Theorem cdleme35sn3a

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. (Contributed by NM, 19-Mar-2013)

	Ref	Expression
Hypotheses	cdleme32s.b	|- B = ( Base ` K )
	cdleme32s.l	|- .<_ = ( le ` K )
	cdleme32s.j	|- .\/ = ( join ` K )
	cdleme32s.m	|- ./\ = ( meet ` K )
	cdleme32s.a	|- A = ( Atoms ` K )
	cdleme32s.h	|- H = ( LHyp ` K )
	cdleme32s.u	|- U = ( ( P .\/ Q ) ./\ W )
	cdleme32s.d	|- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
	cdleme32s.n	|- N = if ( s .<_ ( P .\/ Q ) , I , D )
Assertion	cdleme35sn3a	|- ( ( ( ( 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 .<_ ( P .\/ Q ) )

Proof

Step	Hyp	Ref	Expression
1		cdleme32s.b	|- B = ( Base ` K )
2		cdleme32s.l	|- .<_ = ( le ` K )
3		cdleme32s.j	|- .\/ = ( join ` K )
4		cdleme32s.m	|- ./\ = ( meet ` K )
5		cdleme32s.a	|- A = ( Atoms ` K )
6		cdleme32s.h	|- H = ( LHyp ` K )
7		cdleme32s.u	|- U = ( ( P .\/ Q ) ./\ W )
8		cdleme32s.d	|- D = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9		cdleme32s.n	|- N = if ( s .<_ ( P .\/ Q ) , I , D )
10		eqid	|- ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) )
11	2 3 4 5 6 7 10	cdleme35fnpq	|- ( ( ( ( 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 .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) .<_ ( P .\/ Q ) )
12		simp2rl	|- ( ( ( ( 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 e. A )
13		simp3	|- ( ( ( ( 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 .<_ ( P .\/ Q ) )
14	8 9 10	cdleme31sn2	|- ( ( R e. A /\ -. R .<_ ( P .\/ Q ) ) -> [_ R / s ]_ N = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
15	12 13 14	syl2anc	|- ( ( ( ( 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 = ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) )
16	15	breq1d	|- ( ( ( ( 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 .<_ ( P .\/ Q ) <-> ( ( R .\/ U ) ./\ ( Q .\/ ( ( P .\/ R ) ./\ W ) ) ) .<_ ( P .\/ Q ) ) )
17	11 16	mtbird	|- ( ( ( ( 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 .<_ ( P .\/ Q ) )