Metamath Proof Explorer

Theorem cvlatexch3

Description: Atom exchange property. (Contributed by NM, 29-Nov-2012)

Ref Expression
Hypotheses cvlatexch.l 
|- .<_ = ( le ` K )
cvlatexch.j 
|- .\/ = ( join ` K )
cvlatexch.a 
|- A = ( Atoms ` K )
Assertion cvlatexch3 
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) -> ( P .<_ ( Q .\/ R ) -> ( P .\/ Q ) = ( P .\/ R ) ) )

Proof

Step Hyp Ref Expression
1 cvlatexch.l 
 |-  .<_ = ( le ` K )
2 cvlatexch.j 
 |-  .\/ = ( join ` K )
3 cvlatexch.a 
 |-  A = ( Atoms ` K )
4 simp1 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) -> K e. CvLat )
5 simp21 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) -> P e. A )
6 simp23 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) -> R e. A )
7 simp22 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) -> Q e. A )
8 simp3l 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) -> P =/= Q )
9 1 2 3 cvlatexchb1 
 |-  ( ( K e. CvLat /\ ( P e. A /\ R e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .<_ ( Q .\/ R ) <-> ( Q .\/ P ) = ( Q .\/ R ) ) )
10 4 5 6 7 8 9 syl131anc 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) -> ( P .<_ ( Q .\/ R ) <-> ( Q .\/ P ) = ( Q .\/ R ) ) )
11 10 biimpa 
 |-  ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) /\ P .<_ ( Q .\/ R ) ) -> ( Q .\/ P ) = ( Q .\/ R ) )
12 simpl1 
 |-  ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) /\ P .<_ ( Q .\/ R ) ) -> K e. CvLat )
13 cvllat 
 |-  ( K e. CvLat -> K e. Lat )
14 12 13 syl 
 |-  ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) /\ P .<_ ( Q .\/ R ) ) -> K e. Lat )
15 simpl21 
 |-  ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) /\ P .<_ ( Q .\/ R ) ) -> P e. A )
16 eqid 
 |-  ( Base ` K ) = ( Base ` K )
17 16 3 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
18 15 17 syl 
 |-  ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) /\ P .<_ ( Q .\/ R ) ) -> P e. ( Base ` K ) )
19 simpl22 
 |-  ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) /\ P .<_ ( Q .\/ R ) ) -> Q e. A )
20 16 3 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
21 19 20 syl 
 |-  ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) /\ P .<_ ( Q .\/ R ) ) -> Q e. ( Base ` K ) )
22 16 2 latjcom 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
23 14 18 21 22 syl3anc 
 |-  ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) /\ P .<_ ( Q .\/ R ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
24 1 2 3 cvlatexchb2 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( Q .\/ R ) <-> ( P .\/ R ) = ( Q .\/ R ) ) )
25 24 3adant3l 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) -> ( P .<_ ( Q .\/ R ) <-> ( P .\/ R ) = ( Q .\/ R ) ) )
26 25 biimpa 
 |-  ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) /\ P .<_ ( Q .\/ R ) ) -> ( P .\/ R ) = ( Q .\/ R ) )
27 11 23 26 3eqtr4d 
 |-  ( ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) /\ P .<_ ( Q .\/ R ) ) -> ( P .\/ Q ) = ( P .\/ R ) )
28 27 ex 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ P =/= R ) ) -> ( P .<_ ( Q .\/ R ) -> ( P .\/ Q ) = ( P .\/ R ) ) )