  e.  ( ( MetOpen `  ( normh  o.  -h  ) )  Cn  ( TopOpen ` fld ) ) )
2217, 21lmcn 16865 . . . . 5  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T  o.  f )
( ~~> t `  ( TopOpen
` fld
) ) ( T `
 x ) )
231lnfnfi 22451 . . . . . . . . . 10  |-  T : ~H
--> CC
24 ffvelrn 5515 . . . . . . . . . . 11  |-  ( ( f : NN --> ( null `  T )  /\  n  e.  NN )  ->  (
f `  n )  e.  ( null `  T
) )
2524adantlr 698 . . . . . . . . . 10  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( f `  n )  e.  (
null `  T )
)
26 elnlfn2 22339 . . . . . . . . . 10  |-  ( ( T : ~H --> CC  /\  ( f `  n
)  e.  ( null `  T ) )  -> 
( T `  (
f `  n )
)  =  0 )
2723, 25, 26sylancr 647 . . . . . . . . 9  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( T `  ( f `  n
) )  =  0 )
28 fvco3 5448 . . . . . . . . . 10  |-  ( ( f : NN --> ( null `  T )  /\  n  e.  NN )  ->  (
( T  o.  f
) `  n )  =  ( T `  ( f `  n
) ) )
2928adantlr 698 . . . . . . . . 9  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( ( T  o.  f ) `  n )  =  ( T `  ( f `
 n ) ) )
30 c0ex 8712 . . . . . . . . . . 11  |-  0  e.  _V
3130fvconst2 5581 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (
( NN  X.  {
0 } ) `  n )  =  0 )
3231adantl 454 . . . . . . . . 9  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( ( NN 
X.  { 0 } ) `  n )  =  0 )
3327, 29, 323eqtr4d 2295 . . . . . . . 8  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( ( T  o.  f ) `  n )  =  ( ( NN  X.  {
0 } ) `  n ) )
3433ralrimiva 2588 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  A. n  e.  NN  ( ( T  o.  f ) `  n )  =  ( ( NN  X.  {
0 } ) `  n ) )
35 ffn 5246 . . . . . . . . . 10  |-  ( T : ~H --> CC  ->  T  Fn  ~H )
3623, 35ax-mp 10 . . . . . . . . 9  |-  T  Fn  ~H
37 simpl 445 . . . . . . . . . 10  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  f : NN --> ( null `  T
) )
382shssii 21622 . . . . . . . . . 10  |-  ( null `  T )  C_  ~H
39 fss 5254 . . . . . . . . . 10  |-  ( ( f : NN --> ( null `  T )  /\  ( null `  T )  C_  ~H )  ->  f : NN --> ~H )
4037, 38, 39sylancl 646 . . . . . . . . 9  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  f : NN --> ~H )
41 fnfco 5264 . . . . . . . . 9  |-  ( ( T  Fn  ~H  /\  f : NN --> ~H )  ->  ( T  o.  f
)  Fn  NN )
4236, 40, 41sylancr 647 . . . . . . . 8  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T  o.  f )  Fn  NN )
4330fconst 5284 . . . . . . . . 9  |-  ( NN 
X.  { 0 } ) : NN --> { 0 }
44 ffn 5246 . . . . . . . . 9  |-  ( ( NN  X.  { 0 } ) : NN --> { 0 }  ->  ( NN  X.  { 0 } )  Fn  NN )
4543, 44ax-mp 10 . . . . . . . 8  |-  ( NN 
X.  { 0 } )  Fn  NN
46 eqfnfv 5474 . . . . . . . 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 646 . . . . . . 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 225 . . . . . 6  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T  o.  f )  =  ( NN  X.  { 0 } ) )
496cnfldtopon 18124 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
5049a1i 12 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )
51 0cn 8711 . . . . . . . 8  |-  0  e.  CC
5251a1i 12 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  0  e.  CC )
53 1z 9932 . . . . . . . 8  |-  1  e.  ZZ
5453a1i 12 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  1  e.  ZZ )
55 nnuz 10142 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
5655lmconst 16823 . . . . . . 7  |-  ( ( ( TopOpen ` fld )  e.  (TopOn `  CC )  /\  0  e.  CC  /\  1  e.  ZZ )  ->  ( NN  X.  { 0 } ) ( ~~> t `  ( TopOpen ` fld ) ) 0 )
5750, 52, 54, 56syl3anc 1187 . . . . . 6  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( NN  X.  { 0 } ) ( ~~> t `  ( TopOpen ` fld ) ) 0 )
5848, 57eqbrtrd 3940 . . . . 5  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T  o.  f )
( ~~> t `  ( TopOpen
` fld
) ) 0 )
598, 22, 58lmmo 16940 . . . 4  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T `  x )  =  0 )
60 elnlfn 22338 . . . . 5  |-  ( T : ~H --> CC  ->  ( x  e.  ( null `  T )  <->  ( x  e.  ~H  /\  ( T `
 x )  =  0 ) ) )
6123, 60ax-mp 10 . . . 4  |-  ( x  e.  ( null `  T
)  <->  ( x  e. 
~H  /\  ( T `  x )  =  0 ) )
625, 59, 61sylanbrc 648 . . 3  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  x  e.  ( null `  T
) )
6362gen2 1541 . 2  |-  A. f A. x ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  x  e.  ( null `  T
) )
64 isch2 21633 . 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 891 1  |-  ( null `  T )  e.  CH
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   A.wal 1532    = wceq 1619    e. wcel 1621   A.wral 2509    C_ wss 3078   {csn 3544   <.cop 3547   class class class wbr 3920    X. cxp 4578    |` cres 4582    o. ccom 4584    Fn wfn 4587   -->wf 4588   ` cfv 4592  (class class class)co 5710    ^m cmap 6658   CCcc 8615   0cc0 8617   1c1 8618   NNcn 9626   ZZcz 9903   TopOpenctopn 13200   MetOpencmopn 16204  ℂfldccnfld 16209  TopOnctopon 16464    Cn ccn 16786   ~~> tclm 16788   Hauscha 16868   ~Hchil 21329    +h cva 21330    .h csm 21331   normhcno 21333    -h cmv 21335    ~~>v chli 21337   SHcsh 21338   CHcch 21339   nullcnl 21362   ConFnccnfn 21363   LinFnclf 21364
This theorem is referenced by:  riesz3i  22472
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-addf 8696  ax-mulf 8697  ax-hilex 21409  ax-hfvadd 21410  ax-hvcom 21411  ax-hvass 21412  ax-hv0cl 21413  ax-hvaddid 21414  ax-hfvmul 21415  ax-hvmulid 21416  ax-hvmulass 21417  ax-hvdistr1 21418  ax-hvdistr2 21419  ax-hvmul0 21420  ax-hfi 21488  ax-his1 21491  ax-his2 21492  ax-his3 21493  ax-his4 21494
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-pm 6661  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-7 9689  df-8 9690  df-9 9691  df-10 9692  df-n0 9845  df-z 9904  df-dec 10004  df-uz 10110  df-q 10196  df-rp 10234  df-xneg 10331  df-xadd 10332  df-xmul 10333  df-icc 10541  df-fz 10661  df-seq 10925  df-exp 10983  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-plusg 13095  df-mulr 13096  df-starv 13097  df-tset 13101  df-ple 13102  df-ds 13104  df-rest 13201  df-topn 13202  df-topgen 13218  df-xmet 16205  df-met 16206  df-bl 16207  df-mopn 16208  df-cnfld 16210  df-top 16468  df-bases 16470  df-topon 16471  df-topsp 16472  df-cn 16789  df-cnp 16790  df-lm 16791  df-haus 16875  df-xms 17717  df-ms 17718  df-grpo 20688  df-gid 20689  df-ginv 20690  df-gdiv 20691  df-ablo 20779  df-vc 20932  df-nv 20978  df-va 20981  df-ba 20982  df-sm 20983  df-0v 20984  df-vs 20985  df-nmcv 20986  df-ims 20987  df-hnorm 21378  df-hvsub 21381  df-hlim 21382  df-sh 21616  df-ch 21631  df-nlfn 22256  df-cnfn 22257  df-lnfn 22258
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