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Metamath Proof Explorer

Theorem atnem0

Description: The meet of distinct atoms is zero. ( atnemeq0 analog.) (Contributed by NM, 5-Nov-2012)

Ref Expression
Hypotheses atnem0.m 
|- ./\ = ( meet ` K )
atnem0.z 
|- .0. = ( 0. ` K )
atnem0.a 
|- A = ( Atoms ` K )
Assertion atnem0 
|- ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> ( P ./\ Q ) = .0. ) )

Proof

Step Hyp Ref Expression
1 atnem0.m 
 |-  ./\ = ( meet ` K )
2 atnem0.z 
 |-  .0. = ( 0. ` K )
3 atnem0.a 
 |-  A = ( Atoms ` K )
4 eqid 
 |-  ( le ` K ) = ( le ` K )
5 4 3 atncmp 
 |-  ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( -. P ( le ` K ) Q <-> P =/= Q ) )
6 eqid 
 |-  ( Base ` K ) = ( Base ` K )
7 6 3 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
8 6 4 1 2 3 atnle 
 |-  ( ( K e. AtLat /\ P e. A /\ Q e. ( Base ` K ) ) -> ( -. P ( le ` K ) Q <-> ( P ./\ Q ) = .0. ) )
9 7 8 syl3an3 
 |-  ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( -. P ( le ` K ) Q <-> ( P ./\ Q ) = .0. ) )
10 5 9 bitr3d 
 |-  ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> ( P ./\ Q ) = .0. ) )