4atlem11b

Description: Lemma for 4at . Substitute U for Q (cont.). (Contributed by NM, 10-Jul-2012)

	Ref	Expression
Hypotheses	4at.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	4at.j	$⊢ \underset{˙}{\lor} = join (K)$
	4at.a	$⊢ A = Atoms (K)$
Assertion	4atlem11b	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} (R \underset{˙}{\lor} S) = (P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)$

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		simp11	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to (K \in HL \land P \in A \land Q \in A)$
5		simp12	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to (R \in A \land S \in A)$
6		simp132	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to V \in A$
7		simp133	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to W \in A$
8	5 6 7	3jca	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to ((R \in A \land S \in A) \land V \in A \land W \in A)$
9		simp2l	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R))$
10	4 8 9	3jca	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to ((K \in HL \land P \in A \land Q \in A) \land ((R \in A \land S \in A) \land V \in A \land W \in A) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)))$
11		simp32	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W))$
12		simp33	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W))$
13		simp111	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to K \in HL$
14	13	hllatd	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to K \in Lat$
15		simp12l	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to R \in A$
16		eqid	$⊢ {Base}_{K} = {Base}_{K}$
17	16 3	atbase	$⊢ R \in A \to R \in {Base}_{K}$
18	15 17	syl	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to R \in {Base}_{K}$
19		simp12r	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to S \in A$
20	16 3	atbase	$⊢ S \in A \to S \in {Base}_{K}$
21	19 20	syl	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to S \in {Base}_{K}$
22		simp112	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to P \in A$
23		simp131	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to U \in A$
24	16 2 3	hlatjcl	$⊢ (K \in HL \land P \in A \land U \in A) \to P \underset{˙}{\lor} U \in {Base}_{K}$
25	13 22 23 24	syl3anc	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to P \underset{˙}{\lor} U \in {Base}_{K}$
26	16 2 3	hlatjcl	$⊢ (K \in HL \land V \in A \land W \in A) \to V \underset{˙}{\lor} W \in {Base}_{K}$
27	13 6 7 26	syl3anc	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to V \underset{˙}{\lor} W \in {Base}_{K}$
28	16 2	latjcl	$⊢ (K \in Lat \land P \underset{˙}{\lor} U \in {Base}_{K} \land V \underset{˙}{\lor} W \in {Base}_{K}) \to (P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W) \in {Base}_{K}$
29	14 25 27 28	syl3anc	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to (P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W) \in {Base}_{K}$
30	16 1 2	latjle12	$⊢ (K \in Lat \land (R \in {Base}_{K} \land S \in {Base}_{K} \land (P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W) \in {Base}_{K})) \to ((R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W))) \leftrightarrow (R \underset{˙}{\lor} S) \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))$
31	14 18 21 29 30	syl13anc	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to ((R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W))) \leftrightarrow (R \underset{˙}{\lor} S) \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))$
32	11 12 31	mpbi2and	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to (R \underset{˙}{\lor} S) \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W))$
33		simp31	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W))$
34		simp13	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to (U \in A \land V \in A \land W \in A)$
35		simp2r	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)$
36	1 2 3	4atlem11a	$⊢ ((K \in HL \land P \in A \land Q \in A) \land (U \in A \land V \in A \land W \in A) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \to (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \leftrightarrow (P \underset{˙}{\lor} Q) \underset{˙}{\lor} (V \underset{˙}{\lor} W) = (P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W))$
37	4 34 35 36	syl3anc	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \leftrightarrow (P \underset{˙}{\lor} Q) \underset{˙}{\lor} (V \underset{˙}{\lor} W) = (P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W))$
38	33 37	mpbid	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} (V \underset{˙}{\lor} W) = (P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)$
39	32 38	breqtrrd	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to (R \underset{˙}{\lor} S) \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} (V \underset{˙}{\lor} W))$
40	1 2 3	4atlem10	$⊢ ((K \in HL \land P \in A \land Q \in A) \land ((R \in A \land S \in A) \land V \in A \land W \in A) \land (P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R))) \to ((R \underset{˙}{\lor} S) \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} (R \underset{˙}{\lor} S) = (P \underset{˙}{\lor} Q) \underset{˙}{\lor} (V \underset{˙}{\lor} W))$
41	10 39 40	sylc	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} (R \underset{˙}{\lor} S) = (P \underset{˙}{\lor} Q) \underset{˙}{\lor} (V \underset{˙}{\lor} W)$
42	41 38	eqtrd	$⊢ (((K \in HL \land P \in A \land Q \in A) \land (R \in A \land S \in A) \land (U \in A \land V \in A \land W \in A)) \land ((P \neq Q \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\lor} R)) \land \neg Q \underset{˙}{\leq} ((P \underset{˙}{\lor} V) \underset{˙}{\lor} W)) \land (Q \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land R \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)) \land S \underset{˙}{\leq} ((P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)))) \to (P \underset{˙}{\lor} Q) \underset{˙}{\lor} (R \underset{˙}{\lor} S) = (P \underset{˙}{\lor} U) \underset{˙}{\lor} (V \underset{˙}{\lor} W)$

Metamath Proof Explorer

Theorem 4atlem11b

Proof