Theorem ellkr 34376

Description: Membership in the kernel of a functional. (elnlfn 28787 analog.) (Contributed by NM, 16-Apr-2014.)

Hypotheses

Ref	Expression
lkrfval2.v	|- V = ( Base ` W )
lkrfval2.d	|- D = (Scalar ` W )
lkrfval2.o	|- .0. = ( 0g ` D )
lkrfval2.f	|- F = (LFnl ` W )
lkrfval2.k	|- K = (LKer ` W )

Assertion

Ref	Expression
ellkr	|- ( ( W e. Y /\ G e. F ) -> ( X e. ( K ` G ) <-> ( X e. V /\ ( G ` X ) = .0. ) ) )

Proof of Theorem ellkr

Step	Hyp	Ref	Expression
1		lkrfval2.d	. . . 4 |- D = (Scalar ` W )
2		lkrfval2.o	. . . 4 |- .0. = ( 0g ` D )
3		lkrfval2.f	. . . 4 |- F = (LFnl ` W )
4		lkrfval2.k	. . . 4 |- K = (LKer ` W )
5	1, 2, 3, 4	lkrval 34375	. . 3 |- ( ( W e. Y /\ G e. F ) -> ( K ` G ) = ( `' G " { .0. } ) )
6	5	eleq2d 2687	. 2 |- ( ( W e. Y /\ G e. F ) -> ( X e. ( K ` G ) <-> X e. ( `' G " { .0. } ) ) )
7		eqid 2622	. . . . 5 |- ( Base ` D ) = ( Base ` D )
8		lkrfval2.v	. . . . 5 |- V = ( Base ` W )
9	1, 7, 8, 3	lflf 34350	. . . 4 |- ( ( W e. Y /\ G e. F ) -> G : V --> ( Base ` D ) )
10		ffn 6045	. . . 4 |- ( G : V --> ( Base ` D ) -> G Fn V )
11		elpreima 6337	. . . 4 |- ( G Fn V -> ( X e. ( `' G " { .0. } ) <-> ( X e. V /\ ( G ` X ) e. { .0. } ) ) )
12	9, 10, 11	3syl 18	. . 3 |- ( ( W e. Y /\ G e. F ) -> ( X e. ( `' G " { .0. } ) <-> ( X e. V /\ ( G ` X ) e. { .0. } ) ) )
13		fvex 6201	. . . . 5 |- ( G ` X ) e. _V
14	13	elsn 4192	. . . 4 |- ( ( G ` X ) e. { .0. } <-> ( G ` X ) = .0. )
15	14	anbi2i 730	. . 3 |- ( ( X e. V /\ ( G ` X ) e. { .0. } ) <-> ( X e. V /\ ( G ` X ) = .0. ) )
16	12, 15	syl6bb 276	. 2 |- ( ( W e. Y /\ G e. F ) -> ( X e. ( `' G " { .0. } ) <-> ( X e. V /\ ( G ` X ) = .0. ) ) )
17	6, 16	bitrd 268	1 |- ( ( W e. Y /\ G e. F ) -> ( X e. ( K ` G ) <-> ( X e. V /\ ( G ` X ) = .0. ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888   Basecbs 15857  Scalarcsca 15944   0gc0g 16100  LFnlclfn 34344  LKerclk 34372

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

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-lfl 34345  df-lkr 34373

This theorem is referenced by:  lkrval2  34377  ellkr2  34378  lkrcl  34379  lkrf0  34380  lkrlss  34382  lkrsc  34384  eqlkr  34386  lkrlsp  34389  lkrlsp2  34390  lshpkr  34404  lkrin  34451  dochfln0  36766