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Metamath Proof Explorer

Theorem atlt

Description: Two atoms are unequal iff their join is greater than one of them. (Contributed by NM, 6-May-2012)

Ref Expression
Hypotheses atlt.s 
|- .< = ( lt ` K )
atlt.j 
|- .\/ = ( join ` K )
atlt.a 
|- A = ( Atoms ` K )
Assertion atlt 
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .< ( P .\/ Q ) <-> P =/= Q ) )

Proof

Step Hyp Ref Expression
1 atlt.s 
 |-  .< = ( lt ` K )
2 atlt.j 
 |-  .\/ = ( join ` K )
3 atlt.a 
 |-  A = ( Atoms ` K )
4 simp1 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> K e. HL )
5 simp2 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> P e. A )
6 simp3 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> Q e. A )
7 eqid 
 |-  (
8 1 2 3 7 atltcvr 
 |-  ( ( K e. HL /\ ( P e. A /\ P e. A /\ Q e. A ) ) -> ( P .< ( P .\/ Q ) <-> P (
9 4 5 5 6 8 syl13anc 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .< ( P .\/ Q ) <-> P (
10 2 7 3 atcvr1 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P (
11 9 10 bitr4d 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .< ( P .\/ Q ) <-> P =/= Q ) )