Theorem lfl1sc

Description: The (right vector space) scalar product of a functional with one is the functional. (Contributed by NM, 21-Oct-2014)

	Ref	Expression
Hypotheses	lfl1sc.v	|- V = ( Base ` W )
	lfl1sc.d	|- D = ( Scalar ` W )
	lfl1sc.f	|- F = ( LFnl ` W )
	lfl1sc.k	|- K = ( Base ` D )
	lfl1sc.t	|- .x. = ( .r ` D )
	lfl1sc.i	|- .1. = ( 1r ` D )
	lfl1sc.w	|- ( ph -> W e. LMod )
	lfl1sc.g	|- ( ph -> G e. F )
Assertion	lfl1sc	|- ( ph -> ( G oF .x. ( V X. { .1. } ) ) = G )

Proof

Step	Hyp	Ref	Expression
1		lfl1sc.v	|- V = ( Base ` W )
2		lfl1sc.d	|- D = ( Scalar ` W )
3		lfl1sc.f	|- F = ( LFnl ` W )
4		lfl1sc.k	|- K = ( Base ` D )
5		lfl1sc.t	|- .x. = ( .r ` D )
6		lfl1sc.i	|- .1. = ( 1r ` D )
7		lfl1sc.w	|- ( ph -> W e. LMod )
8		lfl1sc.g	|- ( ph -> G e. F )
9	1	fvexi	|- V e. _V
10	9	a1i	|- ( ph -> V e. _V )
11	2 4 1 3	lflf	|- ( ( W e. LMod /\ G e. F ) -> G : V --> K )
12	7 8 11	syl2anc	|- ( ph -> G : V --> K )
13	6	fvexi	|- .1. e. _V
14	13	a1i	|- ( ph -> .1. e. _V )
15	2	lmodring	|- ( W e. LMod -> D e. Ring )
16	7 15	syl	|- ( ph -> D e. Ring )
17	4 5 6	ringridm	|- ( ( D e. Ring /\ k e. K ) -> ( k .x. .1. ) = k )
18	16 17	sylan	|- ( ( ph /\ k e. K ) -> ( k .x. .1. ) = k )
19	10 12 14 18	caofid0r	|- ( ph -> ( G oF .x. ( V X. { .1. } ) ) = G )