Theorem lclkrlem2u

Description: Lemma for lclkr . lclkrlem2t with X and Y swapped. (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 } ) )
	lclkrlem2u.n	|- ( ph -> ( ( E .+ G ) ` X ) =/= .0. )
Assertion	lclkrlem2u	|- ( 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		lclkrlem2u.n	|- ( ph -> ( ( E .+ G ) ` X ) =/= .0. )
25	17 19 21	dvhlmod	|- ( ph -> U e. LMod )
26	8 9 10 25 13 14	ldualvaddcom	|- ( ph -> ( E .+ G ) = ( G .+ E ) )
27	26	fveq1d	|- ( ph -> ( ( E .+ G ) ` X ) = ( ( G .+ E ) ` X ) )
28	27 24	eqnetrrd	|- ( ph -> ( ( G .+ E ) ` X ) =/= .0. )
29	1 2 3 4 5 6 7 8 9 10 12 11 14 13 15 16 17 18 19 20 21 23 22 28	lclkrlem2t	|- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( G .+ E ) ) ) ) = ( L ` ( G .+ E ) ) )
30	26	fveq2d	|- ( ph -> ( L ` ( E .+ G ) ) = ( L ` ( G .+ E ) ) )
31	30	fveq2d	|- ( ph -> ( ._|_ ` ( L ` ( E .+ G ) ) ) = ( ._|_ ` ( L ` ( G .+ E ) ) ) )
32	31	fveq2d	|- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( ._|_ ` ( ._|_ ` ( L ` ( G .+ E ) ) ) ) )
33	29 32 30	3eqtr4d	|- ( ph -> ( ._|_ ` ( ._|_ ` ( L ` ( E .+ G ) ) ) ) = ( L ` ( E .+ G ) ) )