cdleme16b

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph on p. 114, first part of 3rd sentence. F and G represent f(s) and f(t) respectively. It is unclear how this follows from s \/ u =/= t \/ u, as the authors state, and we used a different proof. (Note: the antecedent -. T .<_ ( P .\/ Q ) is not used.) (Contributed by NM, 11-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	$⊢ G = (T \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} T) \underset{˙}{\land} W))$
Assertion	cdleme16b	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to F \neq G$

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		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (K \in HL \land W \in H)$
10		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
11		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
12		simp21	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
13		simp23l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to P \neq Q$
14		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
15		eqid	$⊢ (P \underset{˙}{\lor} S) \underset{˙}{\land} W = (P \underset{˙}{\lor} S) \underset{˙}{\land} W$
16	1 2 3 4 5 6 7 15	cdleme3g	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \neq U$
17	9 10 11 12 13 14 16	syl132anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to F \neq U$
18		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to K \in HL$
19	18	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to K \in Lat$
20		simp21l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to S \in A$
21	1 2 3 4 5 6 7	cdleme3fa	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \in A$
22	9 10 11 12 13 14 21	syl132anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to F \in A$
23		eqid	$⊢ {Base}_{K} = {Base}_{K}$
24	23 2 4	hlatjcl	$⊢ (K \in HL \land S \in A \land F \in A) \to S \underset{˙}{\lor} F \in {Base}_{K}$
25	18 20 22 24	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to S \underset{˙}{\lor} F \in {Base}_{K}$
26		simp22l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to T \in A$
27	23 4	atbase	$⊢ T \in A \to T \in {Base}_{K}$
28	26 27	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to T \in {Base}_{K}$
29	23 3	latmcl	$⊢ (K \in Lat \land S \underset{˙}{\lor} F \in {Base}_{K} \land T \in {Base}_{K}) \to (S \underset{˙}{\lor} F) \underset{˙}{\land} T \in {Base}_{K}$
30	19 25 28 29	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (S \underset{˙}{\lor} F) \underset{˙}{\land} T \in {Base}_{K}$
31	23 4	atbase	$⊢ F \in A \to F \in {Base}_{K}$
32	22 31	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to F \in {Base}_{K}$
33	23 1 2	latlej2	$⊢ (K \in Lat \land (S \underset{˙}{\lor} F) \underset{˙}{\land} T \in {Base}_{K} \land F \in {Base}_{K}) \to F \underset{˙}{\leq} (((S \underset{˙}{\lor} F) \underset{˙}{\land} T) \underset{˙}{\lor} F)$
34	19 30 32 33	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to F \underset{˙}{\leq} (((S \underset{˙}{\lor} F) \underset{˙}{\land} T) \underset{˙}{\lor} F)$
35	34	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \land F = G) \to F \underset{˙}{\leq} (((S \underset{˙}{\lor} F) \underset{˙}{\land} T) \underset{˙}{\lor} F)$
36	1 2 4	hlatlej2	$⊢ (K \in HL \land S \in A \land F \in A) \to F \underset{˙}{\leq} (S \underset{˙}{\lor} F)$
37	18 20 22 36	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to F \underset{˙}{\leq} (S \underset{˙}{\lor} F)$
38	23 1 2 3 4	atmod2i1	$⊢ (K \in HL \land (F \in A \land S \underset{˙}{\lor} F \in {Base}_{K} \land T \in {Base}_{K}) \land F \underset{˙}{\leq} (S \underset{˙}{\lor} F)) \to ((S \underset{˙}{\lor} F) \underset{˙}{\land} T) \underset{˙}{\lor} F = (S \underset{˙}{\lor} F) \underset{˙}{\land} (T \underset{˙}{\lor} F)$
39	18 22 25 28 37 38	syl131anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to ((S \underset{˙}{\lor} F) \underset{˙}{\land} T) \underset{˙}{\lor} F = (S \underset{˙}{\lor} F) \underset{˙}{\land} (T \underset{˙}{\lor} F)$
40		oveq2	$⊢ F = G \to T \underset{˙}{\lor} F = T \underset{˙}{\lor} G$
41	40	oveq2d	$⊢ F = G \to (S \underset{˙}{\lor} F) \underset{˙}{\land} (T \underset{˙}{\lor} F) = (S \underset{˙}{\lor} F) \underset{˙}{\land} (T \underset{˙}{\lor} G)$
42	39 41	sylan9eq	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \land F = G) \to ((S \underset{˙}{\lor} F) \underset{˙}{\land} T) \underset{˙}{\lor} F = (S \underset{˙}{\lor} F) \underset{˙}{\land} (T \underset{˙}{\lor} G)$
43		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to W \in H$
44		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to Q \in A$
45		simp22	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (T \in A \land \neg T \underset{˙}{\leq} W)$
46		simp23r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to S \neq T$
47		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)$
48	46 47	jca	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))$
49	1 2 3 4 5 6 7 8	cdleme12	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A \land P \neq Q) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (S \neq T \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T)))) \to (S \underset{˙}{\lor} F) \underset{˙}{\land} (T \underset{˙}{\lor} G) = U$
50	18 43 10 44 13 12 45 48 49	syl233anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (S \underset{˙}{\lor} F) \underset{˙}{\land} (T \underset{˙}{\lor} G) = U$
51	50	adantr	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \land F = G) \to (S \underset{˙}{\lor} F) \underset{˙}{\land} (T \underset{˙}{\lor} G) = U$
52	42 51	eqtrd	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \land F = G) \to ((S \underset{˙}{\lor} F) \underset{˙}{\land} T) \underset{˙}{\lor} F = U$
53	35 52	breqtrd	$⊢ ((((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \land F = G) \to F \underset{˙}{\leq} U$
54	53	ex	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (F = G \to F \underset{˙}{\leq} U)$
55		hlatl	$⊢ K \in HL \to K \in AtLat$
56	18 55	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to K \in AtLat$
57		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to P \in A$
58		simp12r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to \neg P \underset{˙}{\leq} W$
59	1 2 3 4 5 6	lhpat2	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to U \in A$
60	18 43 57 58 44 13 59	syl222anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to U \in A$
61	1 4	atcmp	$⊢ (K \in AtLat \land F \in A \land U \in A) \to (F \underset{˙}{\leq} U \leftrightarrow F = U)$
62	56 22 60 61	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (F \underset{˙}{\leq} U \leftrightarrow F = U)$
63	54 62	sylibd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (F = G \to F = U)$
64	63	necon3d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to (F \neq U \to F \neq G)$
65	17 64	mpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (P \neq Q \land S \neq T)) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg U \underset{˙}{\leq} (S \underset{˙}{\lor} T))) \to F \neq G$

Metamath Proof Explorer

Theorem cdleme16b

Proof