Theorem cdleme6

Description: Part of proof of Lemma E in Crawley p. 113. This expresses (r \/ f_s(r)) /\ w = u at the top of p. 114. (Contributed by NM, 7-Jun-2012)

	Ref	Expression
Hypotheses	cdleme4.l	|- .<_ = ( le ` K )
	cdleme4.j	|- .\/ = ( join ` K )
	cdleme4.m	|- ./\ = ( meet ` K )
	cdleme4.a	|- A = ( Atoms ` K )
	cdleme4.h	|- H = ( LHyp ` K )
	cdleme4.u	|- U = ( ( P .\/ Q ) ./\ W )
	cdleme4.f	|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
	cdleme4.g	|- G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) )
Assertion	cdleme6	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ G ) ./\ W ) = U )

Proof

Step	Hyp	Ref	Expression
1		cdleme4.l	|- .<_ = ( le ` K )
2		cdleme4.j	|- .\/ = ( join ` K )
3		cdleme4.m	|- ./\ = ( meet ` K )
4		cdleme4.a	|- A = ( Atoms ` K )
5		cdleme4.h	|- H = ( LHyp ` K )
6		cdleme4.u	|- U = ( ( P .\/ Q ) ./\ W )
7		cdleme4.f	|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8		cdleme4.g	|- G = ( ( P .\/ Q ) ./\ ( F .\/ ( ( R .\/ S ) ./\ W ) ) )
9	1 2 3 4 5 6 7 8	cdleme5	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) ) ) -> ( R .\/ G ) = ( P .\/ Q ) )
10	9	oveq1d	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ G ) ./\ W ) = ( ( P .\/ Q ) ./\ W ) )
11	10 6	eqtr4di	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A /\ ( R e. A /\ -. R .<_ W ) ) /\ ( ( S e. A /\ -. S .<_ W ) /\ R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ G ) ./\ W ) = U )