Theorem hlatj12

Description: Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 for atoms. (Contributed by NM, 4-Jun-2012)

	Ref	Expression
Hypotheses	hlatjcom.j	|- .\/ = ( join ` K )
	hlatjcom.a	|- A = ( Atoms ` K )
Assertion	hlatj12	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .\/ ( Q .\/ R ) ) = ( Q .\/ ( P .\/ R ) ) )

Proof

Step	Hyp	Ref	Expression
1		hlatjcom.j	|- .\/ = ( join ` K )
2		hlatjcom.a	|- A = ( Atoms ` K )
3	1 2	hlatjcom	|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
4	3	3adant3r3	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
5	4	oveq1d	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) = ( ( Q .\/ P ) .\/ R ) )
6	1 2	hlatjass	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) = ( P .\/ ( Q .\/ R ) ) )
7		simpl	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> K e. HL )
8		simpr2	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> Q e. A )
9		simpr1	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> P e. A )
10		simpr3	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> R e. A )
11	1 2	hlatjass	|- ( ( K e. HL /\ ( Q e. A /\ P e. A /\ R e. A ) ) -> ( ( Q .\/ P ) .\/ R ) = ( Q .\/ ( P .\/ R ) ) )
12	7 8 9 10 11	syl13anc	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( Q .\/ P ) .\/ R ) = ( Q .\/ ( P .\/ R ) ) )
13	5 6 12	3eqtr3d	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .\/ ( Q .\/ R ) ) = ( Q .\/ ( P .\/ R ) ) )