Theorem lclkrlem2q

Description: Lemma for lclkr . The sum has a closed kernel when B is nonzero. (Contributed by NM, 18-Jan-2015)

	Ref	Expression
Hypotheses	lclkrlem2m.v	|- V = ( Base ` U )
	lclkrlem2m.t	|- .x. = ( .s ` U )
	lclkrlem2m.s	|- S = ( Scalar ` U )
	lclkrlem2m.q	|- .X. = ( .r ` S )
	lclkrlem2m.z	|- .0. = ( 0g ` S )
	lclkrlem2m.i	|- I = ( invr ` S )
	lclkrlem2m.m	|- .- = ( -g ` U )
	lclkrlem2m.f	|- F = ( LFnl ` U )
	lclkrlem2m.d	|- D = ( LDual ` U )
	lclkrlem2m.p	|- .+ = ( +g ` D )
	lclkrlem2m.x	|- ( ph -> X e. V )
	lclkrlem2m.y	|- ( ph -> Y e. V )
	lclkrlem2m.e	|- ( ph -> E e. F )
	lclkrlem2m.g	|- ( ph -> G e. F )
	lclkrlem2n.n	|- N = ( LSpan ` U )
	lclkrlem2n.l	|- L = ( LKer ` U )
	lclkrlem2o.h	|- H = ( LHyp ` K )
	lclkrlem2o.o	|- ._|_ = ( ( ocH ` K ) ` W )
	lclkrlem2o.u	|- U = ( ( DVecH ` K ) ` W )
	lclkrlem2o.a	|- .(+) = ( LSSum ` U )
	lclkrlem2o.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	lclkrlem2q.le	|- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )
	lclkrlem2q.lg	|- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )
	lclkrlem2q.b	|- B = ( X .- ( ( ( ( E .+ G ) ` X ) .X. ( I ` ( ( E .+ G ) ` Y ) ) ) .x. Y ) )
	lclkrlem2q.n	|- ( ph -> ( ( E .+ G ) ` Y ) =/= .0. )
	lclkrlem2q.bn	|- ( ph -> B =/= ( 0g ` U ) )
Assertion	lclkrlem2q	|- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )

Proof

Step	Hyp	Ref	Expression
1		lclkrlem2m.v	|- V = ( Base ` U )
2		lclkrlem2m.t	|- .x. = ( .s ` U )
3		lclkrlem2m.s	|- S = ( Scalar ` U )
4		lclkrlem2m.q	|- .X. = ( .r ` S )
5		lclkrlem2m.z	|- .0. = ( 0g ` S )
6		lclkrlem2m.i	|- I = ( invr ` S )
7		lclkrlem2m.m	|- .- = ( -g ` U )
8		lclkrlem2m.f	|- F = ( LFnl ` U )
9		lclkrlem2m.d	|- D = ( LDual ` U )
10		lclkrlem2m.p	|- .+ = ( +g ` D )
11		lclkrlem2m.x	|- ( ph -> X e. V )
12		lclkrlem2m.y	|- ( ph -> Y e. V )
13		lclkrlem2m.e	|- ( ph -> E e. F )
14		lclkrlem2m.g	|- ( ph -> G e. F )
15		lclkrlem2n.n	|- N = ( LSpan ` U )
16		lclkrlem2n.l	|- L = ( LKer ` U )
17		lclkrlem2o.h	|- H = ( LHyp ` K )
18		lclkrlem2o.o	|- ._|_ = ( ( ocH ` K ) ` W )
19		lclkrlem2o.u	|- U = ( ( DVecH ` K ) ` W )
20		lclkrlem2o.a	|- .(+) = ( LSSum ` U )
21		lclkrlem2o.k	|- ( ph -> ( K e. HL /\ W e. H ) )
22		lclkrlem2q.le	|- ( ph -> ( L ` E ) = ( ._|_ ` { X } ) )
23		lclkrlem2q.lg	|- ( ph -> ( L ` G ) = ( ._|_ ` { Y } ) )
24		lclkrlem2q.b	|- B = ( X .- ( ( ( ( E .+ G ) ` X ) .X. ( I ` ( ( E .+ G ) ` Y ) ) ) .x. Y ) )
25		lclkrlem2q.n	|- ( ph -> ( ( E .+ G ) ` Y ) =/= .0. )
26		lclkrlem2q.bn	|- ( ph -> B =/= ( 0g ` U ) )
27		eqid	|- ( 0g ` U ) = ( 0g ` U )
28		eqid	|- ( LSHyp ` U ) = ( LSHyp ` U )
29	17 19 21	dvhlvec	|- ( ph -> U e. LVec )
30	1 2 3 4 5 6 7 8 9 10 11 12 13 14 29 24 25	lclkrlem2m	|- ( ph -> ( B e. V /\ ( ( E .+ G ) ` B ) = .0. ) )
31	30	simpld	|- ( ph -> B e. V )
32		eldifsn	|- ( B e. ( V \ { ( 0g ` U ) } ) <-> ( B e. V /\ B =/= ( 0g ` U ) ) )
33	31 26 32	sylanbrc	|- ( ph -> B e. ( V \ { ( 0g ` U ) } ) )
34	30	simprd	|- ( ph -> ( ( E .+ G ) ` B ) = .0. )
35	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 24 25 26	lclkrlem2o	|- ( ph -> ( -. X e. ( ._|_ ` { B } ) \/ -. Y e. ( ._|_ ` { B } ) ) )
36	17 18 19 1 3 5 27 20 15 8 28 16 9 10 21 33 13 14 22 23 34 35 11 12	lclkrlem2l	|- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )