Metamath Proof Explorer

Theorem cvlatexchb1

Description: A version of cvlexchb1 for atoms. (Contributed by NM, 5-Nov-2012)

Ref Expression
Hypotheses cvlatexch.l 
|- .<_ = ( le ` K )
cvlatexch.j 
|- .\/ = ( join ` K )
cvlatexch.a 
|- A = ( Atoms ` K )
Assertion cvlatexchb1 
|- ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( R .\/ Q ) <-> ( R .\/ P ) = ( R .\/ Q ) ) )

Proof

Step Hyp Ref Expression
1 cvlatexch.l 
 |-  .<_ = ( le ` K )
2 cvlatexch.j 
 |-  .\/ = ( join ` K )
3 cvlatexch.a 
 |-  A = ( Atoms ` K )
4 cvlatl 
 |-  ( K e. CvLat -> K e. AtLat )
5 4 adantr 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> K e. AtLat )
6 simpr1 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> P e. A )
7 simpr3 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> R e. A )
8 1 3 atncmp 
 |-  ( ( K e. AtLat /\ P e. A /\ R e. A ) -> ( -. P .<_ R <-> P =/= R ) )
9 5 6 7 8 syl3anc 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( -. P .<_ R <-> P =/= R ) )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 3 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
12 10 1 2 3 cvlexchb1 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. ( Base ` K ) ) /\ -. P .<_ R ) -> ( P .<_ ( R .\/ Q ) <-> ( R .\/ P ) = ( R .\/ Q ) ) )
13 12 3expia 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. ( Base ` K ) ) ) -> ( -. P .<_ R -> ( P .<_ ( R .\/ Q ) <-> ( R .\/ P ) = ( R .\/ Q ) ) ) )
14 11 13 syl3anr3 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( -. P .<_ R -> ( P .<_ ( R .\/ Q ) <-> ( R .\/ P ) = ( R .\/ Q ) ) ) )
15 9 14 sylbird 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P =/= R -> ( P .<_ ( R .\/ Q ) <-> ( R .\/ P ) = ( R .\/ Q ) ) ) )
16 15 3impia 
 |-  ( ( K e. CvLat /\ ( P e. A /\ Q e. A /\ R e. A ) /\ P =/= R ) -> ( P .<_ ( R .\/ Q ) <-> ( R .\/ P ) = ( R .\/ Q ) ) )