cdleme15b

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph on p. 114, showing, in their notation, (p \/ s_1) /\ (q \/ s_1)=s_1. We represent s_1 with C . (Contributed by NM, 10-Oct-2012)

	Ref	Expression
Hypotheses	cdleme12.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme12.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme12.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme12.a	$⊢ A = Atoms (K)$
	cdleme12.h	$⊢ H = LHyp (K)$
	cdleme12.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme12.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
	cdleme12.g	$⊢ G = (T \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} T) \underset{˙}{\land} W))$
	cdleme15.c	$⊢ C = (P \underset{˙}{\lor} S) \underset{˙}{\land} W$
	cdleme15.x	$⊢ X = (P \underset{˙}{\lor} T) \underset{˙}{\land} W$
Assertion	cdleme15b	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (P \underset{˙}{\lor} C) \underset{˙}{\land} (Q \underset{˙}{\lor} C) = C$

Proof

Step	Hyp	Ref	Expression
1		cdleme12.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme12.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme12.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme12.a	$⊢ A = Atoms (K)$
5		cdleme12.h	$⊢ H = LHyp (K)$
6		cdleme12.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme12.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
8		cdleme12.g	$⊢ G = (T \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} T) \underset{˙}{\land} W))$
9		cdleme15.c	$⊢ C = (P \underset{˙}{\lor} S) \underset{˙}{\land} W$
10		cdleme15.x	$⊢ X = (P \underset{˙}{\lor} T) \underset{˙}{\land} W$
11	9	oveq2i	$⊢ P \underset{˙}{\lor} C = P \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W)$
12		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to K \in HL$
13		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to P \in A$
14		simp21l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to S \in A$
15		eqid	$⊢ {Base}_{K} = {Base}_{K}$
16	15 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land S \in A) \to P \underset{˙}{\lor} S \in {Base}_{K}$
17	12 13 14 16	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to P \underset{˙}{\lor} S \in {Base}_{K}$
18		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to W \in H$
19	15 5	lhpbase	$⊢ W \in H \to W \in {Base}_{K}$
20	18 19	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to W \in {Base}_{K}$
21	1 2 4	hlatlej1	$⊢ (K \in HL \land P \in A \land S \in A) \to P \underset{˙}{\leq} (P \underset{˙}{\lor} S)$
22	12 13 14 21	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to P \underset{˙}{\leq} (P \underset{˙}{\lor} S)$
23	15 1 2 3 4	atmod3i1	$⊢ (K \in HL \land (P \in A \land P \underset{˙}{\lor} S \in {Base}_{K} \land W \in {Base}_{K}) \land P \underset{˙}{\leq} (P \underset{˙}{\lor} S)) \to P \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W) = (P \underset{˙}{\lor} S) \underset{˙}{\land} (P \underset{˙}{\lor} W)$
24	12 13 17 20 22 23	syl131anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to P \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W) = (P \underset{˙}{\lor} S) \underset{˙}{\land} (P \underset{˙}{\lor} W)$
25	11 24	syl5eq	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to P \underset{˙}{\lor} C = (P \underset{˙}{\lor} S) \underset{˙}{\land} (P \underset{˙}{\lor} W)$
26	25	oveq1d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (P \underset{˙}{\lor} C) \underset{˙}{\land} Q = ((P \underset{˙}{\lor} S) \underset{˙}{\land} (P \underset{˙}{\lor} W)) \underset{˙}{\land} Q$
27		hlol	$⊢ K \in HL \to K \in OL$
28	12 27	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to K \in OL$
29	12	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to K \in Lat$
30	15 4	atbase	$⊢ P \in A \to P \in {Base}_{K}$
31	13 30	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to P \in {Base}_{K}$
32	15 2	latjcl	$⊢ (K \in Lat \land P \in {Base}_{K} \land W \in {Base}_{K}) \to P \underset{˙}{\lor} W \in {Base}_{K}$
33	29 31 20 32	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to P \underset{˙}{\lor} W \in {Base}_{K}$
34		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to Q \in A$
35	15 4	atbase	$⊢ Q \in A \to Q \in {Base}_{K}$
36	34 35	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to Q \in {Base}_{K}$
37	15 3	latmrot	$⊢ (K \in OL \land (P \underset{˙}{\lor} S \in {Base}_{K} \land P \underset{˙}{\lor} W \in {Base}_{K} \land Q \in {Base}_{K})) \to ((P \underset{˙}{\lor} S) \underset{˙}{\land} (P \underset{˙}{\lor} W)) \underset{˙}{\land} Q = (Q \underset{˙}{\land} (P \underset{˙}{\lor} S)) \underset{˙}{\land} (P \underset{˙}{\lor} W)$
38	28 17 33 36 37	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to ((P \underset{˙}{\lor} S) \underset{˙}{\land} (P \underset{˙}{\lor} W)) \underset{˙}{\land} Q = (Q \underset{˙}{\land} (P \underset{˙}{\lor} S)) \underset{˙}{\land} (P \underset{˙}{\lor} W)$
39		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
40		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to P \neq Q$
41	40	necomd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to Q \neq P$
42	1 2 4	hlatexch1	$⊢ (K \in HL \land (Q \in A \land S \in A \land P \in A) \land Q \neq P) \to (Q \underset{˙}{\leq} (P \underset{˙}{\lor} S) \to S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
43	12 34 14 13 41 42	syl131anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (Q \underset{˙}{\leq} (P \underset{˙}{\lor} S) \to S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
44	39 43	mtod	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to \neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} S)$
45		hlatl	$⊢ K \in HL \to K \in AtLat$
46	12 45	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to K \in AtLat$
47		eqid	$⊢ 0. (K) = 0. (K)$
48	15 1 3 47 4	atnle	$⊢ (K \in AtLat \land Q \in A \land P \underset{˙}{\lor} S \in {Base}_{K}) \to (\neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} S) \leftrightarrow Q \underset{˙}{\land} (P \underset{˙}{\lor} S) = 0. (K))$
49	46 34 17 48	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (\neg Q \underset{˙}{\leq} (P \underset{˙}{\lor} S) \leftrightarrow Q \underset{˙}{\land} (P \underset{˙}{\lor} S) = 0. (K))$
50	44 49	mpbid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to Q \underset{˙}{\land} (P \underset{˙}{\lor} S) = 0. (K)$
51	50	oveq1d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (Q \underset{˙}{\land} (P \underset{˙}{\lor} S)) \underset{˙}{\land} (P \underset{˙}{\lor} W) = 0. (K) \underset{˙}{\land} (P \underset{˙}{\lor} W)$
52	15 3 47	olm02	$⊢ (K \in OL \land P \underset{˙}{\lor} W \in {Base}_{K}) \to 0. (K) \underset{˙}{\land} (P \underset{˙}{\lor} W) = 0. (K)$
53	28 33 52	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to 0. (K) \underset{˙}{\land} (P \underset{˙}{\lor} W) = 0. (K)$
54	38 51 53	3eqtrrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to 0. (K) = ((P \underset{˙}{\lor} S) \underset{˙}{\land} (P \underset{˙}{\lor} W)) \underset{˙}{\land} Q$
55	26 54	eqtr4d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (P \underset{˙}{\lor} C) \underset{˙}{\land} Q = 0. (K)$
56	55	oveq1d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to ((P \underset{˙}{\lor} C) \underset{˙}{\land} Q) \underset{˙}{\lor} C = 0. (K) \underset{˙}{\lor} C$
57	15 2 3 4 5 9	cdleme9b	$⊢ (K \in HL \land (P \in A \land S \in A \land W \in H)) \to C \in {Base}_{K}$
58	12 13 14 18 57	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to C \in {Base}_{K}$
59	15 2	latjcl	$⊢ (K \in Lat \land P \in {Base}_{K} \land C \in {Base}_{K}) \to P \underset{˙}{\lor} C \in {Base}_{K}$
60	29 31 58 59	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to P \underset{˙}{\lor} C \in {Base}_{K}$
61	15 1 2	latlej2	$⊢ (K \in Lat \land P \in {Base}_{K} \land C \in {Base}_{K}) \to C \underset{˙}{\leq} (P \underset{˙}{\lor} C)$
62	29 31 58 61	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to C \underset{˙}{\leq} (P \underset{˙}{\lor} C)$
63	15 1 2 3 4	atmod2i2	$⊢ (K \in HL \land (Q \in A \land P \underset{˙}{\lor} C \in {Base}_{K} \land C \in {Base}_{K}) \land C \underset{˙}{\leq} (P \underset{˙}{\lor} C)) \to ((P \underset{˙}{\lor} C) \underset{˙}{\land} Q) \underset{˙}{\lor} C = (P \underset{˙}{\lor} C) \underset{˙}{\land} (Q \underset{˙}{\lor} C)$
64	12 34 60 58 62 63	syl131anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to ((P \underset{˙}{\lor} C) \underset{˙}{\land} Q) \underset{˙}{\lor} C = (P \underset{˙}{\lor} C) \underset{˙}{\land} (Q \underset{˙}{\lor} C)$
65	15 2 47	olj02	$⊢ (K \in OL \land C \in {Base}_{K}) \to 0. (K) \underset{˙}{\lor} C = C$
66	28 58 65	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to 0. (K) \underset{˙}{\lor} C = C$
67	56 64 66	3eqtr3d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (P \underset{˙}{\lor} C) \underset{˙}{\land} (Q \underset{˙}{\lor} C) = C$

Metamath Proof Explorer

Theorem cdleme15b

Proof