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Theorem nlelchi 23525
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 23524 . 2  |-  ( null `  T )  e.  SH
3 vex 2927 . . . . . 6  |-  x  e. 
_V
43hlimveci 22653 . . . . 5  |-  ( f 
~~>v  x  ->  x  e.  ~H )
54adantl 453 . . . 4  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  x  e.  ~H )
6 eqid 2412 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
76cnfldhaus 18780 . . . . . 6  |-  ( TopOpen ` fld )  e.  Haus
87a1i 11 . . . . 5  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( TopOpen
` fld
)  e.  Haus )
9 eqid 2412 . . . . . . . . . 10  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
10 eqid 2412 . . . . . . . . . . 11  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
119, 10hhims 22635 . . . . . . . . . 10  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
12 eqid 2412 . . . . . . . . . 10  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
139, 11, 12hhlm 22662 . . . . . . . . 9  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
14 resss 5137 . . . . . . . . 9  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
1513, 14eqsstri 3346 . . . . . . . 8  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
1615ssbri 4222 . . . . . . 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 23369 . . . . . . . 8  |-  ConFn  =  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( TopOpen ` fld ) )
2018, 19eleqtri 2484 . . . . . . 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 17331 . . . . 5  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T  o.  f )
( ~~> t `  ( TopOpen
` fld
) ) ( T `
 x ) )
231lnfnfi 23505 . . . . . . . . . 10  |-  T : ~H
--> CC
24 ffvelrn 5835 . . . . . . . . . . 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 23393 . . . . . . . . . 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 5767 . . . . . . . . . 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 9049 . . . . . . . . . . 11  |-  0  e.  _V
3130fvconst2 5914 . . . . . . . . . 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 2454 . . . . . . . 8  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( ( T  o.  f ) `  n )  =  ( ( NN  X.  {
0 } ) `  n ) )
3433ralrimiva 2757 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  A. n  e.  NN  ( ( T  o.  f ) `  n )  =  ( ( NN  X.  {
0 } ) `  n ) )
35 ffn 5558 . . . . . . . . . 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 22676 . . . . . . . . . 10  |-  ( null `  T )  C_  ~H
39 fss 5566 . . . . . . . . . 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 5576 . . . . . . . . 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 5596 . . . . . . . . 9  |-  ( NN 
X.  { 0 } ) : NN --> { 0 }
44 ffn 5558 . . . . . . . . 9  |-  ( ( NN  X.  { 0 } ) : NN --> { 0 }  ->  ( NN  X.  { 0 } )  Fn  NN )
4543, 44ax-mp 8 . . . . . . . 8  |-  ( NN 
X.  { 0 } )  Fn  NN
46 eqfnfv 5794 . . . . . . . 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 18778 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
5049a1i 11 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )
51 0cn 9048 . . . . . . . 8  |-  0  e.  CC
5251a1i 11 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  0  e.  CC )
53 1z 10275 . . . . . . . 8  |-  1  e.  ZZ
5453a1i 11 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  1  e.  ZZ )
55 nnuz 10485 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
5655lmconst 17287 . . . . . . 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 4200 . . . . 5  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T  o.  f )
( ~~> t `  ( TopOpen
` fld
) ) 0 )
598, 22, 58lmmo 17406 . . . 4  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T `  x )  =  0 )
60 elnlfn 23392 . . . . 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 22687 . 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 2674    C_ wss 3288   {csn 3782   <.cop 3785   class class class wbr 4180    X. cxp 4843    |` cres 4847    o. ccom 4849    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6048    ^m cmap 6985   CCcc 8952   0cc0 8954   1c1 8955   NNcn 9964   ZZcz 10246   TopOpenctopn 13612   MetOpencmopn 16654  ℂfldccnfld 16666  TopOnctopon 16922    Cn ccn 17250   ~~> tclm 17252   Hauscha 17334   ~Hchil 22383    +h cva 22384    .h csm 22385   normhcno 22387    -h cmv 22389    ~~>v chli 22391   SHcsh 22392   CHcch 22393   nullcnl 22416   ConFnccnfn 22417   LinFnclf 22418
This theorem is referenced by:  riesz3i  23526
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033  ax-mulf 9034  ax-hilex 22463  ax-hfvadd 22464  ax-hvcom 22465  ax-hvass 22466  ax-hv0cl 22467  ax-hvaddid 22468  ax-hfvmul 22469  ax-hvmulid 22470  ax-hvmulass 22471  ax-hvdistr1 22472  ax-hvdistr2 22473  ax-hvmul0 22474  ax-hfi 22542  ax-his1 22545  ax-his2 22546  ax-his3 22547  ax-his4 22548
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-7 10027  df-8 10028  df-9 10029  df-10 10030  df-n0 10186  df-z 10247  df-dec 10347  df-uz 10453  df-q 10539  df-rp 10577  df-xneg 10674  df-xadd 10675  df-xmul 10676  df-icc 10887  df-fz 11008  df-seq 11287  df-exp 11346  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-plusg 13505  df-mulr 13506  df-starv 13507  df-tset 13511  df-ple 13512  df-ds 13514  df-unif 13515  df-rest 13613  df-topn 13614  df-topgen 13630  df-psmet 16657  df-xmet 16658  df-met 16659  df-bl 16660  df-mopn 16661  df-cnfld 16667  df-top 16926  df-bases 16928  df-topon 16929  df-topsp 16930  df-cn 17253  df-cnp 17254  df-lm 17255  df-haus 17341  df-xms 18311  df-ms 18312  df-grpo 21740  df-gid 21741  df-ginv 21742  df-gdiv 21743  df-ablo 21831  df-vc 21986  df-nv 22032  df-va 22035  df-ba 22036  df-sm 22037  df-0v 22038  df-vs 22039  df-nmcv 22040  df-ims 22041  df-hnorm 22432  df-hvsub 22435  df-hlim 22436  df-sh 22670  df-ch 22685  df-nlfn 23310  df-cnfn 23311  df-lnfn 23312
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