Metamath Proof Explorer

Theorem cdleme21a

Description: Part of proof of Lemma E in Crawley p. 115. (Contributed by NM, 28-Nov-2012)

Ref Expression
Hypotheses cdleme21a.l 
|- .<_ = ( le ` K )
cdleme21a.j 
|- .\/ = ( join ` K )
cdleme21a.a 
|- A = ( Atoms ` K )
Assertion cdleme21a 
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> S =/= z )

Proof

Step Hyp Ref Expression
1 cdleme21a.l 
 |-  .<_ = ( le ` K )
2 cdleme21a.j 
 |-  .\/ = ( join ` K )
3 cdleme21a.a 
 |-  A = ( Atoms ` K )
4 simp11 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> K e. HL )
5 hlcvl 
 |-  ( K e. HL -> K e. CvLat )
6 4 5 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> K e. CvLat )
7 simp12 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> P e. A )
8 simp2l 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> S e. A )
9 simp3l 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> z e. A )
10 simp13 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> Q e. A )
11 simp2r 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> -. S .<_ ( P .\/ Q ) )
12 1 2 3 atnlej1 
 |-  ( ( K e. HL /\ ( S e. A /\ P e. A /\ Q e. A ) /\ -. S .<_ ( P .\/ Q ) ) -> S =/= P )
13 12 necomd 
 |-  ( ( K e. HL /\ ( S e. A /\ P e. A /\ Q e. A ) /\ -. S .<_ ( P .\/ Q ) ) -> P =/= S )
14 4 8 7 10 11 13 syl131anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> P =/= S )
15 simp3r 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> ( P .\/ z ) = ( S .\/ z ) )
16 3 2 cvlsupr6 
 |-  ( ( K e. CvLat /\ ( P e. A /\ S e. A /\ z e. A ) /\ ( P =/= S /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> z =/= S )
17 16 necomd 
 |-  ( ( K e. CvLat /\ ( P e. A /\ S e. A /\ z e. A ) /\ ( P =/= S /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> S =/= z )
18 6 7 8 9 14 15 17 syl132anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( S e. A /\ -. S .<_ ( P .\/ Q ) ) /\ ( z e. A /\ ( P .\/ z ) = ( S .\/ z ) ) ) -> S =/= z )