Theorem cdleme17a

Description: Part of proof of Lemma E in Crawley p. 114, first part of 4th paragraph. F , G , and C represent f(s), f_s(p), and s_1 respectively. We show, in their notation, f_s(p)=(p \/ q) /\ (q \/ s_1). (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 ) ) )
	cdleme17.c	|- C = ( ( P .\/ S ) ./\ W )
Assertion	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 .\/ C ) ) )

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		cdleme17.c	|- C = ( ( P .\/ S ) ./\ W )
10	1 2 3 4 5 6 7 8 9	cdleme7a	|- G = ( ( P .\/ Q ) ./\ ( F .\/ C ) )
11	1 2 3 4 5 6 7 9	cdleme9	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( F .\/ C ) = ( Q .\/ C ) )
12	11	oveq2d	|- ( ( ( K e. HL /\ W e. H ) /\ ( ( P e. A /\ -. P .<_ W ) /\ Q e. A /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( P .\/ Q ) ./\ ( F .\/ C ) ) = ( ( P .\/ Q ) ./\ ( Q .\/ C ) ) )
13	10 12	eqtrid	|- ( ( ( 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 .\/ C ) ) )