Theorem lcfrlem15

Description: Lemma for lcfr . (Contributed by NM, 9-Mar-2015)

	Ref	Expression
Hypotheses	lcf1o.h	|- H = ( LHyp ` K )
	lcf1o.o	|- ._|_ = ( ( ocH ` K ) ` W )
	lcf1o.u	|- U = ( ( DVecH ` K ) ` W )
	lcf1o.v	|- V = ( Base ` U )
	lcf1o.a	|- .+ = ( +g ` U )
	lcf1o.t	|- .x. = ( .s ` U )
	lcf1o.s	|- S = ( Scalar ` U )
	lcf1o.r	|- R = ( Base ` S )
	lcf1o.z	|- .0. = ( 0g ` U )
	lcf1o.f	|- F = ( LFnl ` U )
	lcf1o.l	|- L = ( LKer ` U )
	lcf1o.d	|- D = ( LDual ` U )
	lcf1o.q	|- Q = ( 0g ` D )
	lcf1o.c	|- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
	lcf1o.j	|- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )
	lcflo.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	lcfrlem10.x	|- ( ph -> X e. ( V \ { .0. } ) )
Assertion	lcfrlem15	|- ( ph -> X e. ( ._|_ ` ( L ` ( J ` X ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		lcf1o.h	|- H = ( LHyp ` K )
2		lcf1o.o	|- ._|_ = ( ( ocH ` K ) ` W )
3		lcf1o.u	|- U = ( ( DVecH ` K ) ` W )
4		lcf1o.v	|- V = ( Base ` U )
5		lcf1o.a	|- .+ = ( +g ` U )
6		lcf1o.t	|- .x. = ( .s ` U )
7		lcf1o.s	|- S = ( Scalar ` U )
8		lcf1o.r	|- R = ( Base ` S )
9		lcf1o.z	|- .0. = ( 0g ` U )
10		lcf1o.f	|- F = ( LFnl ` U )
11		lcf1o.l	|- L = ( LKer ` U )
12		lcf1o.d	|- D = ( LDual ` U )
13		lcf1o.q	|- Q = ( 0g ` D )
14		lcf1o.c	|- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
15		lcf1o.j	|- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )
16		lcflo.k	|- ( ph -> ( K e. HL /\ W e. H ) )
17		lcfrlem10.x	|- ( ph -> X e. ( V \ { .0. } ) )
18	1 3 16	dvhlmod	|- ( ph -> U e. LMod )
19	17	eldifad	|- ( ph -> X e. V )
20		eqid	|- ( LSpan ` U ) = ( LSpan ` U )
21	4 20	lspsnid	|- ( ( U e. LMod /\ X e. V ) -> X e. ( ( LSpan ` U ) ` { X } ) )
22	18 19 21	syl2anc	|- ( ph -> X e. ( ( LSpan ` U ) ` { X } ) )
23	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 20	lcfrlem14	|- ( ph -> ( ._|_ ` ( L ` ( J ` X ) ) ) = ( ( LSpan ` U ) ` { X } ) )
24	22 23	eleqtrrd	|- ( ph -> X e. ( ._|_ ` ( L ` ( J ` X ) ) ) )