Metamath Proof Explorer

Theorem 4atlem4d

Description: Lemma for 4at . Frequently used associative law. (Contributed by NM, 9-Jul-2012)

		Ref	Expression
	Hypotheses	4at.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
		4at.j	$⊢ \underset{˙}{\lor} = join (K)$
		4at.a	$⊢ A = Atoms (K)$
	Assertion	4atlem4d	$⊢ ((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} (R \underset{˙}{\lor} S) = S \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)$

Proof

Step	Hyp	Ref	Expression
1		4at.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		4at.j	$⊢ \underset{˙}{\lor} = join (K)$
3		4at.a	$⊢ A = Atoms (K)$
4		simpl1	$⊢ ((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A)) \to K \in HL$
5	4	hllatd	$⊢ ((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A)) \to K \in Lat$
6		eqid	$⊢ {Base}_{K} = {Base}_{K}$
7	6 2 3	hlatjcl	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
8	7	adantr	$⊢ ((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A)) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
9	6 3	atbase	$⊢ R \in A \to R \in {Base}_{K}$
10	9	ad2antrl	$⊢ ((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A)) \to R \in {Base}_{K}$
11	6 3	atbase	$⊢ S \in A \to S \in {Base}_{K}$
12	11	ad2antll	$⊢ ((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A)) \to S \in {Base}_{K}$
13	6 2	latjass	$⊢ (K \in Lat \land (P \underset{˙}{\lor} Q \in {Base}_{K} \land R \in {Base}_{K} \land S \in {Base}_{K})) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R) \underset{˙}{\lor} S = (P \underset{˙}{\lor} Q) \underset{˙}{\lor} (R \underset{˙}{\lor} S)$
14	5 8 10 12 13	syl13anc	$⊢ ((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A)) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R) \underset{˙}{\lor} S = (P \underset{˙}{\lor} Q) \underset{˙}{\lor} (R \underset{˙}{\lor} S)$
15	6 2	latjcl	$⊢ (K \in Lat \land P \underset{˙}{\lor} Q \in {Base}_{K} \land R \in {Base}_{K}) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} R \in {Base}_{K}$
16	5 8 10 15	syl3anc	$⊢ ((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} R \in {Base}_{K}$
17	6 2	latjcom	$⊢ (K \in Lat \land (P \underset{˙}{\lor} Q) \underset{˙}{\lor} R \in {Base}_{K} \land S \in {Base}_{K}) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R) \underset{˙}{\lor} S = S \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)$
18	5 16 12 17	syl3anc	$⊢ ((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A)) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R) \underset{˙}{\lor} S = S \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)$
19	14 18	eqtr3d	$⊢ ((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} (R \underset{˙}{\lor} S) = S \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)$