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Metamath Proof Explorer

Theorem hlatjrot

Description: Rotate lattice join of 3 classes. Frequently-used special case of latjrot for atoms. (Contributed by NM, 2-Aug-2012)

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

Proof

Step Hyp Ref Expression
1 hlatjcom.j 
 |-  .\/ = ( join ` K )
2 hlatjcom.a 
 |-  A = ( Atoms ` K )
3 1 2 hlatj32 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( P .\/ R ) .\/ Q ) )
4 1 2 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ R e. A ) -> ( P .\/ R ) = ( R .\/ P ) )
5 4 3adant3r2 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .\/ R ) = ( R .\/ P ) )
6 5 oveq1d 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ R ) .\/ Q ) = ( ( R .\/ P ) .\/ Q ) )
7 3 6 eqtrd 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( R .\/ P ) .\/ Q ) )