dalawlem5

Description: Lemma for dalaw . Special case to eliminate the requirement -. ( P .\/ S ) ./\ ( Q .\/ T ) ) .<_ ( P .\/ Q ) in dalawlem1 . (Contributed by NM, 4-Oct-2012)

	Ref	Expression
Hypotheses	dalawlem.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	dalawlem.j	$⊢ \underset{˙}{\lor} = join (K)$
	dalawlem.m	$⊢ \underset{˙}{\land} = meet (K)$
	dalawlem.a	$⊢ A = Atoms (K)$
Assertion	dalawlem5	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\land} (S \underset{˙}{\lor} T)) \underset{˙}{\leq} (((Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)) \underset{˙}{\lor} ((R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S)))$

Proof

Step	Hyp	Ref	Expression
1		dalawlem.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		dalawlem.j	$⊢ \underset{˙}{\lor} = join (K)$
3		dalawlem.m	$⊢ \underset{˙}{\land} = meet (K)$
4		dalawlem.a	$⊢ A = Atoms (K)$
5		eqid	$⊢ {Base}_{K} = {Base}_{K}$
6		simp11	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to K \in HL$
7	6	hllatd	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to K \in Lat$
8		simp21	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to P \in A$
9		simp22	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to Q \in A$
10	5 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
11	6 8 9 10	syl3anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
12		simp31	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to S \in A$
13		simp32	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to T \in A$
14	5 2 4	hlatjcl	$⊢ (K \in HL \land S \in A \land T \in A) \to S \underset{˙}{\lor} T \in {Base}_{K}$
15	6 12 13 14	syl3anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to S \underset{˙}{\lor} T \in {Base}_{K}$
16	5 3	latmcl	$⊢ (K \in Lat \land P \underset{˙}{\lor} Q \in {Base}_{K} \land S \underset{˙}{\lor} T \in {Base}_{K}) \to (P \underset{˙}{\lor} Q) \underset{˙}{\land} (S \underset{˙}{\lor} T) \in {Base}_{K}$
17	7 11 15 16	syl3anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\land} (S \underset{˙}{\lor} T) \in {Base}_{K}$
18	5 4	atbase	$⊢ T \in A \to T \in {Base}_{K}$
19	13 18	syl	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to T \in {Base}_{K}$
20	5 2	latjcl	$⊢ (K \in Lat \land P \underset{˙}{\lor} Q \in {Base}_{K} \land T \in {Base}_{K}) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} T \in {Base}_{K}$
21	7 11 19 20	syl3anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} T \in {Base}_{K}$
22	5 4	atbase	$⊢ S \in A \to S \in {Base}_{K}$
23	12 22	syl	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to S \in {Base}_{K}$
24	5 3	latmcl	$⊢ (K \in Lat \land (P \underset{˙}{\lor} Q) \underset{˙}{\lor} T \in {Base}_{K} \land S \in {Base}_{K}) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} T) \underset{˙}{\land} S \in {Base}_{K}$
25	7 21 23 24	syl3anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} T) \underset{˙}{\land} S \in {Base}_{K}$
26	5 2	latjcl	$⊢ (K \in Lat \land P \underset{˙}{\lor} Q \in {Base}_{K} \land S \in {Base}_{K}) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} S \in {Base}_{K}$
27	7 11 23 26	syl3anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} S \in {Base}_{K}$
28	5 3	latmcl	$⊢ (K \in Lat \land (P \underset{˙}{\lor} Q) \underset{˙}{\lor} S \in {Base}_{K} \land T \in {Base}_{K}) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} S) \underset{˙}{\land} T \in {Base}_{K}$
29	7 27 19 28	syl3anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} S) \underset{˙}{\land} T \in {Base}_{K}$
30	5 2	latjcl	$⊢ (K \in Lat \land ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} T) \underset{˙}{\land} S \in {Base}_{K} \land ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} S) \underset{˙}{\land} T \in {Base}_{K}) \to (((P \underset{˙}{\lor} Q) \underset{˙}{\lor} T) \underset{˙}{\land} S) \underset{˙}{\lor} (((P \underset{˙}{\lor} Q) \underset{˙}{\lor} S) \underset{˙}{\land} T) \in {Base}_{K}$
31	7 25 29 30	syl3anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to (((P \underset{˙}{\lor} Q) \underset{˙}{\lor} T) \underset{˙}{\land} S) \underset{˙}{\lor} (((P \underset{˙}{\lor} Q) \underset{˙}{\lor} S) \underset{˙}{\land} T) \in {Base}_{K}$
32		simp23	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to R \in A$
33	5 2 4	hlatjcl	$⊢ (K \in HL \land Q \in A \land R \in A) \to Q \underset{˙}{\lor} R \in {Base}_{K}$
34	6 9 32 33	syl3anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to Q \underset{˙}{\lor} R \in {Base}_{K}$
35		simp33	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to U \in A$
36	5 2 4	hlatjcl	$⊢ (K \in HL \land T \in A \land U \in A) \to T \underset{˙}{\lor} U \in {Base}_{K}$
37	6 13 35 36	syl3anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to T \underset{˙}{\lor} U \in {Base}_{K}$
38	5 3	latmcl	$⊢ (K \in Lat \land Q \underset{˙}{\lor} R \in {Base}_{K} \land T \underset{˙}{\lor} U \in {Base}_{K}) \to (Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U) \in {Base}_{K}$
39	7 34 37 38	syl3anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to (Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U) \in {Base}_{K}$
40	5 2 4	hlatjcl	$⊢ (K \in HL \land R \in A \land P \in A) \to R \underset{˙}{\lor} P \in {Base}_{K}$
41	6 32 8 40	syl3anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to R \underset{˙}{\lor} P \in {Base}_{K}$
42	5 2 4	hlatjcl	$⊢ (K \in HL \land U \in A \land S \in A) \to U \underset{˙}{\lor} S \in {Base}_{K}$
43	6 35 12 42	syl3anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to U \underset{˙}{\lor} S \in {Base}_{K}$
44	5 3	latmcl	$⊢ (K \in Lat \land R \underset{˙}{\lor} P \in {Base}_{K} \land U \underset{˙}{\lor} S \in {Base}_{K}) \to (R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S) \in {Base}_{K}$
45	7 41 43 44	syl3anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to (R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S) \in {Base}_{K}$
46	5 2	latjcl	$⊢ (K \in Lat \land (Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U) \in {Base}_{K} \land (R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S) \in {Base}_{K}) \to ((Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)) \underset{˙}{\lor} ((R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S)) \in {Base}_{K}$
47	7 39 45 46	syl3anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to ((Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)) \underset{˙}{\lor} ((R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S)) \in {Base}_{K}$
48	1 2 3 4	dalawlem2	$⊢ (K \in HL \land (P \in A \land Q \in A) \land (S \in A \land T \in A)) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\land} (S \underset{˙}{\lor} T)) \underset{˙}{\leq} ((((P \underset{˙}{\lor} Q) \underset{˙}{\lor} T) \underset{˙}{\land} S) \underset{˙}{\lor} (((P \underset{˙}{\lor} Q) \underset{˙}{\lor} S) \underset{˙}{\land} T))$
49	6 8 9 12 13 48	syl122anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\land} (S \underset{˙}{\lor} T)) \underset{˙}{\leq} ((((P \underset{˙}{\lor} Q) \underset{˙}{\lor} T) \underset{˙}{\land} S) \underset{˙}{\lor} (((P \underset{˙}{\lor} Q) \underset{˙}{\lor} S) \underset{˙}{\land} T))$
50	2 4	hlatjcom	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q = Q \underset{˙}{\lor} P$
51	6 8 9 50	syl3anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to P \underset{˙}{\lor} Q = Q \underset{˙}{\lor} P$
52	51	oveq1d	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} T = (Q \underset{˙}{\lor} P) \underset{˙}{\lor} T$
53	2 4	hlatj32	$⊢ (K \in HL \land (Q \in A \land P \in A \land T \in A)) \to (Q \underset{˙}{\lor} P) \underset{˙}{\lor} T = (Q \underset{˙}{\lor} T) \underset{˙}{\lor} P$
54	6 9 8 13 53	syl13anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to (Q \underset{˙}{\lor} P) \underset{˙}{\lor} T = (Q \underset{˙}{\lor} T) \underset{˙}{\lor} P$
55	52 54	eqtrd	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} T = (Q \underset{˙}{\lor} T) \underset{˙}{\lor} P$
56	55	oveq1d	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} T) \underset{˙}{\land} S = ((Q \underset{˙}{\lor} T) \underset{˙}{\lor} P) \underset{˙}{\land} S$
57	1 2 3 4	dalawlem3	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to (((Q \underset{˙}{\lor} T) \underset{˙}{\lor} P) \underset{˙}{\land} S) \underset{˙}{\leq} (((Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)) \underset{˙}{\lor} ((R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S)))$
58	56 57	eqbrtrd	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to (((P \underset{˙}{\lor} Q) \underset{˙}{\lor} T) \underset{˙}{\land} S) \underset{˙}{\leq} (((Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)) \underset{˙}{\lor} ((R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S)))$
59	2 4	hlatj32	$⊢ (K \in HL \land (P \in A \land Q \in A \land S \in A)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} S = (P \underset{˙}{\lor} S) \underset{˙}{\lor} Q$
60	6 8 9 12 59	syl13anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} S = (P \underset{˙}{\lor} S) \underset{˙}{\lor} Q$
61	60	oveq1d	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} S) \underset{˙}{\land} T = ((P \underset{˙}{\lor} S) \underset{˙}{\lor} Q) \underset{˙}{\land} T$
62	1 2 3 4	dalawlem4	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to (((P \underset{˙}{\lor} S) \underset{˙}{\lor} Q) \underset{˙}{\land} T) \underset{˙}{\leq} (((Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)) \underset{˙}{\lor} ((R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S)))$
63	61 62	eqbrtrd	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to (((P \underset{˙}{\lor} Q) \underset{˙}{\lor} S) \underset{˙}{\land} T) \underset{˙}{\leq} (((Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)) \underset{˙}{\lor} ((R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S)))$
64	5 1 2	latjle12	$⊢ (K \in Lat \land (((P \underset{˙}{\lor} Q) \underset{˙}{\lor} T) \underset{˙}{\land} S \in {Base}_{K} \land ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} S) \underset{˙}{\land} T \in {Base}_{K} \land ((Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)) \underset{˙}{\lor} ((R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S)) \in {Base}_{K})) \to (((((P \underset{˙}{\lor} Q) \underset{˙}{\lor} T) \underset{˙}{\land} S) \underset{˙}{\leq} (((Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)) \underset{˙}{\lor} ((R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S))) \land (((P \underset{˙}{\lor} Q) \underset{˙}{\lor} S) \underset{˙}{\land} T) \underset{˙}{\leq} (((Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)) \underset{˙}{\lor} ((R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S)))) \leftrightarrow ((((P \underset{˙}{\lor} Q) \underset{˙}{\lor} T) \underset{˙}{\land} S) \underset{˙}{\lor} (((P \underset{˙}{\lor} Q) \underset{˙}{\lor} S) \underset{˙}{\land} T)) \underset{˙}{\leq} (((Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)) \underset{˙}{\lor} ((R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S))))$
65	7 25 29 47 64	syl13anc	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to (((((P \underset{˙}{\lor} Q) \underset{˙}{\lor} T) \underset{˙}{\land} S) \underset{˙}{\leq} (((Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)) \underset{˙}{\lor} ((R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S))) \land (((P \underset{˙}{\lor} Q) \underset{˙}{\lor} S) \underset{˙}{\land} T) \underset{˙}{\leq} (((Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)) \underset{˙}{\lor} ((R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S)))) \leftrightarrow ((((P \underset{˙}{\lor} Q) \underset{˙}{\lor} T) \underset{˙}{\land} S) \underset{˙}{\lor} (((P \underset{˙}{\lor} Q) \underset{˙}{\lor} S) \underset{˙}{\land} T)) \underset{˙}{\leq} (((Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)) \underset{˙}{\lor} ((R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S))))$
66	58 63 65	mpbi2and	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to ((((P \underset{˙}{\lor} Q) \underset{˙}{\lor} T) \underset{˙}{\land} S) \underset{˙}{\lor} (((P \underset{˙}{\lor} Q) \underset{˙}{\lor} S) \underset{˙}{\land} T)) \underset{˙}{\leq} (((Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)) \underset{˙}{\lor} ((R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S)))$
67	5 1 7 17 31 47 49 66	lattrd	$⊢ ((K \in HL \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((P \underset{˙}{\lor} S) \underset{˙}{\land} (Q \underset{˙}{\lor} T)) \underset{˙}{\leq} (R \underset{˙}{\lor} U)) \land (P \in A \land Q \in A \land R \in A) \land (S \in A \land T \in A \land U \in A)) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\land} (S \underset{˙}{\lor} T)) \underset{˙}{\leq} (((Q \underset{˙}{\lor} R) \underset{˙}{\land} (T \underset{˙}{\lor} U)) \underset{˙}{\lor} ((R \underset{˙}{\lor} P) \underset{˙}{\land} (U \underset{˙}{\lor} S)))$

Metamath Proof Explorer

Theorem dalawlem5

Proof