Theorem lkrfval 34374

Description: The kernel of a functional. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Hypotheses

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

Assertion

Ref	Expression
lkrfval	|- ( W e. X -> K = ( f e. F |-> ( `' f " { .0. } ) ) )

Distinct variable groups:   f, F   f, W

Allowed substitution hints:   D( f)   K( f)   X( f)   .0. ( f)

Proof of Theorem lkrfval

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( W e. X -> W e. _V )
2		lkrfval.k	. . 3 |- K = (LKer ` W )
3		fveq2 6191	. . . . . 6 |- ( w = W -> (LFnl ` w ) = (LFnl ` W ) )
4		lkrfval.f	. . . . . 6 |- F = (LFnl ` W )
5	3, 4	syl6eqr 2674	. . . . 5 |- ( w = W -> (LFnl ` w ) = F )
6		fveq2 6191	. . . . . . . . . 10 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
7		lkrfval.d	. . . . . . . . . 10 |- D = (Scalar ` W )
8	6, 7	syl6eqr 2674	. . . . . . . . 9 |- ( w = W -> (Scalar ` w ) = D )
9	8	fveq2d 6195	. . . . . . . 8 |- ( w = W -> ( 0g ` (Scalar ` w ) ) = ( 0g ` D ) )
10		lkrfval.o	. . . . . . . 8 |- .0. = ( 0g ` D )
11	9, 10	syl6eqr 2674	. . . . . . 7 |- ( w = W -> ( 0g ` (Scalar ` w ) ) = .0. )
12	11	sneqd 4189	. . . . . 6 |- ( w = W -> { ( 0g ` (Scalar ` w ) ) } = { .0. } )
13	12	imaeq2d 5466	. . . . 5 |- ( w = W -> ( `' f " { ( 0g ` (Scalar ` w ) ) } ) = ( `' f " { .0. } ) )
14	5, 13	mpteq12dv 4733	. . . 4 |- ( w = W -> ( f e. (LFnl ` w ) |-> ( `' f " { ( 0g ` (Scalar ` w ) ) } ) ) = ( f e. F |-> ( `' f " { .0. } ) ) )
15		df-lkr 34373	. . . 4 |- LKer = ( w e. _V |-> ( f e. (LFnl ` w ) |-> ( `' f " { ( 0g ` (Scalar ` w ) ) } ) ) )
16		fvex 6201	. . . . . 6 |- (LFnl ` W ) e. _V
17	4, 16	eqeltri 2697	. . . . 5 |- F e. _V
18	17	mptex 6486	. . . 4 |- ( f e. F |-> ( `' f " { .0. } ) ) e. _V
19	14, 15, 18	fvmpt 6282	. . 3 |- ( W e. _V -> (LKer ` W ) = ( f e. F |-> ( `' f " { .0. } ) ) )
20	2, 19	syl5eq 2668	. 2 |- ( W e. _V -> K = ( f e. F |-> ( `' f " { .0. } ) ) )
21	1, 20	syl 17	1 |- ( W e. X -> K = ( f e. F |-> ( `' f " { .0. } ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   |-> cmpt 4729   `'ccnv 5113   "cima 5117   ` cfv 5888  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-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-pr 4906

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-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-lkr 34373

This theorem is referenced by:  lkrval  34375