Theorem lcfrlem22

Description: Lemma for lcfr . (Contributed by NM, 24-Feb-2015)

	Ref	Expression
Hypotheses	lcfrlem17.h	|- H = ( LHyp ` K )
	lcfrlem17.o	|- ._|_ = ( ( ocH ` K ) ` W )
	lcfrlem17.u	|- U = ( ( DVecH ` K ) ` W )
	lcfrlem17.v	|- V = ( Base ` U )
	lcfrlem17.p	|- .+ = ( +g ` U )
	lcfrlem17.z	|- .0. = ( 0g ` U )
	lcfrlem17.n	|- N = ( LSpan ` U )
	lcfrlem17.a	|- A = ( LSAtoms ` U )
	lcfrlem17.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	lcfrlem17.x	|- ( ph -> X e. ( V \ { .0. } ) )
	lcfrlem17.y	|- ( ph -> Y e. ( V \ { .0. } ) )
	lcfrlem17.ne	|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
	lcfrlem22.b	|- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )
Assertion	lcfrlem22	|- ( ph -> B e. A )

Proof

Step	Hyp	Ref	Expression
1		lcfrlem17.h	|- H = ( LHyp ` K )
2		lcfrlem17.o	|- ._|_ = ( ( ocH ` K ) ` W )
3		lcfrlem17.u	|- U = ( ( DVecH ` K ) ` W )
4		lcfrlem17.v	|- V = ( Base ` U )
5		lcfrlem17.p	|- .+ = ( +g ` U )
6		lcfrlem17.z	|- .0. = ( 0g ` U )
7		lcfrlem17.n	|- N = ( LSpan ` U )
8		lcfrlem17.a	|- A = ( LSAtoms ` U )
9		lcfrlem17.k	|- ( ph -> ( K e. HL /\ W e. H ) )
10		lcfrlem17.x	|- ( ph -> X e. ( V \ { .0. } ) )
11		lcfrlem17.y	|- ( ph -> Y e. ( V \ { .0. } ) )
12		lcfrlem17.ne	|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
13		lcfrlem22.b	|- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )
14	1 2 3 4 5 6 7 8 9 10 11 12	lcfrlem21	|- ( ph -> ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) ) e. A )
15	13 14	eqeltrid	|- ( ph -> B e. A )