Theorem lfli 34348

Description: Property of a linear functional. (lnfnli 28899 analog.) (Contributed by NM, 16-Apr-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
lfli	|- ( ( W e. Z /\ G e. F /\ ( R e. K /\ X e. V /\ Y e. V ) ) -> ( G ` ( ( R .x. X ) .+ Y ) ) = ( ( R .X. ( G ` X ) ) .+^ ( G ` Y ) ) )

Proof of Theorem lfli

Dummy variables r x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lflset.v	. . . . 5 |- V = ( Base ` W )
2		lflset.a	. . . . 5 |- .+ = ( +g ` W )
3		lflset.d	. . . . 5 |- D = (Scalar ` W )
4		lflset.s	. . . . 5 |- .x. = ( .s ` W )
5		lflset.k	. . . . 5 |- K = ( Base ` D )
6		lflset.p	. . . . 5 |- .+^ = ( +g ` D )
7		lflset.t	. . . . 5 |- .X. = ( .r ` D )
8		lflset.f	. . . . 5 |- F = (LFnl ` W )
9	1, 2, 3, 4, 5, 6, 7, 8	islfl 34347	. . . 4 |- ( W e. Z -> ( G e. F <-> ( G : V --> K /\ A. r e. K A. x e. V A. y e. V ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) ) ) )
10	9	simplbda 654	. . 3 |- ( ( W e. Z /\ G e. F ) -> A. r e. K A. x e. V A. y e. V ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) )
11	10	3adant3 1081	. 2 |- ( ( W e. Z /\ G e. F /\ ( R e. K /\ X e. V /\ Y e. V ) ) -> A. r e. K A. x e. V A. y e. V ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) )
12		oveq1 6657	. . . . . . 7 |- ( r = R -> ( r .x. x ) = ( R .x. x ) )
13	12	oveq1d 6665	. . . . . 6 |- ( r = R -> ( ( r .x. x ) .+ y ) = ( ( R .x. x ) .+ y ) )
14	13	fveq2d 6195	. . . . 5 |- ( r = R -> ( G ` ( ( r .x. x ) .+ y ) ) = ( G ` ( ( R .x. x ) .+ y ) ) )
15		oveq1 6657	. . . . . 6 |- ( r = R -> ( r .X. ( G ` x ) ) = ( R .X. ( G ` x ) ) )
16	15	oveq1d 6665	. . . . 5 |- ( r = R -> ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) = ( ( R .X. ( G ` x ) ) .+^ ( G ` y ) ) )
17	14, 16	eqeq12d 2637	. . . 4 |- ( r = R -> ( ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) <-> ( G ` ( ( R .x. x ) .+ y ) ) = ( ( R .X. ( G ` x ) ) .+^ ( G ` y ) ) ) )
18		oveq2 6658	. . . . . . 7 |- ( x = X -> ( R .x. x ) = ( R .x. X ) )
19	18	oveq1d 6665	. . . . . 6 |- ( x = X -> ( ( R .x. x ) .+ y ) = ( ( R .x. X ) .+ y ) )
20	19	fveq2d 6195	. . . . 5 |- ( x = X -> ( G ` ( ( R .x. x ) .+ y ) ) = ( G ` ( ( R .x. X ) .+ y ) ) )
21		fveq2 6191	. . . . . . 7 |- ( x = X -> ( G ` x ) = ( G ` X ) )
22	21	oveq2d 6666	. . . . . 6 |- ( x = X -> ( R .X. ( G ` x ) ) = ( R .X. ( G ` X ) ) )
23	22	oveq1d 6665	. . . . 5 |- ( x = X -> ( ( R .X. ( G ` x ) ) .+^ ( G ` y ) ) = ( ( R .X. ( G ` X ) ) .+^ ( G ` y ) ) )
24	20, 23	eqeq12d 2637	. . . 4 |- ( x = X -> ( ( G ` ( ( R .x. x ) .+ y ) ) = ( ( R .X. ( G ` x ) ) .+^ ( G ` y ) ) <-> ( G ` ( ( R .x. X ) .+ y ) ) = ( ( R .X. ( G ` X ) ) .+^ ( G ` y ) ) ) )
25		oveq2 6658	. . . . . 6 |- ( y = Y -> ( ( R .x. X ) .+ y ) = ( ( R .x. X ) .+ Y ) )
26	25	fveq2d 6195	. . . . 5 |- ( y = Y -> ( G ` ( ( R .x. X ) .+ y ) ) = ( G ` ( ( R .x. X ) .+ Y ) ) )
27		fveq2 6191	. . . . . 6 |- ( y = Y -> ( G ` y ) = ( G ` Y ) )
28	27	oveq2d 6666	. . . . 5 |- ( y = Y -> ( ( R .X. ( G ` X ) ) .+^ ( G ` y ) ) = ( ( R .X. ( G ` X ) ) .+^ ( G ` Y ) ) )
29	26, 28	eqeq12d 2637	. . . 4 |- ( y = Y -> ( ( G ` ( ( R .x. X ) .+ y ) ) = ( ( R .X. ( G ` X ) ) .+^ ( G ` y ) ) <-> ( G ` ( ( R .x. X ) .+ Y ) ) = ( ( R .X. ( G ` X ) ) .+^ ( G ` Y ) ) ) )
30	17, 24, 29	rspc3v 3325	. . 3 |- ( ( R e. K /\ X e. V /\ Y e. V ) -> ( A. r e. K A. x e. V A. y e. V ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) -> ( G ` ( ( R .x. X ) .+ Y ) ) = ( ( R .X. ( G ` X ) ) .+^ ( G ` Y ) ) ) )
31	30	3ad2ant3 1084	. 2 |- ( ( W e. Z /\ G e. F /\ ( R e. K /\ X e. V /\ Y e. V ) ) -> ( A. r e. K A. x e. V A. y e. V ( G ` ( ( r .x. x ) .+ y ) ) = ( ( r .X. ( G ` x ) ) .+^ ( G ` y ) ) -> ( G ` ( ( R .x. X ) .+ Y ) ) = ( ( R .X. ( G ` X ) ) .+^ ( G ` Y ) ) ) )
32	11, 31	mpd 15	1 |- ( ( W e. Z /\ G e. F /\ ( R e. K /\ X e. V /\ Y e. V ) ) -> ( G ` ( ( R .x. X ) .+ Y ) ) = ( ( R .X. ( G ` X ) ) .+^ ( G ` Y ) ) )

Syntax hints:   -> wi 4   /\ 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:  lfl0  34352  lfladd  34353  lflsub  34354  lflmul  34355  lflnegcl  34362  lflvscl  34364  lkrlss  34382  hdmapln1  37198