Theorem hvmaplfl

Description: The vector to functional map value is a functional. (Contributed by NM, 28-Mar-2015)

	Ref	Expression
Hypotheses	hvmaplfl.h	|- H = ( LHyp ` K )
	hvmaplfl.u	|- U = ( ( DVecH ` K ) ` W )
	hvmaplfl.v	|- V = ( Base ` U )
	hvmaplfl.z	|- .0. = ( 0g ` U )
	hvmaplfl.f	|- F = ( LFnl ` U )
	hvmaplfl.m	|- M = ( ( HVMap ` K ) ` W )
	hvmaplfl.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	hvmaplfl.x	|- ( ph -> X e. ( V \ { .0. } ) )
Assertion	hvmaplfl	|- ( ph -> ( M ` X ) e. F )

Proof

Step	Hyp	Ref	Expression
1		hvmaplfl.h	|- H = ( LHyp ` K )
2		hvmaplfl.u	|- U = ( ( DVecH ` K ) ` W )
3		hvmaplfl.v	|- V = ( Base ` U )
4		hvmaplfl.z	|- .0. = ( 0g ` U )
5		hvmaplfl.f	|- F = ( LFnl ` U )
6		hvmaplfl.m	|- M = ( ( HVMap ` K ) ` W )
7		hvmaplfl.k	|- ( ph -> ( K e. HL /\ W e. H ) )
8		hvmaplfl.x	|- ( ph -> X e. ( V \ { .0. } ) )
9		eqid	|- ( ( LCDual ` K ) ` W ) = ( ( LCDual ` K ) ` W )
10		eqid	|- ( Base ` ( ( LCDual ` K ) ` W ) ) = ( Base ` ( ( LCDual ` K ) ` W ) )
11		eqid	|- ( 0g ` ( ( LCDual ` K ) ` W ) ) = ( 0g ` ( ( LCDual ` K ) ` W ) )
12	1 2 3 4 9 10 11 6 7 8	hvmapcl2	|- ( ph -> ( M ` X ) e. ( ( Base ` ( ( LCDual ` K ) ` W ) ) \ { ( 0g ` ( ( LCDual ` K ) ` W ) ) } ) )
13	12	eldifad	|- ( ph -> ( M ` X ) e. ( Base ` ( ( LCDual ` K ) ` W ) ) )
14	1 9 10 2 5 7 13	lcdvbaselfl	|- ( ph -> ( M ` X ) e. F )