cdleme35b

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013)

	Ref	Expression
Hypotheses	cdleme35.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme35.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme35.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme35.a	$⊢ A = Atoms (K)$
	cdleme35.h	$⊢ H = LHyp (K)$
	cdleme35.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme35.f	$⊢ F = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
Assertion	cdleme35b	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)) \underset{˙}{\leq} (Q \underset{˙}{\lor} (R \underset{˙}{\lor} U))$

Proof

Step	Hyp	Ref	Expression
1		cdleme35.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme35.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme35.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme35.a	$⊢ A = Atoms (K)$
5		cdleme35.h	$⊢ H = LHyp (K)$
6		cdleme35.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme35.f	$⊢ F = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
8		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to K \in HL$
9	8	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to K \in Lat$
10		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to Q \in A$
11		eqid	$⊢ {Base}_{K} = {Base}_{K}$
12	11 4	atbase	$⊢ Q \in A \to Q \in {Base}_{K}$
13	10 12	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to Q \in {Base}_{K}$
14		simp2rl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to R \in A$
15		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (K \in HL \land W \in H)$
16		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
17		simp2l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \neq Q$
18	1 2 3 4 5 6	cdleme0a	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to U \in A$
19	15 16 10 17 18	syl112anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to U \in A$
20	11 2 4	hlatjcl	$⊢ (K \in HL \land R \in A \land U \in A) \to R \underset{˙}{\lor} U \in {Base}_{K}$
21	8 14 19 20	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to R \underset{˙}{\lor} U \in {Base}_{K}$
22	11 1 2	latlej1	$⊢ (K \in Lat \land Q \in {Base}_{K} \land R \underset{˙}{\lor} U \in {Base}_{K}) \to Q \underset{˙}{\leq} (Q \underset{˙}{\lor} (R \underset{˙}{\lor} U))$
23	9 13 21 22	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to Q \underset{˙}{\leq} (Q \underset{˙}{\lor} (R \underset{˙}{\lor} U))$
24		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \in A$
25	11 4	atbase	$⊢ P \in A \to P \in {Base}_{K}$
26	24 25	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \in {Base}_{K}$
27	11 4	atbase	$⊢ R \in A \to R \in {Base}_{K}$
28	14 27	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to R \in {Base}_{K}$
29	11 2	latjcl	$⊢ (K \in Lat \land P \in {Base}_{K} \land R \in {Base}_{K}) \to P \underset{˙}{\lor} R \in {Base}_{K}$
30	9 26 28 29	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} R \in {Base}_{K}$
31		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to W \in H$
32	11 5	lhpbase	$⊢ W \in H \to W \in {Base}_{K}$
33	31 32	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to W \in {Base}_{K}$
34	11 3	latmcl	$⊢ (K \in Lat \land P \underset{˙}{\lor} R \in {Base}_{K} \land W \in {Base}_{K}) \to (P \underset{˙}{\lor} R) \underset{˙}{\land} W \in {Base}_{K}$
35	9 30 33 34	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} R) \underset{˙}{\land} W \in {Base}_{K}$
36	11 2	latjcl	$⊢ (K \in Lat \land P \underset{˙}{\lor} R \in {Base}_{K} \land Q \in {Base}_{K}) \to (P \underset{˙}{\lor} R) \underset{˙}{\lor} Q \in {Base}_{K}$
37	9 30 13 36	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} R) \underset{˙}{\lor} Q \in {Base}_{K}$
38	11 1 3	latmle1	$⊢ (K \in Lat \land P \underset{˙}{\lor} R \in {Base}_{K} \land W \in {Base}_{K}) \to ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) \underset{˙}{\leq} (P \underset{˙}{\lor} R)$
39	9 30 33 38	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) \underset{˙}{\leq} (P \underset{˙}{\lor} R)$
40	11 1 2	latlej1	$⊢ (K \in Lat \land P \underset{˙}{\lor} R \in {Base}_{K} \land Q \in {Base}_{K}) \to (P \underset{˙}{\lor} R) \underset{˙}{\leq} ((P \underset{˙}{\lor} R) \underset{˙}{\lor} Q)$
41	9 30 13 40	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} R) \underset{˙}{\leq} ((P \underset{˙}{\lor} R) \underset{˙}{\lor} Q)$
42	11 1 9 35 30 37 39 41	lattrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) \underset{˙}{\leq} ((P \underset{˙}{\lor} R) \underset{˙}{\lor} Q)$
43	6	oveq2i	$⊢ Q \underset{˙}{\lor} U = Q \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W)$
44	11 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
45	8 24 10 44	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
46	1 2 4	hlatlej2	$⊢ (K \in HL \land P \in A \land Q \in A) \to Q \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
47	8 24 10 46	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to Q \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
48	11 1 2 3 4	atmod3i1	$⊢ (K \in HL \land (Q \in A \land P \underset{˙}{\lor} Q \in {Base}_{K} \land W \in {Base}_{K}) \land Q \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to Q \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W) = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Q \underset{˙}{\lor} W)$
49	8 10 45 33 47 48	syl131anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to Q \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W) = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Q \underset{˙}{\lor} W)$
50		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
51		eqid	$⊢ 1. (K) = 1. (K)$
52	1 2 51 4 5	lhpjat2	$⊢ ((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to Q \underset{˙}{\lor} W = 1. (K)$
53	15 50 52	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to Q \underset{˙}{\lor} W = 1. (K)$
54	53	oveq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\land} (Q \underset{˙}{\lor} W) = (P \underset{˙}{\lor} Q) \underset{˙}{\land} 1. (K)$
55		hlol	$⊢ K \in HL \to K \in OL$
56	8 55	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to K \in OL$
57	11 3 51	olm11	$⊢ (K \in OL \land P \underset{˙}{\lor} Q \in {Base}_{K}) \to (P \underset{˙}{\lor} Q) \underset{˙}{\land} 1. (K) = P \underset{˙}{\lor} Q$
58	56 45 57	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\land} 1. (K) = P \underset{˙}{\lor} Q$
59	49 54 58	3eqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to Q \underset{˙}{\lor} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W) = P \underset{˙}{\lor} Q$
60	43 59	eqtrid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to Q \underset{˙}{\lor} U = P \underset{˙}{\lor} Q$
61	60	oveq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to R \underset{˙}{\lor} (Q \underset{˙}{\lor} U) = R \underset{˙}{\lor} (P \underset{˙}{\lor} Q)$
62	2 4	hlatj12	$⊢ (K \in HL \land (Q \in A \land R \in A \land U \in A)) \to Q \underset{˙}{\lor} (R \underset{˙}{\lor} U) = R \underset{˙}{\lor} (Q \underset{˙}{\lor} U)$
63	8 10 14 19 62	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to Q \underset{˙}{\lor} (R \underset{˙}{\lor} U) = R \underset{˙}{\lor} (Q \underset{˙}{\lor} U)$
64	2 4	hlatjcom	$⊢ (K \in HL \land P \in A \land R \in A) \to P \underset{˙}{\lor} R = R \underset{˙}{\lor} P$
65	8 24 14 64	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} R = R \underset{˙}{\lor} P$
66	65	oveq1d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} R) \underset{˙}{\lor} Q = (R \underset{˙}{\lor} P) \underset{˙}{\lor} Q$
67	2 4	hlatjass	$⊢ (K \in HL \land (R \in A \land P \in A \land Q \in A)) \to (R \underset{˙}{\lor} P) \underset{˙}{\lor} Q = R \underset{˙}{\lor} (P \underset{˙}{\lor} Q)$
68	8 14 24 10 67	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (R \underset{˙}{\lor} P) \underset{˙}{\lor} Q = R \underset{˙}{\lor} (P \underset{˙}{\lor} Q)$
69	66 68	eqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} R) \underset{˙}{\lor} Q = R \underset{˙}{\lor} (P \underset{˙}{\lor} Q)$
70	61 63 69	3eqtr4rd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} R) \underset{˙}{\lor} Q = Q \underset{˙}{\lor} (R \underset{˙}{\lor} U)$
71	42 70	breqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) \underset{˙}{\leq} (Q \underset{˙}{\lor} (R \underset{˙}{\lor} U))$
72	11 2	latjcl	$⊢ (K \in Lat \land Q \in {Base}_{K} \land R \underset{˙}{\lor} U \in {Base}_{K}) \to Q \underset{˙}{\lor} (R \underset{˙}{\lor} U) \in {Base}_{K}$
73	9 13 21 72	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to Q \underset{˙}{\lor} (R \underset{˙}{\lor} U) \in {Base}_{K}$
74	11 1 2	latjle12	$⊢ (K \in Lat \land (Q \in {Base}_{K} \land (P \underset{˙}{\lor} R) \underset{˙}{\land} W \in {Base}_{K} \land Q \underset{˙}{\lor} (R \underset{˙}{\lor} U) \in {Base}_{K})) \to ((Q \underset{˙}{\leq} (Q \underset{˙}{\lor} (R \underset{˙}{\lor} U)) \land ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) \underset{˙}{\leq} (Q \underset{˙}{\lor} (R \underset{˙}{\lor} U))) \leftrightarrow (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)) \underset{˙}{\leq} (Q \underset{˙}{\lor} (R \underset{˙}{\lor} U)))$
75	9 13 35 73 74	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ((Q \underset{˙}{\leq} (Q \underset{˙}{\lor} (R \underset{˙}{\lor} U)) \land ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) \underset{˙}{\leq} (Q \underset{˙}{\lor} (R \underset{˙}{\lor} U))) \leftrightarrow (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)) \underset{˙}{\leq} (Q \underset{˙}{\lor} (R \underset{˙}{\lor} U)))$
76	23 71 75	mpbi2and	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)) \underset{˙}{\leq} (Q \underset{˙}{\lor} (R \underset{˙}{\lor} U))$

Metamath Proof Explorer

Theorem cdleme35b

Proof