Theorem lflvsdi2a 34367

Description: Reverse distributive law for (right vector space) scalar product of functionals. (Contributed by NM, 21-Oct-2014.)

Hypotheses

Ref	Expression
lfldi.v	|- V = ( Base ` W )
lfldi.r	|- R = (Scalar ` W )
lfldi.k	|- K = ( Base ` R )
lfldi.p	|- .+ = ( +g ` R )
lfldi.t	|- .x. = ( .r ` R )
lfldi.f	|- F = (LFnl ` W )
lfldi.w	|- ( ph -> W e. LMod )
lfldi.x	|- ( ph -> X e. K )
lfldi2.y	|- ( ph -> Y e. K )
lfldi2.g	|- ( ph -> G e. F )

Assertion

Ref	Expression
lflvsdi2a	|- ( ph -> ( G oF .x. ( V X. { ( X .+ Y ) } ) ) = ( ( G oF .x. ( V X. { X } ) ) oF .+ ( G oF .x. ( V X. { Y } ) ) ) )

Proof of Theorem lflvsdi2a

Step	Hyp	Ref	Expression
1		lfldi.v	. . . . . 6 |- V = ( Base ` W )
2		fvex 6201	. . . . . 6 |- ( Base ` W ) e. _V
3	1, 2	eqeltri 2697	. . . . 5 |- V e. _V
4	3	a1i 11	. . . 4 |- ( ph -> V e. _V )
5		lfldi.x	. . . 4 |- ( ph -> X e. K )
6		lfldi2.y	. . . 4 |- ( ph -> Y e. K )
7	4, 5, 6	ofc12 6922	. . 3 |- ( ph -> ( ( V X. { X } ) oF .+ ( V X. { Y } ) ) = ( V X. { ( X .+ Y ) } ) )
8	7	oveq2d 6666	. 2 |- ( ph -> ( G oF .x. ( ( V X. { X } ) oF .+ ( V X. { Y } ) ) ) = ( G oF .x. ( V X. { ( X .+ Y ) } ) ) )
9		lfldi.r	. . 3 |- R = (Scalar ` W )
10		lfldi.k	. . 3 |- K = ( Base ` R )
11		lfldi.p	. . 3 |- .+ = ( +g ` R )
12		lfldi.t	. . 3 |- .x. = ( .r ` R )
13		lfldi.f	. . 3 |- F = (LFnl ` W )
14		lfldi.w	. . 3 |- ( ph -> W e. LMod )
15		lfldi2.g	. . 3 |- ( ph -> G e. F )
16	1, 9, 10, 11, 12, 13, 14, 5, 6, 15	lflvsdi2 34366	. 2 |- ( ph -> ( G oF .x. ( ( V X. { X } ) oF .+ ( V X. { Y } ) ) ) = ( ( G oF .x. ( V X. { X } ) ) oF .+ ( G oF .x. ( V X. { Y } ) ) ) )
17	8, 16	eqtr3d 2658	1 |- ( ph -> ( G oF .x. ( V X. { ( X .+ Y ) } ) ) = ( ( G oF .x. ( V X. { X } ) ) oF .+ ( G oF .x. ( V X. { Y } ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   X. cxp 5112   ` cfv 5888  (class class class)co 6650   oFcof 6895   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   LModclmod 18863  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-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-of 6897  df-map 7859  df-ring 18549  df-lmod 18865  df-lfl 34345

This theorem is referenced by:  ldualvsdi2  34431