cdleme22f2

Description: Part of proof of Lemma E in Crawley p. 113. cdleme22f with s and t swapped (this case is not mentioned by them). If s <_ t \/ v, then f(s) <_ f_s(t) \/ v. (Contributed by NM, 7-Dec-2012)

	Ref	Expression
Hypotheses	cdleme22.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme22.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme22.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme22.a	$⊢ A = Atoms (K)$
	cdleme22.h	$⊢ H = LHyp (K)$
	cdleme22f2.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme22f2.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
	cdleme22f2.n	$⊢ N = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((T \underset{˙}{\lor} S) \underset{˙}{\land} W))$
Assertion	cdleme22f2	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to F \underset{˙}{\leq} (N \underset{˙}{\lor} V)$

Proof

Step	Hyp	Ref	Expression
1		cdleme22.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme22.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme22.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme22.a	$⊢ A = Atoms (K)$
5		cdleme22.h	$⊢ H = LHyp (K)$
6		cdleme22f2.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme22f2.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
8		cdleme22f2.n	$⊢ N = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((T \underset{˙}{\lor} S) \underset{˙}{\land} W))$
9		simp11	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (K \in HL \land W \in H)$
10		simp2l	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
11		simp2r	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
12	9 10 11	3jca	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W))$
13		simp12	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (T \in A \land \neg T \underset{˙}{\leq} W)$
14		simp31l	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to S \in A$
15		simp33	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (V \in A \land V \underset{˙}{\leq} W)$
16		simp32l	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to S \neq T$
17	16	necomd	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to T \neq S$
18		simp32r	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to S \underset{˙}{\leq} (T \underset{˙}{\lor} V)$
19		simp11l	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to K \in HL$
20		hlcvl	$⊢ K \in HL \to K \in CvLat$
21	19 20	syl	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to K \in CvLat$
22		simp12l	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to T \in A$
23		simp33l	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to V \in A$
24		simp33r	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to V \underset{˙}{\leq} W$
25		simp31r	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to \neg S \underset{˙}{\leq} W$
26		nbrne2	$⊢ (V \underset{˙}{\leq} W \land \neg S \underset{˙}{\leq} W) \to V \neq S$
27	26	necomd	$⊢ (V \underset{˙}{\leq} W \land \neg S \underset{˙}{\leq} W) \to S \neq V$
28	24 25 27	syl2anc	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to S \neq V$
29	1 2 4	cvlatexch2	$⊢ (K \in CvLat \land (S \in A \land T \in A \land V \in A) \land S \neq V) \to (S \underset{˙}{\leq} (T \underset{˙}{\lor} V) \to T \underset{˙}{\leq} (S \underset{˙}{\lor} V))$
30	21 14 22 23 28 29	syl131anc	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (S \underset{˙}{\leq} (T \underset{˙}{\lor} V) \to T \underset{˙}{\leq} (S \underset{˙}{\lor} V))$
31	18 30	mpd	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to T \underset{˙}{\leq} (S \underset{˙}{\lor} V)$
32	1 2 3 4 5 6 7 8	cdleme22f	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((T \in A \land \neg T \underset{˙}{\leq} W) \land S \in A \land (V \in A \land V \underset{˙}{\leq} W)) \land (T \neq S \land T \underset{˙}{\leq} (S \underset{˙}{\lor} V))) \to N \underset{˙}{\leq} (F \underset{˙}{\lor} V)$
33	12 13 14 15 17 31 32	syl132anc	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to N \underset{˙}{\leq} (F \underset{˙}{\lor} V)$
34		simp31	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
35		simp133	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to P \neq Q$
36		simp132	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to T \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
37		simp131	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
38	1 2 3 4 5 6 7 8	cdleme7ga	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((T \in A \land \neg T \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to N \in A$
39	12 13 34 35 36 37 38	syl123anc	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to N \in A$
40	1 2 3 4 5 6 7	cdleme3fa	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \in A$
41	9 10 11 34 35 37 40	syl132anc	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to F \in A$
42	1 2 3 4 5 6 7 8	cdleme7	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((T \in A \land \neg T \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg N \underset{˙}{\leq} W$
43	12 13 34 35 36 37 42	syl123anc	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to \neg N \underset{˙}{\leq} W$
44		nbrne2	$⊢ (V \underset{˙}{\leq} W \land \neg N \underset{˙}{\leq} W) \to V \neq N$
45	44	necomd	$⊢ (V \underset{˙}{\leq} W \land \neg N \underset{˙}{\leq} W) \to N \neq V$
46	24 43 45	syl2anc	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to N \neq V$
47	1 2 4	cvlatexch2	$⊢ (K \in CvLat \land (N \in A \land F \in A \land V \in A) \land N \neq V) \to (N \underset{˙}{\leq} (F \underset{˙}{\lor} V) \to F \underset{˙}{\leq} (N \underset{˙}{\lor} V))$
48	21 39 41 23 46 47	syl131anc	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to (N \underset{˙}{\leq} (F \underset{˙}{\lor} V) \to F \underset{˙}{\leq} (N \underset{˙}{\lor} V))$
49	33 48	mpd	$⊢ (((K \in HL \land W \in H) \land (T \in A \land \neg T \underset{˙}{\leq} W) \land (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land T \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land P \neq Q)) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land (S \neq T \land S \underset{˙}{\leq} (T \underset{˙}{\lor} V)) \land (V \in A \land V \underset{˙}{\leq} W))) \to F \underset{˙}{\leq} (N \underset{˙}{\lor} V)$

Metamath Proof Explorer

Theorem cdleme22f2

Proof