cdleme22gb

Description: Utility lemma for Lemma E in Crawley p. 115. (Contributed by NM, 5-Dec-2012)

	Ref	Expression
Hypotheses	cdleme18d.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme18d.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme18d.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme18d.a	$⊢ A = Atoms (K)$
	cdleme18d.h	$⊢ H = LHyp (K)$
	cdleme18d.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme18d.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
	cdleme18d.g	$⊢ G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$
	cdleme22.b	$⊢ B = {Base}_{K}$
Assertion	cdleme22gb	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to G \in B$

Proof

Step	Hyp	Ref	Expression
1		cdleme18d.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme18d.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme18d.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme18d.a	$⊢ A = Atoms (K)$
5		cdleme18d.h	$⊢ H = LHyp (K)$
6		cdleme18d.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme18d.f	$⊢ F = (S \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} S) \underset{˙}{\land} W))$
8		cdleme18d.g	$⊢ G = (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))$
9		cdleme22.b	$⊢ B = {Base}_{K}$
10		simp1l	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to K \in HL$
11	10	hllatd	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to K \in Lat$
12		simp2l	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to P \in A$
13		simp2r	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to Q \in A$
14	9 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q \in B$
15	10 12 13 14	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to P \underset{˙}{\lor} Q \in B$
16		simp1	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to (K \in HL \land W \in H)$
17		simp3r	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to S \in A$
18	1 2 3 4 5 6 7 9	cdleme1b	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land S \in A)) \to F \in B$
19	16 12 13 17 18	syl13anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to F \in B$
20		simp3l	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to R \in A$
21	9 2 4	hlatjcl	$⊢ (K \in HL \land R \in A \land S \in A) \to R \underset{˙}{\lor} S \in B$
22	10 20 17 21	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to R \underset{˙}{\lor} S \in B$
23		simp1r	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to W \in H$
24	9 5	lhpbase	$⊢ W \in H \to W \in B$
25	23 24	syl	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to W \in B$
26	9 3	latmcl	$⊢ (K \in Lat \land R \underset{˙}{\lor} S \in B \land W \in B) \to (R \underset{˙}{\lor} S) \underset{˙}{\land} W \in B$
27	11 22 25 26	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to (R \underset{˙}{\lor} S) \underset{˙}{\land} W \in B$
28	9 2	latjcl	$⊢ (K \in Lat \land F \in B \land (R \underset{˙}{\lor} S) \underset{˙}{\land} W \in B) \to F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \in B$
29	11 19 27 28	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \in B$
30	9 3	latmcl	$⊢ (K \in Lat \land P \underset{˙}{\lor} Q \in B \land F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \in B) \to (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)) \in B$
31	11 15 29 30	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to (P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W)) \in B$
32	8 31	eqeltrid	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A) \land (R \in A \land S \in A)) \to G \in B$

Metamath Proof Explorer

Theorem cdleme22gb

Proof