Metamath Proof Explorer

Theorem hlatjass

Description: Lattice join is associative. Frequently-used special case of latjass for atoms. (Contributed by NM, 27-Jul-2012)

Ref Expression
Hypotheses hlatjcom.j 
|- .\/ = ( join ` K )
hlatjcom.a 
|- A = ( Atoms ` K )
Assertion hlatjass 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) = ( P .\/ ( Q .\/ R ) ) )

Proof

Step Hyp Ref Expression
1 hlatjcom.j 
 |-  .\/ = ( join ` K )
2 hlatjcom.a 
 |-  A = ( Atoms ` K )
3 hllat 
 |-  ( K e. HL -> K e. Lat )
4 3 adantr 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> K e. Lat )
5 simpr1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> P e. A )
6 eqid 
 |-  ( Base ` K ) = ( Base ` K )
7 6 2 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
8 5 7 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> P e. ( Base ` K ) )
9 simpr2 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> Q e. A )
10 6 2 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
11 9 10 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> Q e. ( Base ` K ) )
12 simpr3 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> R e. A )
13 6 2 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
14 12 13 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> R e. ( Base ` K ) )
15 6 1 latjass 
 |-  ( ( K e. Lat /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) /\ R e. ( Base ` K ) ) ) -> ( ( P .\/ Q ) .\/ R ) = ( P .\/ ( Q .\/ R ) ) )
16 4 8 11 14 15 syl13anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) = ( P .\/ ( Q .\/ R ) ) )