Theorem cdleme46frvlpq

Description: Show that ( FS ) is not under P .\/ Q when S isn't. (Contributed by NM, 1-Apr-2013)

	Ref	Expression
Hypotheses	cdlemef46.b	|- B = ( Base ` K )
	cdlemef46.l	|- .<_ = ( le ` K )
	cdlemef46.j	|- .\/ = ( join ` K )
	cdlemef46.m	|- ./\ = ( meet ` K )
	cdlemef46.a	|- A = ( Atoms ` K )
	cdlemef46.h	|- H = ( LHyp ` K )
	cdlemef46.u	|- U = ( ( P .\/ Q ) ./\ W )
	cdlemef46.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
	cdlemefs46.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
	cdlemef46.f	|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
Assertion	cdleme46frvlpq	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> -. ( F ` S ) .<_ ( P .\/ Q ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemef46.b	|- B = ( Base ` K )
2		cdlemef46.l	|- .<_ = ( le ` K )
3		cdlemef46.j	|- .\/ = ( join ` K )
4		cdlemef46.m	|- ./\ = ( meet ` K )
5		cdlemef46.a	|- A = ( Atoms ` K )
6		cdlemef46.h	|- H = ( LHyp ` K )
7		cdlemef46.u	|- U = ( ( P .\/ Q ) ./\ W )
8		cdlemef46.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9		cdlemefs46.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
10		cdlemef46.f	|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( if ( s .<_ ( P .\/ Q ) , ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) ) , [_ s / t ]_ D ) .\/ ( x ./\ W ) ) ) ) , x ) )
11		eqid	|- ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
12	2 3 4 5 6 7 11	cdleme35fnpq	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> -. ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .<_ ( P .\/ Q ) )
13	1 2 3 4 5 6 7 8 10	cdlemefr45e	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( F ` S ) = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) )
14	13	breq1d	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> ( ( F ` S ) .<_ ( P .\/ Q ) <-> ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .<_ ( P .\/ Q ) ) )
15	12 14	mtbird	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( S e. A /\ -. S .<_ W ) ) /\ -. S .<_ ( P .\/ Q ) ) -> -. ( F ` S ) .<_ ( P .\/ Q ) )