Theorem nlelchi 28920

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. ContFn

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.

Step	Hyp	Ref	Expression
1		nlelch.1	. . 3 |- T e. LinFn
2	1	nlelshi 28919	. 2 |- ( null ` T ) e. SH
3		vex 3203	. . . . . 6 |- x e. _V
4	3	hlimveci 28047	. . . . 5 |- ( f ~~>v x -> x e. ~H )
5	4	adantl 482	. . . 4 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> x e. ~H )
6		eqid 2622	. . . . . . 7 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
7	6	cnfldhaus 22588	. . . . . 6 |- ( TopOpen ` ℂfld ) e. Haus
8	7	a1i 11	. . . . 5 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> ( TopOpen ` ℂfld ) e. Haus )
9		eqid 2622	. . . . . . . . . 10 |- <. <. +h , .h >. , normh >. = <. <. +h , .h >. , normh >.
10		eqid 2622	. . . . . . . . . . 11 |- ( normh o. -h ) = ( normh o. -h )
11	9, 10	hhims 28029	. . . . . . . . . 10 |- ( normh o. -h ) = ( IndMet ` <. <. +h , .h >. , normh >. )
12		eqid 2622	. . . . . . . . . 10 |- ( MetOpen ` ( normh o. -h ) ) = ( MetOpen ` ( normh o. -h ) )
13	9, 11, 12	hhlm 28056	. . . . . . . . 9 |- ~~>v = ( ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) |` ( ~H ^m NN ) )
14		resss 5422	. . . . . . . . 9 |- ( ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) |` ( ~H ^m NN ) ) C_ ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) )
15	13, 14	eqsstri 3635	. . . . . . . 8 |- ~~>v C_ ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) )
16	15	ssbri 4697	. . . . . . 7 |- ( f ~~>v x -> f ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) x )
17	16	adantl 482	. . . . . 6 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> f ( ~~> t ` ( MetOpen ` ( normh o. -h ) ) ) x )
18		nlelch.2	. . . . . . . 8 |- T e. ContFn
19	10, 12, 6	hhcnf 28764	. . . . . . . 8 |- ContFn = ( ( MetOpen ` ( normh o. -h ) ) Cn ( TopOpen ` ℂfld ) )
20	18, 19	eleqtri 2699	. . . . . . 7 |- T e. ( ( MetOpen ` ( normh o. -h ) ) Cn ( TopOpen ` ℂfld ) )
21	20	a1i 11	. . . . . 6 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> T e. ( ( MetOpen ` ( normh o. -h ) ) Cn ( TopOpen ` ℂfld ) ) )
22	17, 21	lmcn 21109	. . . . 5 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> ( T o. f ) ( ~~> t ` ( TopOpen ` ℂfld ) ) ( T ` x ) )
23	1	lnfnfi 28900	. . . . . . . . . 10 |- T : ~H --> CC
24		ffvelrn 6357	. . . . . . . . . . 11 |- ( ( f : NN --> ( null ` T ) /\ n e. NN ) -> ( f ` n ) e. ( null ` T ) )
25	24	adantlr 751	. . . . . . . . . 10 |- ( ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) /\ n e. NN ) -> ( f ` n ) e. ( null ` T ) )
26		elnlfn2 28788	. . . . . . . . . 10 |- ( ( T : ~H --> CC /\ ( f ` n ) e. ( null ` T ) ) -> ( T ` ( f ` n ) ) = 0 )
27	23, 25, 26	sylancr 695	. . . . . . . . 9 |- ( ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) /\ n e. NN ) -> ( T ` ( f ` n ) ) = 0 )
28		fvco3 6275	. . . . . . . . . 10 |- ( ( f : NN --> ( null ` T ) /\ n e. NN ) -> ( ( T o. f ) ` n ) = ( T ` ( f ` n ) ) )
29	28	adantlr 751	. . . . . . . . 9 |- ( ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) /\ n e. NN ) -> ( ( T o. f ) ` n ) = ( T ` ( f ` n ) ) )
30		c0ex 10034	. . . . . . . . . . 11 |- 0 e. _V
31	30	fvconst2 6469	. . . . . . . . . 10 |- ( n e. NN -> ( ( NN X. { 0 } ) ` n ) = 0 )
32	31	adantl 482	. . . . . . . . 9 |- ( ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) /\ n e. NN ) -> ( ( NN X. { 0 } ) ` n ) = 0 )
33	27, 29, 32	3eqtr4d 2666	. . . . . . . 8 |- ( ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) /\ n e. NN ) -> ( ( T o. f ) ` n ) = ( ( NN X. { 0 } ) ` n ) )
34	33	ralrimiva 2966	. . . . . . 7 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> A. n e. NN ( ( T o. f ) ` n ) = ( ( NN X. { 0 } ) ` n ) )
35		ffn 6045	. . . . . . . . . 10 |- ( T : ~H --> CC -> T Fn ~H )
36	23, 35	ax-mp 5	. . . . . . . . 9 |- T Fn ~H
37		simpl 473	. . . . . . . . . 10 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> f : NN --> ( null ` T ) )
38	2	shssii 28070	. . . . . . . . . 10 |- ( null ` T ) C_ ~H
39		fss 6056	. . . . . . . . . 10 |- ( ( f : NN --> ( null ` T ) /\ ( null ` T ) C_ ~H ) -> f : NN --> ~H )
40	37, 38, 39	sylancl 694	. . . . . . . . 9 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> f : NN --> ~H )
41		fnfco 6069	. . . . . . . . 9 |- ( ( T Fn ~H /\ f : NN --> ~H ) -> ( T o. f ) Fn NN )
42	36, 40, 41	sylancr 695	. . . . . . . 8 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> ( T o. f ) Fn NN )
43	30	fconst 6091	. . . . . . . . 9 |- ( NN X. { 0 } ) : NN --> { 0 }
44		ffn 6045	. . . . . . . . 9 |- ( ( NN X. { 0 } ) : NN --> { 0 } -> ( NN X. { 0 } ) Fn NN )
45	43, 44	ax-mp 5	. . . . . . . 8 |- ( NN X. { 0 } ) Fn NN
46		eqfnfv 6311	. . . . . . . 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 ) ) )
47	42, 45, 46	sylancl 694	. . . . . . 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 ) ) )
48	34, 47	mpbird 247	. . . . . 6 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> ( T o. f ) = ( NN X. { 0 } ) )
49	6	cnfldtopon 22586	. . . . . . . 8 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
50	49	a1i 11	. . . . . . 7 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> ( TopOpen ` ℂfld ) e. (TopOn ` CC ) )
51		0cnd 10033	. . . . . . 7 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> 0 e. CC )
52		1zzd 11408	. . . . . . 7 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> 1 e. ZZ )
53		nnuz 11723	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
54	53	lmconst 21065	. . . . . . 7 |- ( ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) /\ 0 e. CC /\ 1 e. ZZ ) -> ( NN X. { 0 } ) ( ~~> t ` ( TopOpen ` ℂfld ) ) 0 )
55	50, 51, 52, 54	syl3anc 1326	. . . . . 6 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> ( NN X. { 0 } ) ( ~~> t ` ( TopOpen ` ℂfld ) ) 0 )
56	48, 55	eqbrtrd 4675	. . . . 5 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> ( T o. f ) ( ~~> t ` ( TopOpen ` ℂfld ) ) 0 )
57	8, 22, 56	lmmo 21184	. . . 4 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> ( T ` x ) = 0 )
58		elnlfn 28787	. . . . 5 |- ( T : ~H --> CC -> ( x e. ( null ` T ) <-> ( x e. ~H /\ ( T ` x ) = 0 ) ) )
59	23, 58	ax-mp 5	. . . 4 |- ( x e. ( null ` T ) <-> ( x e. ~H /\ ( T ` x ) = 0 ) )
60	5, 57, 59	sylanbrc 698	. . 3 |- ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> x e. ( null ` T ) )
61	60	gen2 1723	. 2 |- A. f A. x ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> x e. ( null ` T ) )
62		isch2 28080	. 2 |- ( ( null ` T ) e. CH <-> ( ( null ` T ) e. SH /\ A. f A. x ( ( f : NN --> ( null ` T ) /\ f ~~>v x ) -> x e. ( null ` T ) ) ) )
63	2, 61, 62	mpbir2an 955	1 |- ( null ` T ) e. CH

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   {csn 4177   <.cop 4183   class class class wbr 4653   X. cxp 5112   |` cres 5116   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   0cc0 9936   1c1 9937   NNcn 11020   ZZcz 11377   TopOpenctopn 16082   MetOpencmopn 19736  ℂ_fldccnfld 19746  TopOnctopon 20715   Cn ccn 21028   ~~> tclm 21030   Hauscha 21112   ~Hchil 27776   +h cva 27777   .h csm 27778   normhcno 27780   -h cmv 27782   ~~>v chli 27784   SHcsh 27785   CHcch 27786   nullcnl 27809   ContFnccnfn 27810   LinFnclf 27811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016  ax-hilex 27856  ax-hfvadd 27857  ax-hvcom 27858  ax-hvass 27859  ax-hv0cl 27860  ax-hvaddid 27861  ax-hfvmul 27862  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvdistr1 27865  ax-hvdistr2 27866  ax-hvmul0 27867  ax-hfi 27936  ax-his1 27939  ax-his2 27940  ax-his3 27941  ax-his4 27942

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-topn 16084  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-lm 21033  df-haus 21119  df-xms 22125  df-ms 22126  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-vs 27454  df-nmcv 27455  df-ims 27456  df-hnorm 27825  df-hvsub 27828  df-hlim 27829  df-sh 28064  df-ch 28078  df-nlfn 28705  df-cnfn 28706  df-lnfn 28707

This theorem is referenced by:  riesz3i  28921