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Theorem nlelchi 22602
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 22601 . 2  |-  ( null `  T )  e.  SH
3 vex 2766 . . . . . 6  |-  x  e. 
_V
43hlimveci 21730 . . . . 5  |-  ( f 
~~>v  x  ->  x  e.  ~H )
54adantl 454 . . . 4  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  x  e.  ~H )
6 eqid 2258 . . . . . . 7  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
76cnfldhaus 18257 . . . . . 6  |-  ( TopOpen ` fld )  e.  Haus
87a1i 12 . . . . 5  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( TopOpen
` fld
)  e.  Haus )
9 eqid 2258 . . . . . . . . . 10  |-  <. <.  +h  ,  .h  >. ,  normh >.  =  <. <.  +h  ,  .h  >. ,  normh >.
10 eqid 2258 . . . . . . . . . . 11  |-  ( normh  o. 
-h  )  =  (
normh  o.  -h  )
119, 10hhims 21712 . . . . . . . . . 10  |-  ( normh  o. 
-h  )  =  (
IndMet `  <. <.  +h  ,  .h  >. ,  normh >. )
12 eqid 2258 . . . . . . . . . 10  |-  ( MetOpen `  ( normh  o.  -h  )
)  =  ( MetOpen `  ( normh  o.  -h  )
)
139, 11, 12hhlm 21739 . . . . . . . . 9  |-  ~~>v  =  ( ( ~~> t `  ( MetOpen
`  ( normh  o.  -h  ) ) )  |`  ( ~H  ^m  NN ) )
14 resss 4967 . . . . . . . . 9  |-  ( ( ~~> t `  ( MetOpen `  ( normh  o.  -h  )
) )  |`  ( ~H  ^m  NN ) ) 
C_  ( ~~> t `  ( MetOpen `  ( normh  o. 
-h  ) ) )
1513, 14eqsstri 3183 . . . . . . . 8  |-  ~~>v  C_  ( ~~> t `  ( MetOpen `  ( normh  o.  -h  ) ) )
1615ssbri 4039 . . . . . . 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 22446 . . . . . . . 8  |-  ConFn  =  ( ( MetOpen `  ( normh  o. 
-h  ) )  Cn  ( TopOpen ` fld ) )
2018, 19eleqtri 2330 . . . . . . 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 16996 . . . . 5  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T  o.  f )
( ~~> t `  ( TopOpen
` fld
) ) ( T `
 x ) )
231lnfnfi 22582 . . . . . . . . . 10  |-  T : ~H
--> CC
24 ffvelrn 5597 . . . . . . . . . . 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 22470 . . . . . . . . . 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 5530 . . . . . . . . . 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 8800 . . . . . . . . . . 11  |-  0  e.  _V
3130fvconst2 5663 . . . . . . . . . 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 2300 . . . . . . . 8  |-  ( ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  /\  n  e.  NN )  ->  ( ( T  o.  f ) `  n )  =  ( ( NN  X.  {
0 } ) `  n ) )
3433ralrimiva 2601 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  A. n  e.  NN  ( ( T  o.  f ) `  n )  =  ( ( NN  X.  {
0 } ) `  n ) )
35 ffn 5327 . . . . . . . . . 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 21753 . . . . . . . . . 10  |-  ( null `  T )  C_  ~H
39 fss 5335 . . . . . . . . . 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 5345 . . . . . . . . 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 5365 . . . . . . . . 9  |-  ( NN 
X.  { 0 } ) : NN --> { 0 }
44 ffn 5327 . . . . . . . . 9  |-  ( ( NN  X.  { 0 } ) : NN --> { 0 }  ->  ( NN  X.  { 0 } )  Fn  NN )
4543, 44ax-mp 10 . . . . . . . 8  |-  ( NN 
X.  { 0 } )  Fn  NN
46 eqfnfv 5556 . . . . . . . 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 18255 . . . . . . . 8  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
5049a1i 12 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( TopOpen
` fld
)  e.  (TopOn `  CC ) )
51 0cn 8799 . . . . . . . 8  |-  0  e.  CC
5251a1i 12 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  0  e.  CC )
53 1z 10021 . . . . . . . 8  |-  1  e.  ZZ
5453a1i 12 . . . . . . 7  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  1  e.  ZZ )
55 nnuz 10231 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
5655lmconst 16954 . . . . . . 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 4017 . . . . 5  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T  o.  f )
( ~~> t `  ( TopOpen
` fld
) ) 0 )
598, 22, 58lmmo 17071 . . . 4  |-  ( ( f : NN --> ( null `  T )  /\  f  ~~>v  x )  ->  ( T `  x )  =  0 )
60 elnlfn 22469 . . . . 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 21764 . 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 2518    C_ wss 3127   {csn 3614   <.cop 3617   class class class wbr 3997    X. cxp 4659    |` cres 4663    o. ccom 4665    Fn wfn 4668   -->wf 4669   ` cfv 4673  (class class class)co 5792    ^m cmap 6740   CCcc 8703   0cc0 8705   1c1 8706   NNcn 9714   ZZcz 9992   TopOpenctopn 13289   MetOpencmopn 16335  ℂfldccnfld 16340  TopOnctopon 16595    Cn ccn 16917   ~~> tclm 16919   Hauscha 16999   ~Hchil 21460    +h cva 21461    .h csm 21462   normhcno 21464    -h cmv 21466    ~~>v chli 21468   SHcsh 21469   CHcch 21470   nullcnl 21493   ConFnccnfn 21494   LinFnclf 21495
This theorem is referenced by:  riesz3i  22603
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 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785  ax-hilex 21540  ax-hfvadd 21541  ax-hvcom 21542  ax-hvass 21543  ax-hv0cl 21544  ax-hvaddid 21545  ax-hfvmul 21546  ax-hvmulid 21547  ax-hvmulass 21548  ax-hvdistr1 21549  ax-hvdistr2 21550  ax-hvmul0 21551  ax-hfi 21619  ax-his1 21622  ax-his2 21623  ax-his3 21624  ax-his4 21625
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-pm 6743  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-div 9392  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9934  df-z 9993  df-dec 10093  df-uz 10199  df-q 10285  df-rp 10323  df-xneg 10420  df-xadd 10421  df-xmul 10422  df-icc 10630  df-fz 10750  df-seq 11014  df-exp 11072  df-cj 11550  df-re 11551  df-im 11552  df-sqr 11686  df-abs 11687  df-struct 13113  df-ndx 13114  df-slot 13115  df-base 13116  df-plusg 13184  df-mulr 13185  df-starv 13186  df-tset 13190  df-ple 13191  df-ds 13193  df-rest 13290  df-topn 13291  df-topgen 13307  df-xmet 16336  df-met 16337  df-bl 16338  df-mopn 16339  df-cnfld 16341  df-top 16599  df-bases 16601  df-topon 16602  df-topsp 16603  df-cn 16920  df-cnp 16921  df-lm 16922  df-haus 17006  df-xms 17848  df-ms 17849  df-grpo 20819  df-gid 20820  df-ginv 20821  df-gdiv 20822  df-ablo 20910  df-vc 21063  df-nv 21109  df-va 21112  df-ba 21113  df-sm 21114  df-0v 21115  df-vs 21116  df-nmcv 21117  df-ims 21118  df-hnorm 21509  df-hvsub 21512  df-hlim 21513  df-sh 21747  df-ch 21762  df-nlfn 22387  df-cnfn 22388  df-lnfn 22389
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