Theorem cdleme17d1

Description: Part of proof of Lemma E in Crawley p. 114, first part of 4th paragraph. F , G represent f(s), f_s(p) respectively. We show, in their notation, f_s(p)=q. (Contributed by NM, 11-Oct-2012)

	Ref	Expression
Hypotheses	cdleme17.l	|- .<_ = ( le ` K )
	cdleme17.j	|- .\/ = ( join ` K )
	cdleme17.m	|- ./\ = ( meet ` K )
	cdleme17.a	|- A = ( Atoms ` K )
	cdleme17.h	|- H = ( LHyp ` K )
	cdleme17.u	|- U = ( ( P .\/ Q ) ./\ W )
	cdleme17.f	|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
	cdleme17.g	|- G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( P .\/ S ) ./\ W ) ) )
Assertion	cdleme17d1	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> G = Q )

Proof

Step	Hyp	Ref	Expression
1		cdleme17.l	|- .<_ = ( le ` K )
2		cdleme17.j	|- .\/ = ( join ` K )
3		cdleme17.m	|- ./\ = ( meet ` K )
4		cdleme17.a	|- A = ( Atoms ` K )
5		cdleme17.h	|- H = ( LHyp ` K )
6		cdleme17.u	|- U = ( ( P .\/ Q ) ./\ W )
7		cdleme17.f	|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8		cdleme17.g	|- G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( P .\/ S ) ./\ W ) ) )
9		eqid	|- ( ( P .\/ S ) ./\ W ) = ( ( P .\/ S ) ./\ W )
10	1 2 3 4 5 6 7 8 9	cdleme17a	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> G = ( ( P .\/ Q ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) )
11		simp1l	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> K e. HL )
12		simp1r	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> W e. H )
13		simp21l	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> P e. A )
14		simp21r	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> -. P .<_ W )
15		simp22	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> Q e. A )
16		simp23l	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> S e. A )
17		simp3	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> -. S .<_ ( P .\/ Q ) )
18	1 2 3 4 5 6 7 8 9	cdleme17c	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ S e. A /\ -. S .<_ ( P .\/ Q ) ) ) -> ( ( P .\/ Q ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = Q )
19	11 12 13 14 15 16 17 18	syl223anc	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( P .\/ Q ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = Q )
20	10 19	eqtrd	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> G = Q )