Theorem cdleme32sn2awN

Description: Show that [_ R / s ]_ N is an atom not under W when -. R .<_ ( P .\/ Q ) . (Contributed by NM, 6-Mar-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdleme32.b	|- B = ( Base ` K )
	cdleme32.l	|- .<_ = ( le ` K )
	cdleme32.j	|- .\/ = ( join ` K )
	cdleme32.m	|- ./\ = ( meet ` K )
	cdleme32.a	|- A = ( Atoms ` K )
	cdleme32.h	|- H = ( LHyp ` K )
	cdleme32.u	|- U = ( ( P .\/ Q ) ./\ W )
	cdleme32.c	|- C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
	cdleme32.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
	cdleme32.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
	cdleme32.i	|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
	cdleme32.n	|- N = if ( s .<_ ( P .\/ Q ) , I , C )
Assertion	cdleme32sn2awN	|- ( ( ( ( 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. A /\ -. [_ R / s ]_ N .<_ W ) )

Proof

Step	Hyp	Ref	Expression
1		cdleme32.b	|- B = ( Base ` K )
2		cdleme32.l	|- .<_ = ( le ` K )
3		cdleme32.j	|- .\/ = ( join ` K )
4		cdleme32.m	|- ./\ = ( meet ` K )
5		cdleme32.a	|- A = ( Atoms ` K )
6		cdleme32.h	|- H = ( LHyp ` K )
7		cdleme32.u	|- U = ( ( P .\/ Q ) ./\ W )
8		cdleme32.c	|- C = ( ( s .\/ U ) ./\ ( Q .\/ ( ( P .\/ s ) ./\ W ) ) )
9		cdleme32.d	|- D = ( ( t .\/ U ) ./\ ( Q .\/ ( ( P .\/ t ) ./\ W ) ) )
10		cdleme32.e	|- E = ( ( P .\/ Q ) ./\ ( D .\/ ( ( s .\/ t ) ./\ W ) ) )
11		cdleme32.i	|- I = ( iota_ y e. B A. t e. A ( ( -. t .<_ W /\ -. t .<_ ( P .\/ Q ) ) -> y = E ) )
12		cdleme32.n	|- N = if ( s .<_ ( P .\/ Q ) , I , C )
13	1 2 3 4 5 6 7 8 12	cdlemefr32sn2aw	|- ( ( ( ( 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. A /\ -. [_ R / s ]_ N .<_ W ) )