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 ) )