Theorem cdlemg17j

Description: TODO: fix comment. (Contributed by NM, 11-May-2013)

	Ref	Expression
Hypotheses	cdlemg12.l	|- .<_ = ( le ` K )
	cdlemg12.j	|- .\/ = ( join ` K )
	cdlemg12.m	|- ./\ = ( meet ` K )
	cdlemg12.a	|- A = ( Atoms ` K )
	cdlemg12.h	|- H = ( LHyp ` K )
	cdlemg12.t	|- T = ( ( LTrn ` K ) ` W )
	cdlemg12b.r	|- R = ( ( trL ` K ) ` W )
Assertion	cdlemg17j	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( G ` ( F ` P ) ) = ( F ` ( G ` P ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemg12.l	|- .<_ = ( le ` K )
2		cdlemg12.j	|- .\/ = ( join ` K )
3		cdlemg12.m	|- ./\ = ( meet ` K )
4		cdlemg12.a	|- A = ( Atoms ` K )
5		cdlemg12.h	|- H = ( LHyp ` K )
6		cdlemg12.t	|- T = ( ( LTrn ` K ) ` W )
7		cdlemg12b.r	|- R = ( ( trL ` K ) ` W )
8	1 2 3 4 5 6 7	cdlemg17i	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( G ` ( F ` P ) ) = ( F ` Q ) )
9	1 2 3 4 5 6 7	cdlemg17ir	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( F ` ( G ` P ) ) = ( F ` Q ) )
10	8 9	eqtr4d	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( F e. T /\ G e. T /\ P =/= Q ) /\ ( ( G ` P ) =/= P /\ ( R ` G ) .<_ ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( G ` ( F ` P ) ) = ( F ` ( G ` P ) ) )