Theorem cdlemg18

Description: Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-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	cdlemg18	|- ( ( ( ( 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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) .<_ W )

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		simp11	|- ( ( ( ( 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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( K e. HL /\ W e. H ) )
9		simp21r	|- ( ( ( ( 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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> G e. T )
10		simp12	|- ( ( ( ( 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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( P e. A /\ -. P .<_ W ) )
11	1 2 3 4 5 6 7	cdlemg18d	|- ( ( ( ( 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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) e. A )
12		simp23	|- ( ( ( ( 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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( G ` P ) =/= P )
13		simp1	|- ( ( ( ( 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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
14		simp21l	|- ( ( ( ( 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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> F e. T )
15		simp22	|- ( ( ( ( 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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> P =/= Q )
16		simp31	|- ( ( ( ( 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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( R ` G ) .<_ ( P .\/ Q ) )
17		simp33	|- ( ( ( ( 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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) )
18	1 2 3 4 5 6 7	cdlemg17	|- ( ( ( ( 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 ` ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) ) = ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) )
19	13 14 9 15 12 16 17 18	syl133anc	|- ( ( ( ( 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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( G ` ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) ) = ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) )
20	1 4 5 6	ltrnatlw	|- ( ( ( K e. HL /\ W e. H ) /\ ( G e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) e. A ) /\ ( ( G ` P ) =/= P /\ ( G ` ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) ) = ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) .<_ W )
21	8 9 10 11 12 19 20	syl132anc	|- ( ( ( ( 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 ) /\ ( ( F ` ( G ` P ) ) .\/ ( F ` ( G ` Q ) ) ) =/= ( P .\/ Q ) /\ -. E. r e. A ( -. r .<_ W /\ ( P .\/ r ) = ( Q .\/ r ) ) ) ) -> ( ( P .\/ ( F ` ( G ` P ) ) ) ./\ ( Q .\/ ( F ` ( G ` Q ) ) ) ) .<_ W )