Theorem elnlfn 28787

Description: Membership in the null space of a Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
elnlfn	|- ( T : ~H --> CC -> ( A e. ( null ` T ) <-> ( A e. ~H /\ ( T ` A ) = 0 ) ) )

Proof of Theorem elnlfn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nlfnval 28740	. . . . . 6 |- ( T : ~H --> CC -> ( null ` T ) = ( `' T " { 0 } ) )
2		cnvimass 5485	. . . . . 6 |- ( `' T " { 0 } ) C_ dom T
3	1, 2	syl6eqss 3655	. . . . 5 |- ( T : ~H --> CC -> ( null ` T ) C_ dom T )
4		fdm 6051	. . . . 5 |- ( T : ~H --> CC -> dom T = ~H )
5	3, 4	sseqtrd 3641	. . . 4 |- ( T : ~H --> CC -> ( null ` T ) C_ ~H )
6	5	sseld 3602	. . 3 |- ( T : ~H --> CC -> ( A e. ( null ` T ) -> A e. ~H ) )
7	6	pm4.71rd 667	. 2 |- ( T : ~H --> CC -> ( A e. ( null ` T ) <-> ( A e. ~H /\ A e. ( null ` T ) ) ) )
8	1	eleq2d 2687	. . . . 5 |- ( T : ~H --> CC -> ( A e. ( null ` T ) <-> A e. ( `' T " { 0 } ) ) )
9	8	adantr 481	. . . 4 |- ( ( T : ~H --> CC /\ A e. ~H ) -> ( A e. ( null ` T ) <-> A e. ( `' T " { 0 } ) ) )
10		ffn 6045	. . . . 5 |- ( T : ~H --> CC -> T Fn ~H )
11		eleq1 2689	. . . . . . . 8 |- ( x = A -> ( x e. ( `' T " { 0 } ) <-> A e. ( `' T " { 0 } ) ) )
12		fveq2 6191	. . . . . . . . 9 |- ( x = A -> ( T ` x ) = ( T ` A ) )
13	12	eqeq1d 2624	. . . . . . . 8 |- ( x = A -> ( ( T ` x ) = 0 <-> ( T ` A ) = 0 ) )
14	11, 13	bibi12d 335	. . . . . . 7 |- ( x = A -> ( ( x e. ( `' T " { 0 } ) <-> ( T ` x ) = 0 ) <-> ( A e. ( `' T " { 0 } ) <-> ( T ` A ) = 0 ) ) )
15	14	imbi2d 330	. . . . . 6 |- ( x = A -> ( ( T Fn ~H -> ( x e. ( `' T " { 0 } ) <-> ( T ` x ) = 0 ) ) <-> ( T Fn ~H -> ( A e. ( `' T " { 0 } ) <-> ( T ` A ) = 0 ) ) ) )
16		fnbrfvb 6236	. . . . . . . 8 |- ( ( T Fn ~H /\ x e. ~H ) -> ( ( T ` x ) = 0 <-> x T 0 ) )
17		0cn 10032	. . . . . . . . 9 |- 0 e. CC
18		vex 3203	. . . . . . . . . 10 |- x e. _V
19	18	eliniseg 5494	. . . . . . . . 9 |- ( 0 e. CC -> ( x e. ( `' T " { 0 } ) <-> x T 0 ) )
20	17, 19	ax-mp 5	. . . . . . . 8 |- ( x e. ( `' T " { 0 } ) <-> x T 0 )
21	16, 20	syl6rbbr 279	. . . . . . 7 |- ( ( T Fn ~H /\ x e. ~H ) -> ( x e. ( `' T " { 0 } ) <-> ( T ` x ) = 0 ) )
22	21	expcom 451	. . . . . 6 |- ( x e. ~H -> ( T Fn ~H -> ( x e. ( `' T " { 0 } ) <-> ( T ` x ) = 0 ) ) )
23	15, 22	vtoclga 3272	. . . . 5 |- ( A e. ~H -> ( T Fn ~H -> ( A e. ( `' T " { 0 } ) <-> ( T ` A ) = 0 ) ) )
24	10, 23	mpan9 486	. . . 4 |- ( ( T : ~H --> CC /\ A e. ~H ) -> ( A e. ( `' T " { 0 } ) <-> ( T ` A ) = 0 ) )
25	9, 24	bitrd 268	. . 3 |- ( ( T : ~H --> CC /\ A e. ~H ) -> ( A e. ( null ` T ) <-> ( T ` A ) = 0 ) )
26	25	pm5.32da 673	. 2 |- ( T : ~H --> CC -> ( ( A e. ~H /\ A e. ( null ` T ) ) <-> ( A e. ~H /\ ( T ` A ) = 0 ) ) )
27	7, 26	bitrd 268	1 |- ( T : ~H --> CC -> ( A e. ( null ` T ) <-> ( A e. ~H /\ ( T ` A ) = 0 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888   CCcc 9934   0cc0 9936   ~Hchil 27776   nullcnl 27809

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004  ax-hilex 27856

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-nlfn 28705

This theorem is referenced by:  elnlfn2  28788  nlelshi  28919  nlelchi  28920  riesz3i  28921