Theorem cdlemefs32fva1

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. (Contributed by NM, 29-Mar-2013)

	Ref	Expression
Hypotheses	cdlemefs32.b	|- B = ( Base ` K )
	cdlemefs32.l	|- .<_ = ( le ` K )
	cdlemefs32.j	|- .\/ = ( join ` K )
	cdlemefs32.m	|- ./\ = ( meet ` K )
	cdlemefs32.a	|- A = ( Atoms ` K )
	cdlemefs32.h	|- H = ( LHyp ` K )
	cdlemefs32.u	|- U = ( ( P .\/ Q ) ./\ W )
	cdlemefs32.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
	cdlemefs32.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
	cdlemefs32.i	|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
	cdlemefs32.n	|- N = if ( s .<_ ( P .\/ Q ) , I , C )
	cdleme29fs.o	|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
	cdleme29fs.f	|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
Assertion	cdlemefs32fva1	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> ( F ` R ) = [_ R / s ]_ N )

Proof

Step	Hyp	Ref	Expression
1		cdlemefs32.b	|- B = ( Base ` K )
2		cdlemefs32.l	|- .<_ = ( le ` K )
3		cdlemefs32.j	|- .\/ = ( join ` K )
4		cdlemefs32.m	|- ./\ = ( meet ` K )
5		cdlemefs32.a	|- A = ( Atoms ` K )
6		cdlemefs32.h	|- H = ( LHyp ` K )
7		cdlemefs32.u	|- U = ( ( P .\/ Q ) ./\ W )
8		cdlemefs32.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
9		cdlemefs32.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
10		cdlemefs32.i	|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
11		cdlemefs32.n	|- N = if ( s .<_ ( P .\/ Q ) , I , C )
12		cdleme29fs.o	|- O = ( iota_ z e. B A. s e. A ( ( -. s .<_ W /\ ( s .\/ ( x ./\ W ) ) = x ) -> z = ( N .\/ ( x ./\ W ) ) ) )
13		cdleme29fs.f	|- F = ( x e. B |-> if ( ( P =/= Q /\ -. x .<_ W ) , O , x ) )
14		breq1	|- ( s = R -> ( s .<_ ( P .\/ Q ) <-> R .<_ ( P .\/ Q ) ) )
15		simp1	|- ( ( ( ( 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 ) ) ) ) -> ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) )
16		simp3l	|- ( ( ( ( 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 e. A )
17		simp3rl	|- ( ( ( ( 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 .<_ W )
18	16 17	jca	|- ( ( ( ( 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 e. A /\ -. s .<_ W ) )
19		simp3rr	|- ( ( ( ( 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 .<_ ( P .\/ Q ) )
20		simp2	|- ( ( ( ( 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 ) ) ) ) -> P =/= Q )
21	1 2 3 4 5 6 7 8 9 10 11	cdlemefs27cl	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( ( s e. A /\ -. s .<_ W ) /\ s .<_ ( P .\/ Q ) /\ P =/= Q ) ) -> N e. B )
22	15 18 19 20 21	syl13anc	|- ( ( ( ( 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 ) ) ) ) -> N e. B )
23	1 2 3 4 5 6 7 8 9 10 11	cdlemefs32snb	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N e. B )
24	1 2 3 4 5 6 14 22 23 12 13	cdlemefrs32fva1	|- ( ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) /\ ( P =/= Q /\ ( R e. A /\ -. R .<_ W ) ) /\ R .<_ ( P .\/ Q ) ) -> ( F ` R ) = [_ R / s ]_ N )