Theorem cdlemefs45ee

Description: Explicit expansion of cdlemefs45 . TODO: use to shorten cdlemefs45 uses? Should ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) be assigned to a hypothesis letter? TODO: FIX COMMENT. (Contributed by NM, 10-Apr-2013)

	Ref	Expression
Hypotheses	cdlemef45.b	|- B = ( Base ` K )
	cdlemef45.l	|- .<_ = ( le ` K )
	cdlemef45.j	|- .\/ = ( join ` K )
	cdlemef45.m	|- ./\ = ( meet ` K )
	cdlemef45.a	|- A = ( Atoms ` K )
	cdlemef45.h	|- H = ( LHyp ` K )
	cdlemef45.u	|- U = ( ( P .\/ Q ) ./\ W )
	cdlemef45.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
	cdlemef45.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 ) )
	cdlemefs45.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
Assertion	cdlemefs45ee	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F ` R ) = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemef45.b	|- B = ( Base ` K )
2		cdlemef45.l	|- .<_ = ( le ` K )
3		cdlemef45.j	|- .\/ = ( join ` K )
4		cdlemef45.m	|- ./\ = ( meet ` K )
5		cdlemef45.a	|- A = ( Atoms ` K )
6		cdlemef45.h	|- H = ( LHyp ` K )
7		cdlemef45.u	|- U = ( ( P .\/ Q ) ./\ W )
8		cdlemef45.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9		cdlemef45.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 ) )
10		cdlemefs45.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
11	1 2 3 4 5 6 7 8 9 10	cdlemefs45	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F ` R ) = [_ R / s ]_ [_ S / t ]_ E )
12		simp22l	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> R e. A )
13		simp23l	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> S e. A )
14		eqid	|- ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) = ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) )
15		eqid	|- ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) )
16	8 10 14 15	cdleme31sde	|- ( ( R e. A /\ S e. A ) -> [_ R / s ]_ [_ S / t ]_ E = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) )
17	12 13 16	syl2anc	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> [_ R / s ]_ [_ S / t ]_ E = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) )
18	11 17	eqtrd	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) /\ ( S e. A /\ -. S .<_ W ) ) /\ ( R .<_ ( P .\/ Q ) /\ -. S .<_ ( P .\/ Q ) ) ) -> ( F ` R ) = ( ( P .\/ Q ) ./\ ( ( ( S .\/ U ) ./\ ( Q .\/ ( ( P .\/ S ) ./\ W ) ) ) .\/ ( ( R .\/ S ) ./\ W ) ) ) )