Theorem cdlemg4b12

Description: TODO: FIX COMMENT. (Contributed by NM, 24-Apr-2013)

	Ref	Expression
Hypotheses	cdlemg4.l	|- .<_ = ( le ` K )
	cdlemg4.a	|- A = ( Atoms ` K )
	cdlemg4.h	|- H = ( LHyp ` K )
	cdlemg4.t	|- T = ( ( LTrn ` K ) ` W )
	cdlemg4.r	|- R = ( ( trL ` K ) ` W )
	cdlemg4.j	|- .\/ = ( join ` K )
	cdlemg4b.v	|- V = ( R ` G )
Assertion	cdlemg4b12	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( ( G ` P ) .\/ V ) = ( P .\/ V ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemg4.l	|- .<_ = ( le ` K )
2		cdlemg4.a	|- A = ( Atoms ` K )
3		cdlemg4.h	|- H = ( LHyp ` K )
4		cdlemg4.t	|- T = ( ( LTrn ` K ) ` W )
5		cdlemg4.r	|- R = ( ( trL ` K ) ` W )
6		cdlemg4.j	|- .\/ = ( join ` K )
7		cdlemg4b.v	|- V = ( R ` G )
8	1 2 3 4 5 6 7	cdlemg4b2	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( ( G ` P ) .\/ V ) = ( P .\/ ( G ` P ) ) )
9	1 2 3 4 5 6 7	cdlemg4b1	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( P .\/ V ) = ( P .\/ ( G ` P ) ) )
10	8 9	eqtr4d	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ G e. T ) -> ( ( G ` P ) .\/ V ) = ( P .\/ V ) )