Theorem cdleme19f

Description: Part of proof of Lemma E in Crawley p. 113, 5th paragraph on p. 114, line 3. D , F , N , Y , G , O represent s_2, f(s), f_s(r), t_2, f(t), f_t(r). We prove that if r <_ s \/ t, then f_t(r) = f_t(r). (Contributed by NM, 14-Nov-2012)

	Ref	Expression
Hypotheses	cdleme19.l	|- .<_ = ( le ` K )
	cdleme19.j	|- .\/ = ( join ` K )
	cdleme19.m	|- ./\ = ( meet ` K )
	cdleme19.a	|- A = ( Atoms ` K )
	cdleme19.h	|- H = ( LHyp ` K )
	cdleme19.u	|- U = ( ( P .\/ Q ) ./\ W )
	cdleme19.f	|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
	cdleme19.g	|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
	cdleme19.d	|- D = ( ( R .\/ S ) ./\ W )
	cdleme19.y	|- Y = ( ( R .\/ T ) ./\ W )
	cdleme19.n	|- N = ( ( P .\/ Q ) ./\ ( F .\/ D ) )
	cdleme19.o	|- O = ( ( P .\/ Q ) ./\ ( G .\/ Y ) )
Assertion	cdleme19f	|- ( ( ( ( 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 ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> N = O )

Proof

Step	Hyp	Ref	Expression
1		cdleme19.l	|- .<_ = ( le ` K )
2		cdleme19.j	|- .\/ = ( join ` K )
3		cdleme19.m	|- ./\ = ( meet ` K )
4		cdleme19.a	|- A = ( Atoms ` K )
5		cdleme19.h	|- H = ( LHyp ` K )
6		cdleme19.u	|- U = ( ( P .\/ Q ) ./\ W )
7		cdleme19.f	|- F = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
8		cdleme19.g	|- G = ( ( T .\/ U ) ./\ ( Q .\/ ( ( P .\/ T ) ./\ W ) ) )
9		cdleme19.d	|- D = ( ( R .\/ S ) ./\ W )
10		cdleme19.y	|- Y = ( ( R .\/ T ) ./\ W )
11		cdleme19.n	|- N = ( ( P .\/ Q ) ./\ ( F .\/ D ) )
12		cdleme19.o	|- O = ( ( P .\/ Q ) ./\ ( G .\/ Y ) )
13	1 2 3 4 5 6 7 8 9 10	cdleme19e	|- ( ( ( ( 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 ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( F .\/ D ) = ( G .\/ Y ) )
14	13	oveq2d	|- ( ( ( ( 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 ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> ( ( P .\/ Q ) ./\ ( F .\/ D ) ) = ( ( P .\/ Q ) ./\ ( G .\/ Y ) ) )
15	14 11 12	3eqtr4g	|- ( ( ( ( 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 ) /\ R e. A ) /\ ( ( P =/= Q /\ S =/= T ) /\ ( -. S .<_ ( P .\/ Q ) /\ -. T .<_ ( P .\/ Q ) ) /\ ( R .<_ ( P .\/ Q ) /\ R .<_ ( S .\/ T ) ) ) ) -> N = O )