cdleme7ga

Description: Part of proof of Lemma E in Crawley p. 113. See cdleme7 . (Contributed by NM, 8-Jun-2012)

	Ref	Expression
Hypotheses	cdleme4.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme4.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme4.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme4.a	$⊢ A = Atoms (K)$
	cdleme4.h	$⊢ H = LHyp (K)$
	cdleme4.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme4.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
	cdleme4.g	$⊢ G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$
Assertion	cdleme7ga	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G \in A$

Proof

Step	Hyp	Ref	Expression
1		cdleme4.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme4.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme4.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme4.a	$⊢ A = Atoms (K)$
5		cdleme4.h	$⊢ H = LHyp (K)$
6		cdleme4.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme4.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
8		cdleme4.g	$⊢ G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$
9		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in HL$
10		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \in A$
11		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \in A$
12		eqid	$⊢ {Base}_{K} = {Base}_{K}$
13	12 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
14	9 10 11 13	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
15		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (K \in HL \land W \in H)$
16		simp12	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \in A \land \neg P \underset{˙}{\leq} W)$
17		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
18		simp2r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (S \in A \land \neg S \underset{˙}{\leq} W)$
19		simp31	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \neq Q$
20		simp33	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
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	15 16 17 18 19 20 21	syl132anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \in A$
23		simp2l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \in A \land \neg R \underset{˙}{\leq} W)$
24		simp2rl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to S \in A$
25		simp32	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
26		eqid	$⊢ (R \underset{˙}{\lor} S) \underset{˙}{\land} W = (R \underset{˙}{\lor} S) \underset{˙}{\land} W$
27	1 2 3 4 5 6 7 8 26	cdleme7b	$⊢ ((K \in HL \land W \in H) \land (R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \underset{˙}{\lor} S) \underset{˙}{\land} W \in A$
28	15 23 24 20 25 27	syl113anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \underset{˙}{\lor} S) \underset{˙}{\land} W \in A$
29	12 2 4	hlatjcl	$⊢ (K \in HL \land F \in A \land (R \underset{˙}{\lor} S) \underset{˙}{\land} W \in A) \to F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \in {Base}_{K}$
30	9 22 28 29	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \in {Base}_{K}$
31	9	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in Lat$
32		eqid	$⊢ Lines (K) = Lines (K)$
33		eqid	$⊢ pmap (K) = pmap (K)$
34	2 4 32 33	linepmap	$⊢ ((K \in Lat \land P \in A \land Q \in A) \land P \neq Q) \to pmap (K) (P \underset{˙}{\lor} Q) \in Lines (K)$
35	31 10 11 19 34	syl31anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to pmap (K) (P \underset{˙}{\lor} Q) \in Lines (K)$
36		simp2ll	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \in A$
37	12 2 4	hlatjcl	$⊢ (K \in HL \land R \in A \land S \in A) \to R \underset{˙}{\lor} S \in {Base}_{K}$
38	9 36 24 37	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \underset{˙}{\lor} S \in {Base}_{K}$
39		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to W \in H$
40	12 5	lhpbase	$⊢ W \in H \to W \in {Base}_{K}$
41	39 40	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to W \in {Base}_{K}$
42	12 1 3	latmle2	$⊢ (K \in Lat \land R \underset{˙}{\lor} S \in {Base}_{K} \land W \in {Base}_{K}) \to ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} W$
43	31 38 41 42	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} W$
44	1 2 3 4 5 6 7	cdleme3	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg F \underset{˙}{\leq} W$
45	15 16 17 18 19 20 44	syl132anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg F \underset{˙}{\leq} W$
46		nbrne2	$⊢ (((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} W \land \neg F \underset{˙}{\leq} W) \to (R \underset{˙}{\lor} S) \underset{˙}{\land} W \neq F$
47	46	necomd	$⊢ (((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} W \land \neg F \underset{˙}{\leq} W) \to F \neq (R \underset{˙}{\lor} S) \underset{˙}{\land} W$
48	43 45 47	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \neq (R \underset{˙}{\lor} S) \underset{˙}{\land} W$
49	2 4 32 33	linepmap	$⊢ ((K \in Lat \land F \in A \land (R \underset{˙}{\lor} S) \underset{˙}{\land} W \in A) \land F \neq (R \underset{˙}{\lor} S) \underset{˙}{\land} W) \to pmap (K) (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)) \in Lines (K)$
50	31 22 28 48 49	syl31anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to pmap (K) (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)) \in Lines (K)$
51	12 4	atbase	$⊢ F \in A \to F \in {Base}_{K}$
52	22 51	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to F \in {Base}_{K}$
53	12 3	latmcl	$⊢ (K \in Lat \land R \underset{˙}{\lor} S \in {Base}_{K} \land W \in {Base}_{K}) \to (R \underset{˙}{\lor} S) \underset{˙}{\land} W \in {Base}_{K}$
54	31 38 41 53	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \underset{˙}{\lor} S) \underset{˙}{\land} W \in {Base}_{K}$
55	12 1 2	latlej2	$⊢ (K \in Lat \land F \in {Base}_{K} \land (R \underset{˙}{\lor} S) \underset{˙}{\land} W \in {Base}_{K}) \to ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$
56	31 52 54 55	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$
57	1 2 3 4 5 6 7 8 26	cdleme7c	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land Q \in A) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to U \neq (R \underset{˙}{\lor} S) \underset{˙}{\land} W$
58	15 16 11 23 18 19 25 20 57	syl323anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to U \neq (R \underset{˙}{\lor} S) \underset{˙}{\land} W$
59	58	necomd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \underset{˙}{\lor} S) \underset{˙}{\land} W \neq U$
60		hlatl	$⊢ K \in HL \to K \in AtLat$
61	9 60	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in AtLat$
62	1 2 3 4 5 6	lhpat2	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to U \in A$
63	15 16 11 19 62	syl112anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to U \in A$
64	1 4	atncmp	$⊢ (K \in AtLat \land (R \underset{˙}{\lor} S) \underset{˙}{\land} W \in A \land U \in A) \to (\neg ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} U \leftrightarrow (R \underset{˙}{\lor} S) \underset{˙}{\land} W \neq U)$
65	61 28 63 64	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (\neg ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} U \leftrightarrow (R \underset{˙}{\lor} S) \underset{˙}{\land} W \neq U)$
66	59 65	mpbird	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} U$
67	12 1 3	latlem12	$⊢ (K \in Lat \land ((R \underset{˙}{\lor} S) \underset{˙}{\land} W \in {Base}_{K} \land P \underset{˙}{\lor} Q \in {Base}_{K} \land W \in {Base}_{K})) \to ((((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} W) \leftrightarrow ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W))$
68	31 54 14 41 67	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} W) \leftrightarrow ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W))$
69	68	biimpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} W) \to ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W))$
70	43 69	mpan2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W))$
71	6	breq2i	$⊢ ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} U \leftrightarrow ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W)$
72	70 71	syl6ibr	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} U)$
73	66 72	mtod	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
74		nbrne1	$⊢ (((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)) \land \neg ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \neq P \underset{˙}{\lor} Q$
75	74	necomd	$⊢ (((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)) \land \neg ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} Q \neq F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)$
76	56 73 75	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \underset{˙}{\lor} Q \neq F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)$
77	1 2 3 4 5 6 7 8 26	cdleme7e	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G \neq 0. (K)$
78	8 77	eqnetrrid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)) \neq 0. (K)$
79		eqid	$⊢ 0. (K) = 0. (K)$
80	12 3 79 4 32 33	2lnat	$⊢ ((K \in HL \land P \underset{˙}{\lor} Q \in {Base}_{K} \land F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \in {Base}_{K}) \land (pmap (K) (P \underset{˙}{\lor} Q) \in Lines (K) \land pmap (K) (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)) \in Lines (K)) \land (P \underset{˙}{\lor} Q \neq F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \land (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)) \neq 0. (K))) \to (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)) \in A$
81	9 14 30 35 50 76 78 80	syl322anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)) \in A$
82	8 81	eqeltrid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land ((R \in A \land \neg R \underset{˙}{\leq} W) \land (S \in A \land \neg S \underset{˙}{\leq} W)) \land (P \neq Q \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to G \in A$

Metamath Proof Explorer

Theorem cdleme7ga

Proof