Metamath Proof Explorer

Theorem cvlsupr7

Description: Consequence of superposition condition ( P .\/ R ) = ( Q .\/ R ) . (Contributed by NM, 24-Nov-2012)

Ref Expression
Hypotheses cvlsupr5.a 
|- A = ( Atoms ` K )
cvlsupr5.j 
|- .\/ = ( join ` K )
Assertion cvlsupr7 
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> ( P .\/ Q ) = ( R .\/ Q ) )

Proof

Step Hyp Ref Expression
1 cvlsupr5.a 
 |-  A = ( Atoms ` K )
2 cvlsupr5.j 
 |-  .\/ = ( join ` K )
3 cvllat 
 |-  ( K e. CvLat -> K e. Lat )
4 3 3ad2ant1 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> K e. Lat )
5 simp21 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> P e. A )
6 eqid 
 |-  ( Base ` K ) = ( Base ` K )
7 6 1 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
8 5 7 syl 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> P e. ( Base ` K ) )
9 simp23 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> R e. A )
10 6 1 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
11 9 10 syl 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> R e. ( Base ` K ) )
12 eqid 
 |-  ( le ` K ) = ( le ` K )
13 6 12 2 latlej1 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> P ( le ` K ) ( P .\/ R ) )
14 4 8 11 13 syl3anc 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> P ( le ` K ) ( P .\/ R ) )
15 simp3r 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> ( P .\/ R ) = ( Q .\/ R ) )
16 14 15 breqtrd 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> P ( le ` K ) ( Q .\/ R ) )
17 simp22 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> Q e. A )
18 6 1 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
19 17 18 syl 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> Q e. ( Base ` K ) )
20 6 2 latjcom 
 |-  ( ( K e. Lat /\ Q e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( Q .\/ R ) = ( R .\/ Q ) )
21 4 19 11 20 syl3anc 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> ( Q .\/ R ) = ( R .\/ Q ) )
22 16 21 breqtrd 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> P ( le ` K ) ( R .\/ Q ) )
23 simp1 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> K e. CvLat )
24 simp3l 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> P =/= Q )
25 12 2 1 cvlatexchb2 
 |-  ( ( K e. CvLat /\ ( P e. A /\ R e. A /\ Q e. A ) /\ P =/= Q ) -> ( P ( le ` K ) ( R .\/ Q ) <-> ( P .\/ Q ) = ( R .\/ Q ) ) )
26 23 5 9 17 24 25 syl131anc 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> ( P ( le ` K ) ( R .\/ Q ) <-> ( P .\/ Q ) = ( R .\/ Q ) ) )
27 22 26 mpbid 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ ( P .\/ R ) = ( Q .\/ R ) ) ) -> ( P .\/ Q ) = ( R .\/ Q ) )