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	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
	hlatjcom.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
Assertion	hlatjrot	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R=\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }Q$$

Proof

Step	Hyp	Ref	Expression
1		hlatjcom.j	$$⊢\underset{\dot{}}{\vee }=\mathrm{join}\left(K\right)$$
2		hlatjcom.a	$$⊢A=\mathrm{Atoms}\left(K\right)$$
3	1 2	hlatj32	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R=\left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }Q$$
4	1 2	hlatjcom	$$⊢\left(K\in \mathrm{HL}\wedge P\in A\wedge R\in A\right)\to P\underset{\dot{}}{\vee }R=R\underset{\dot{}}{\vee }P$$
5	4	3adant3r2	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\to P\underset{\dot{}}{\vee }R=R\underset{\dot{}}{\vee }P$$
6	5	oveq1d	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\to \left(P\underset{\dot{}}{\vee }R\right)\underset{\dot{}}{\vee }Q=\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }Q$$
7	3 6	eqtrd	$$⊢\left(K\in \mathrm{HL}\wedge \left(P\in A\wedge Q\in A\wedge R\in A\right)\right)\to \left(P\underset{\dot{}}{\vee }Q\right)\underset{\dot{}}{\vee }R=\left(R\underset{\dot{}}{\vee }P\right)\underset{\dot{}}{\vee }Q$$