Theorem lflset 34346

Description: The set of linear functionals in a left module or left vector space. (Contributed by NM, 15-Apr-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Hypotheses

Ref	Expression
lflset.v	|- V = ( Base ` W )
lflset.a	|- .+ = ( +g ` W )
lflset.d	|- D = (Scalar ` W )
lflset.s	|- .x. = ( .s ` W )
lflset.k	|- K = ( Base ` D )
lflset.p	|- .+^ = ( +g ` D )
lflset.t	|- .X. = ( .r ` D )
lflset.f	|- F = (LFnl ` W )

Assertion

Ref	Expression
lflset	|- ( W e. X -> F = { f e. ( K ^m V ) | A. r e. K A. x e. V A. y e. V ( f ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( f ` x ) ) .+^ ( f ` y ) ) } )

Distinct variable groups:   f, r, K   x, f, y, V   f, W   x, r, y, W

Allowed substitution hints:   D( x, y, f, r)   .+ ( x, y, f, r)   .+^ ( x, y, f, r)   .x. ( x, y, f, r)   .X. ( x, y, f, r)   F( x, y, f, r)   K( x, y)   V( r)   X( x, y, f, r)

Proof of Theorem lflset

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( W e. X -> W e. _V )
2		lflset.f	. . 3 |- F = (LFnl ` W )
3		fveq2 6191	. . . . . . . . 9 |- ( w = W -> (Scalar ` w ) = (Scalar ` W ) )
4		lflset.d	. . . . . . . . 9 |- D = (Scalar ` W )
5	3, 4	syl6eqr 2674	. . . . . . . 8 |- ( w = W -> (Scalar ` w ) = D )
6	5	fveq2d 6195	. . . . . . 7 |- ( w = W -> ( Base ` (Scalar ` w ) ) = ( Base ` D ) )
7		lflset.k	. . . . . . 7 |- K = ( Base ` D )
8	6, 7	syl6eqr 2674	. . . . . 6 |- ( w = W -> ( Base ` (Scalar ` w ) ) = K )
9		fveq2 6191	. . . . . . 7 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
10		lflset.v	. . . . . . 7 |- V = ( Base ` W )
11	9, 10	syl6eqr 2674	. . . . . 6 |- ( w = W -> ( Base ` w ) = V )
12	8, 11	oveq12d 6668	. . . . 5 |- ( w = W -> ( ( Base ` (Scalar ` w ) ) ^m ( Base ` w ) ) = ( K ^m V ) )
13		fveq2 6191	. . . . . . . . . . . 12 |- ( w = W -> ( +g ` w ) = ( +g ` W ) )
14		lflset.a	. . . . . . . . . . . 12 |- .+ = ( +g ` W )
15	13, 14	syl6eqr 2674	. . . . . . . . . . 11 |- ( w = W -> ( +g ` w ) = .+ )
16		fveq2 6191	. . . . . . . . . . . . 13 |- ( w = W -> ( .s ` w ) = ( .s ` W ) )
17		lflset.s	. . . . . . . . . . . . 13 |- .x. = ( .s ` W )
18	16, 17	syl6eqr 2674	. . . . . . . . . . . 12 |- ( w = W -> ( .s ` w ) = .x. )
19	18	oveqd 6667	. . . . . . . . . . 11 |- ( w = W -> ( r ( .s ` w ) x ) = ( r .x. x ) )
20		eqidd 2623	. . . . . . . . . . 11 |- ( w = W -> y = y )
21	15, 19, 20	oveq123d 6671	. . . . . . . . . 10 |- ( w = W -> ( ( r ( .s ` w ) x ) ( +g ` w ) y ) = ( ( r .x. x ) .+ y ) )
22	21	fveq2d 6195	. . . . . . . . 9 |- ( w = W -> ( f ` ( ( r ( .s ` w ) x ) ( +g ` w ) y ) ) = ( f ` ( ( r .x. x ) .+ y ) ) )
23	5	fveq2d 6195	. . . . . . . . . . 11 |- ( w = W -> ( +g ` (Scalar ` w ) ) = ( +g ` D ) )
24		lflset.p	. . . . . . . . . . 11 |- .+^ = ( +g ` D )
25	23, 24	syl6eqr 2674	. . . . . . . . . 10 |- ( w = W -> ( +g ` (Scalar ` w ) ) = .+^ )
26	5	fveq2d 6195	. . . . . . . . . . . 12 |- ( w = W -> ( .r ` (Scalar ` w ) ) = ( .r ` D ) )
27		lflset.t	. . . . . . . . . . . 12 |- .X. = ( .r ` D )
28	26, 27	syl6eqr 2674	. . . . . . . . . . 11 |- ( w = W -> ( .r ` (Scalar ` w ) ) = .X. )
29	28	oveqd 6667	. . . . . . . . . 10 |- ( w = W -> ( r ( .r ` (Scalar ` w ) ) ( f ` x ) ) = ( r .X. ( f ` x ) ) )
30		eqidd 2623	. . . . . . . . . 10 |- ( w = W -> ( f ` y ) = ( f ` y ) )
31	25, 29, 30	oveq123d 6671	. . . . . . . . 9 |- ( w = W -> ( ( r ( .r ` (Scalar ` w ) ) ( f ` x ) ) ( +g ` (Scalar ` w ) ) ( f ` y ) ) = ( ( r .X. ( f ` x ) ) .+^ ( f ` y ) ) )
32	22, 31	eqeq12d 2637	. . . . . . . 8 |- ( w = W -> ( ( f ` ( ( r ( .s ` w ) x ) ( +g ` w ) y ) ) = ( ( r ( .r ` (Scalar ` w ) ) ( f ` x ) ) ( +g ` (Scalar ` w ) ) ( f ` y ) ) <-> ( f ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( f ` x ) ) .+^ ( f ` y ) ) ) )
33	11, 32	raleqbidv 3152	. . . . . . 7 |- ( w = W -> ( A. y e. ( Base ` w ) ( f ` ( ( r ( .s ` w ) x ) ( +g ` w ) y ) ) = ( ( r ( .r ` (Scalar ` w ) ) ( f ` x ) ) ( +g ` (Scalar ` w ) ) ( f ` y ) ) <-> A. y e. V ( f ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( f ` x ) ) .+^ ( f ` y ) ) ) )
34	11, 33	raleqbidv 3152	. . . . . 6 |- ( w = W -> ( A. x e. ( Base ` w ) A. y e. ( Base ` w ) ( f ` ( ( r ( .s ` w ) x ) ( +g ` w ) y ) ) = ( ( r ( .r ` (Scalar ` w ) ) ( f ` x ) ) ( +g ` (Scalar ` w ) ) ( f ` y ) ) <-> A. x e. V A. y e. V ( f ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( f ` x ) ) .+^ ( f ` y ) ) ) )
35	8, 34	raleqbidv 3152	. . . . 5 |- ( w = W -> ( A. r e. ( Base ` (Scalar ` w ) ) A. x e. ( Base ` w ) A. y e. ( Base ` w ) ( f ` ( ( r ( .s ` w ) x ) ( +g ` w ) y ) ) = ( ( r ( .r ` (Scalar ` w ) ) ( f ` x ) ) ( +g ` (Scalar ` w ) ) ( f ` y ) ) <-> A. r e. K A. x e. V A. y e. V ( f ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( f ` x ) ) .+^ ( f ` y ) ) ) )
36	12, 35	rabeqbidv 3195	. . . 4 |- ( w = W -> { f e. ( ( Base ` (Scalar ` w ) ) ^m ( Base ` w ) ) | A. r e. ( Base ` (Scalar ` w ) ) A. x e. ( Base ` w ) A. y e. ( Base ` w ) ( f ` ( ( r ( .s ` w ) x ) ( +g ` w ) y ) ) = ( ( r ( .r ` (Scalar ` w ) ) ( f ` x ) ) ( +g ` (Scalar ` w ) ) ( f ` y ) ) } = { f e. ( K ^m V ) | A. r e. K A. x e. V A. y e. V ( f ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( f ` x ) ) .+^ ( f ` y ) ) } )
37		df-lfl 34345	. . . 4 |- LFnl = ( w e. _V |-> { f e. ( ( Base ` (Scalar ` w ) ) ^m ( Base ` w ) ) | A. r e. ( Base ` (Scalar ` w ) ) A. x e. ( Base ` w ) A. y e. ( Base ` w ) ( f ` ( ( r ( .s ` w ) x ) ( +g ` w ) y ) ) = ( ( r ( .r ` (Scalar ` w ) ) ( f ` x ) ) ( +g ` (Scalar ` w ) ) ( f ` y ) ) } )
38		ovex 6678	. . . . 5 |- ( K ^m V ) e. _V
39	38	rabex 4813	. . . 4 |- { f e. ( K ^m V ) | A. r e. K A. x e. V A. y e. V ( f ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( f ` x ) ) .+^ ( f ` y ) ) } e. _V
40	36, 37, 39	fvmpt 6282	. . 3 |- ( W e. _V -> (LFnl ` W ) = { f e. ( K ^m V ) | A. r e. K A. x e. V A. y e. V ( f ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( f ` x ) ) .+^ ( f ` y ) ) } )
41	2, 40	syl5eq 2668	. 2 |- ( W e. _V -> F = { f e. ( K ^m V ) | A. r e. K A. x e. V A. y e. V ( f ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( f ` x ) ) .+^ ( f ` y ) ) } )
42	1, 41	syl 17	1 |- ( W e. X -> F = { f e. ( K ^m V ) | A. r e. K A. x e. V A. y e. V ( f ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( f ` x ) ) .+^ ( f ` y ) ) } )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945  LFnlclfn 34344

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-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-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-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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-lfl 34345

This theorem is referenced by:  islfl  34347