Theorem islfld 34349

Description: Properties that determine a linear functional. TODO: use this in place of islfl 34347 when it shortens the proof. (Contributed by NM, 19-Oct-2014.)

Hypotheses

Ref	Expression
islfld.v	|- ( ph -> V = ( Base ` W ) )
islfld.a	|- ( ph -> .+ = ( +g ` W ) )
islfld.d	|- ( ph -> D = (Scalar ` W ) )
islfld.s	|- ( ph -> .x. = ( .s ` W ) )
islfld.k	|- ( ph -> K = ( Base ` D ) )
islfld.p	|- ( ph -> .+^ = ( +g ` D ) )
islfld.t	|- ( ph -> .X. = ( .r ` D ) )
islfld.f	|- ( ph -> F = (LFnl ` W ) )
islfld.u	|- ( ph -> G : V --> K )
islfld.l	|- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) )
islfld.w	|- ( ph -> W e. X )

Assertion

Ref	Expression
islfld	|- ( ph -> G e. F )

Distinct variable groups:   x, r, y, G   K, r, x, y   x, V, y   W, r, x, y   ph, r, x, y

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

Proof of Theorem islfld

Step	Hyp	Ref	Expression
1		islfld.w	. . 3 |- ( ph -> W e. X )
2		islfld.u	. . . 4 |- ( ph -> G : V --> K )
3		islfld.v	. . . . 5 |- ( ph -> V = ( Base ` W ) )
4		islfld.k	. . . . . 6 |- ( ph -> K = ( Base ` D ) )
5		islfld.d	. . . . . . 7 |- ( ph -> D = (Scalar ` W ) )
6	5	fveq2d 6195	. . . . . 6 |- ( ph -> ( Base ` D ) = ( Base ` (Scalar ` W ) ) )
7	4, 6	eqtrd 2656	. . . . 5 |- ( ph -> K = ( Base ` (Scalar ` W ) ) )
8	3, 7	feq23d 6040	. . . 4 |- ( ph -> ( G : V --> K <-> G : ( Base ` W ) --> ( Base ` (Scalar ` W ) ) ) )
9	2, 8	mpbid 222	. . 3 |- ( ph -> G : ( Base ` W ) --> ( Base ` (Scalar ` W ) ) )
10		islfld.l	. . . . 5 |- ( ( ph /\ ( r e. K /\ x e. V /\ y e. V ) ) -> ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) )
11	10	ralrimivvva 2972	. . . 4 |- ( ph -> A. r e. K A. x e. V A. y e. V ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) )
12		islfld.a	. . . . . . . . . 10 |- ( ph -> .+ = ( +g ` W ) )
13		islfld.s	. . . . . . . . . . 11 |- ( ph -> .x. = ( .s ` W ) )
14	13	oveqd 6667	. . . . . . . . . 10 |- ( ph -> ( r .x. x ) = ( r ( .s ` W ) x ) )
15		eqidd 2623	. . . . . . . . . 10 |- ( ph -> y = y )
16	12, 14, 15	oveq123d 6671	. . . . . . . . 9 |- ( ph -> ( ( r .x. x ) .+ y ) = ( ( r ( .s ` W ) x ) ( +g ` W ) y ) )
17	16	fveq2d 6195	. . . . . . . 8 |- ( ph -> ( G ` ( ( r .x. x ) .+ y ) ) = ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) )
18		islfld.p	. . . . . . . . . 10 |- ( ph -> .+^ = ( +g ` D ) )
19	5	fveq2d 6195	. . . . . . . . . 10 |- ( ph -> ( +g ` D ) = ( +g ` (Scalar ` W ) ) )
20	18, 19	eqtrd 2656	. . . . . . . . 9 |- ( ph -> .+^ = ( +g ` (Scalar ` W ) ) )
21		islfld.t	. . . . . . . . . . 11 |- ( ph -> .X. = ( .r ` D ) )
22	5	fveq2d 6195	. . . . . . . . . . 11 |- ( ph -> ( .r ` D ) = ( .r ` (Scalar ` W ) ) )
23	21, 22	eqtrd 2656	. . . . . . . . . 10 |- ( ph -> .X. = ( .r ` (Scalar ` W ) ) )
24	23	oveqd 6667	. . . . . . . . 9 |- ( ph -> ( r .X. ( G ` x ) ) = ( r ( .r ` (Scalar ` W ) ) ( G ` x ) ) )
25		eqidd 2623	. . . . . . . . 9 |- ( ph -> ( G ` y ) = ( G ` y ) )
26	20, 24, 25	oveq123d 6671	. . . . . . . 8 |- ( ph -> ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) = ( ( r ( .r ` (Scalar ` W ) ) ( G ` x ) ) ( +g ` (Scalar ` W ) ) ( G ` y ) ) )
27	17, 26	eqeq12d 2637	. . . . . . 7 |- ( ph -> ( ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) <-> ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` (Scalar ` W ) ) ( G ` x ) ) ( +g ` (Scalar ` W ) ) ( G ` y ) ) ) )
28	3, 27	raleqbidv 3152	. . . . . 6 |- ( ph -> ( A. y e. V ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) <-> A. y e. ( Base ` W ) ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` (Scalar ` W ) ) ( G ` x ) ) ( +g ` (Scalar ` W ) ) ( G ` y ) ) ) )
29	3, 28	raleqbidv 3152	. . . . 5 |- ( ph -> ( A. x e. V A. y e. V ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) <-> A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` (Scalar ` W ) ) ( G ` x ) ) ( +g ` (Scalar ` W ) ) ( G ` y ) ) ) )
30	7, 29	raleqbidv 3152	. . . 4 |- ( ph -> ( A. r e. K A. x e. V A. y e. V ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) <-> A. r e. ( Base ` (Scalar ` W ) ) A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` (Scalar ` W ) ) ( G ` x ) ) ( +g ` (Scalar ` W ) ) ( G ` y ) ) ) )
31	11, 30	mpbid 222	. . 3 |- ( ph -> A. r e. ( Base ` (Scalar ` W ) ) A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` (Scalar ` W ) ) ( G ` x ) ) ( +g ` (Scalar ` W ) ) ( G ` y ) ) )
32		eqid 2622	. . . . 5 |- ( Base ` W ) = ( Base ` W )
33		eqid 2622	. . . . 5 |- ( +g ` W ) = ( +g ` W )
34		eqid 2622	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
35		eqid 2622	. . . . 5 |- ( .s ` W ) = ( .s ` W )
36		eqid 2622	. . . . 5 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
37		eqid 2622	. . . . 5 |- ( +g ` (Scalar ` W ) ) = ( +g ` (Scalar ` W ) )
38		eqid 2622	. . . . 5 |- ( .r ` (Scalar ` W ) ) = ( .r ` (Scalar ` W ) )
39		eqid 2622	. . . . 5 |- (LFnl ` W ) = (LFnl ` W )
40	32, 33, 34, 35, 36, 37, 38, 39	islfl 34347	. . . 4 |- ( W e. X -> ( G e. (LFnl ` W ) <-> ( G : ( Base ` W ) --> ( Base ` (Scalar ` W ) ) /\ A. r e. ( Base ` (Scalar ` W ) ) A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` (Scalar ` W ) ) ( G ` x ) ) ( +g ` (Scalar ` W ) ) ( G ` y ) ) ) ) )
41	40	biimpar 502	. . 3 |- ( ( W e. X /\ ( G : ( Base ` W ) --> ( Base ` (Scalar ` W ) ) /\ A. r e. ( Base ` (Scalar ` W ) ) A. x e. ( Base ` W ) A. y e. ( Base ` W ) ( G ` ( ( r ( .s ` W ) x ) ( +g ` W ) y ) ) = ( ( r ( .r ` (Scalar ` W ) ) ( G ` x ) ) ( +g ` (Scalar ` W ) ) ( G ` y ) ) ) ) -> G e. (LFnl ` W ) )
42	1, 9, 31, 41	syl12anc 1324	. 2 |- ( ph -> G e. (LFnl ` W ) )
43		islfld.f	. 2 |- ( ph -> F = (LFnl ` W ) )
44	42, 43	eleqtrrd 2704	1 |- ( ph -> G e. F )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   -->wf 5884   ` cfv 5888  (class class class)co 6650   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-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

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-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-lfl 34345

This theorem is referenced by:  lflvscl  34364