cdleme21g

Description: Part of proof of Lemma E in Crawley p. 115. (Contributed by NM, 29-Nov-2012)

	Ref	Expression
Hypotheses	cdleme21.l	\|- .<_ = ( le ` K )
	cdleme21.j	\|- .\/ = ( join ` K )
	cdleme21.m	\|- ./\ = ( meet ` K )
	cdleme21.a	\|- A = ( Atoms ` K )
	cdleme21.h	\|- H = ( LHyp ` K )
	cdleme21.u	\|- U = ( ( P .\/ Q ) ./\ W )
	cdleme21.f	\|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
	cdleme21g.g	\|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
	cdleme21g.d	\|- D = ( ( R .\/ S ) ./\ W )
	cdleme21g.y	\|- Y = ( ( R .\/ T ) ./\ W )
	cdleme21g.n	\|- N = ( ( P .\/ Q ) ./\ ( F .\/ D ) )
	cdleme21g.o	\|- O = ( ( P .\/ Q ) ./\ ( G .\/ Y ) )
Assertion	cdleme21g	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> N = O )

Step	Hyp	Ref	Expression
1		cdleme21.l	\|- .<_ = ( le ` K )
2		cdleme21.j	\|- .\/ = ( join ` K )
3		cdleme21.m	\|- ./\ = ( meet ` K )
4		cdleme21.a	\|- A = ( Atoms ` K )
5		cdleme21.h	\|- H = ( LHyp ` K )
6		cdleme21.u	\|- U = ( ( P .\/ Q ) ./\ W )
7		cdleme21.f	\|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8		cdleme21g.g	\|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9		cdleme21g.d	\|- D = ( ( R .\/ S ) ./\ W )
10		cdleme21g.y	\|- Y = ( ( R .\/ T ) ./\ W )
11		cdleme21g.n	\|- N = ( ( P .\/ Q ) ./\ ( F .\/ D ) )
12		cdleme21g.o	\|- O = ( ( P .\/ Q ) ./\ ( G .\/ Y ) )
13		eqid	\|- ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) ) = ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) )
14		eqid	\|- ( ( R .\/ z ) ./\ W ) = ( ( R .\/ z ) ./\ W )
15		eqid	\|- ( ( P .\/ Q ) ./\ ( ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) ) .\/ ( ( R .\/ z ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( ( ( z .\/ U ) ./\ ( Q .\/ ( ( P .\/ z ) ./\ W ) ) ) .\/ ( ( R .\/ z ) ./\ W ) ) )
16	1 2 3 4 5 6 7 13 9 14 11 15 8 10 12	cdleme21f	\|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ ( T e. A /\ -. T .<_ W ) /\ ( P =/= Q /\ -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) ) /\ ( ( R e. A /\ -. R .<_ W ) /\ ( R .<_ ( P .\/ Q ) /\ U .<_ ( S .\/ T ) ) /\ ( ( z e. A /\ -. z .<_ W ) /\ ( P .\/ z ) = ( S .\/ z ) ) ) ) -> N = O )

Metamath Proof Explorer