cdleme11

Description: Part of proof of Lemma E in Crawley p. 113, 1st sentence of 3rd paragraph on p. 114. F and G represent f(s) and f(t) respectively. Their proof provides no details of our cdleme11a through cdleme11 , so there may be a simpler proof that we have overlooked. (Contributed by NM, 15-Jun-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	$⊢ G = (T \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} T) \underset{˙}{\land} W))$
Assertion	cdleme11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to F \underset{˙}{\lor} G = S \underset{˙}{\lor} T$

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		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to K \in HL$
10	9	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to K \in Lat$
11		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (K \in HL \land W \in H)$
12		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to P \in A$
13		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to Q \in A$
14		eqid	$⊢ {Base}_{K} = {Base}_{K}$
15	1 2 3 4 5 6 14	cdleme0aa	$⊢ ((K \in HL \land W \in H) \land P \in A \land Q \in A) \to U \in {Base}_{K}$
16	11 12 13 15	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to U \in {Base}_{K}$
17	14 2	latjidm	$⊢ (K \in Lat \land U \in {Base}_{K}) \to U \underset{˙}{\lor} U = U$
18	10 16 17	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to U \underset{˙}{\lor} U = U$
19	18	oveq2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (S \underset{˙}{\lor} T) \underset{˙}{\lor} (U \underset{˙}{\lor} U) = (S \underset{˙}{\lor} T) \underset{˙}{\lor} U$
20		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to U \underset{˙}{\leq} (S \underset{˙}{\lor} T)$
21		simp21l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to S \in A$
22	14 4	atbase	$⊢ S \in A \to S \in {Base}_{K}$
23	21 22	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to S \in {Base}_{K}$
24		simp22l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to T \in A$
25	14 4	atbase	$⊢ T \in A \to T \in {Base}_{K}$
26	24 25	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to T \in {Base}_{K}$
27	14 2	latjcl	$⊢ (K \in Lat \land S \in {Base}_{K} \land T \in {Base}_{K}) \to S \underset{˙}{\lor} T \in {Base}_{K}$
28	10 23 26 27	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to S \underset{˙}{\lor} T \in {Base}_{K}$
29	14 1 2	latleeqj2	$⊢ (K \in Lat \land U \in {Base}_{K} \land S \underset{˙}{\lor} T \in {Base}_{K}) \to (U \underset{˙}{\leq} (S \underset{˙}{\lor} T) \leftrightarrow (S \underset{˙}{\lor} T) \underset{˙}{\lor} U = S \underset{˙}{\lor} T)$
30	10 16 28 29	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (U \underset{˙}{\leq} (S \underset{˙}{\lor} T) \leftrightarrow (S \underset{˙}{\lor} T) \underset{˙}{\lor} U = S \underset{˙}{\lor} T)$
31	20 30	mpbid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (S \underset{˙}{\lor} T) \underset{˙}{\lor} U = S \underset{˙}{\lor} T$
32	19 31	eqtr2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to S \underset{˙}{\lor} T = (S \underset{˙}{\lor} T) \underset{˙}{\lor} (U \underset{˙}{\lor} U)$
33		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
34	1 2 3 4 5 6 7	cdleme1	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (S \in A \land \neg S \underset{˙}{\leq} W))) \to S \underset{˙}{\lor} F = S \underset{˙}{\lor} U$
35	11 12 13 33 34	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to S \underset{˙}{\lor} F = S \underset{˙}{\lor} U$
36		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (T \in A \land \neg T \underset{˙}{\leq} W)$
37	1 2 3 4 5 6 8	cdleme1	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (T \in A \land \neg T \underset{˙}{\leq} W))) \to T \underset{˙}{\lor} G = T \underset{˙}{\lor} U$
38	11 12 13 36 37	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to T \underset{˙}{\lor} G = T \underset{˙}{\lor} U$
39	35 38	oveq12d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (S \underset{˙}{\lor} F) \underset{˙}{\lor} (T \underset{˙}{\lor} G) = (S \underset{˙}{\lor} U) \underset{˙}{\lor} (T \underset{˙}{\lor} U)$
40	14 2	latj4	$⊢ (K \in Lat \land (S \in {Base}_{K} \land T \in {Base}_{K}) \land (U \in {Base}_{K} \land U \in {Base}_{K})) \to (S \underset{˙}{\lor} T) \underset{˙}{\lor} (U \underset{˙}{\lor} U) = (S \underset{˙}{\lor} U) \underset{˙}{\lor} (T \underset{˙}{\lor} U)$
41	10 23 26 16 16 40	syl122anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (S \underset{˙}{\lor} T) \underset{˙}{\lor} (U \underset{˙}{\lor} U) = (S \underset{˙}{\lor} U) \underset{˙}{\lor} (T \underset{˙}{\lor} U)$
42	39 41	eqtr4d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (S \underset{˙}{\lor} F) \underset{˙}{\lor} (T \underset{˙}{\lor} G) = (S \underset{˙}{\lor} T) \underset{˙}{\lor} (U \underset{˙}{\lor} U)$
43	32 42	eqtr4d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to S \underset{˙}{\lor} T = (S \underset{˙}{\lor} F) \underset{˙}{\lor} (T \underset{˙}{\lor} G)$
44	1 2 3 4 5 6 7 14	cdleme1b	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land S \in A)) \to F \in {Base}_{K}$
45	11 12 13 21 44	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to F \in {Base}_{K}$
46	1 2 3 4 5 6 8 14	cdleme1b	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land T \in A)) \to G \in {Base}_{K}$
47	11 12 13 24 46	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to G \in {Base}_{K}$
48	14 2	latj4	$⊢ (K \in Lat \land (S \in {Base}_{K} \land F \in {Base}_{K}) \land (T \in {Base}_{K} \land G \in {Base}_{K})) \to (S \underset{˙}{\lor} F) \underset{˙}{\lor} (T \underset{˙}{\lor} G) = (S \underset{˙}{\lor} T) \underset{˙}{\lor} (F \underset{˙}{\lor} G)$
49	10 23 45 26 47 48	syl122anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (S \underset{˙}{\lor} F) \underset{˙}{\lor} (T \underset{˙}{\lor} G) = (S \underset{˙}{\lor} T) \underset{˙}{\lor} (F \underset{˙}{\lor} G)$
50	43 49	eqtr2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (S \underset{˙}{\lor} T) \underset{˙}{\lor} (F \underset{˙}{\lor} G) = S \underset{˙}{\lor} T$
51	14 2	latjcl	$⊢ (K \in Lat \land F \in {Base}_{K} \land G \in {Base}_{K}) \to F \underset{˙}{\lor} G \in {Base}_{K}$
52	10 45 47 51	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to F \underset{˙}{\lor} G \in {Base}_{K}$
53	14 1 2	latleeqj2	$⊢ (K \in Lat \land F \underset{˙}{\lor} G \in {Base}_{K} \land S \underset{˙}{\lor} T \in {Base}_{K}) \to ((F \underset{˙}{\lor} G) \underset{˙}{\leq} (S \underset{˙}{\lor} T) \leftrightarrow (S \underset{˙}{\lor} T) \underset{˙}{\lor} (F \underset{˙}{\lor} G) = S \underset{˙}{\lor} T)$
54	10 52 28 53	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to ((F \underset{˙}{\lor} G) \underset{˙}{\leq} (S \underset{˙}{\lor} T) \leftrightarrow (S \underset{˙}{\lor} T) \underset{˙}{\lor} (F \underset{˙}{\lor} G) = S \underset{˙}{\lor} T)$
55	50 54	mpbird	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (F \underset{˙}{\lor} G) \underset{˙}{\leq} (S \underset{˙}{\lor} T)$
56		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
57		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
58		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to P \neq Q$
59		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
60	1 2 3 4 5 6 7	cdleme3fa	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \in A$
61	11 56 57 33 58 59 60	syl132anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to F \in A$
62		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
63	1 2 3 4 5 6 8	cdleme3fa	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W)) \land (P \neq Q \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G \in A$
64	11 56 57 36 58 62 63	syl132anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to G \in A$
65	1 2 3 4 5 6 7 8	cdleme11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to F \neq G$
66	1 2 4	ps-1	$⊢ (K \in HL \land (F \in A \land G \in A \land F \neq G) \land (S \in A \land T \in A)) \to ((F \underset{˙}{\lor} G) \underset{˙}{\leq} (S \underset{˙}{\lor} T) \leftrightarrow F \underset{˙}{\lor} G = S \underset{˙}{\lor} T)$
67	9 61 64 65 21 24 66	syl132anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to ((F \underset{˙}{\lor} G) \underset{˙}{\leq} (S \underset{˙}{\lor} T) \leftrightarrow F \underset{˙}{\lor} G = S \underset{˙}{\lor} T)$
68	55 67	mpbid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to F \underset{˙}{\lor} G = S \underset{˙}{\lor} T$

Metamath Proof Explorer

Theorem cdleme11

Proof