Metamath Proof Explorer

Theorem 2atjlej

Description: Two atoms are different if their join majorizes the join of two different atoms. (Contributed by NM, 4-Jun-2013)

Ref Expression
Hypotheses ps1.l 
|- .<_ = ( le ` K )
ps1.j 
|- .\/ = ( join ` K )
ps1.a 
|- A = ( Atoms ` K )
Assertion 2atjlej 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) ) -> R =/= S )

Proof

Step Hyp Ref Expression
1 ps1.l 
 |-  .<_ = ( le ` K )
2 ps1.j 
 |-  .\/ = ( join ` K )
3 ps1.a 
 |-  A = ( Atoms ` K )
4 simp33 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) ) -> ( P .\/ Q ) .<_ ( R .\/ S ) )
5 simp1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) ) -> K e. HL )
6 simp21 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) ) -> P e. A )
7 simp22 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) ) -> Q e. A )
8 simp23 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) ) -> P =/= Q )
9 simp31 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) ) -> R e. A )
10 simp32 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) ) -> S e. A )
11 1 2 3 ps-1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A ) ) -> ( ( P .\/ Q ) .<_ ( R .\/ S ) <-> ( P .\/ Q ) = ( R .\/ S ) ) )
12 5 6 7 8 9 10 11 syl132anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) ) -> ( ( P .\/ Q ) .<_ ( R .\/ S ) <-> ( P .\/ Q ) = ( R .\/ S ) ) )
13 4 12 mpbid 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) ) -> ( P .\/ Q ) = ( R .\/ S ) )
14 2 3 lnnat 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> -. ( P .\/ Q ) e. A ) )
15 5 6 7 14 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) ) -> ( P =/= Q <-> -. ( P .\/ Q ) e. A ) )
16 8 15 mpbid 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) ) -> -. ( P .\/ Q ) e. A )
17 13 16 eqneltrrd 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) ) -> -. ( R .\/ S ) e. A )
18 2 3 lnnat 
 |-  ( ( K e. HL /\ R e. A /\ S e. A ) -> ( R =/= S <-> -. ( R .\/ S ) e. A ) )
19 5 9 10 18 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) ) -> ( R =/= S <-> -. ( R .\/ S ) e. A ) )
20 17 19 mpbird 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( R e. A /\ S e. A /\ ( P .\/ Q ) .<_ ( R .\/ S ) ) ) -> R =/= S )