cdleme4a

Description: Part of proof of Lemma E in Crawley p. 114 top. G represents f_s(r). Auxiliary lemma derived from cdleme5 . We show f_s(r) <_ p \/ q. (Contributed by NM, 10-Nov-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	cdleme4a	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A) \land S \in A) \to G \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$

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		simp1l	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A) \land S \in A) \to K \in HL$
10	9	hllatd	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A) \land S \in A) \to K \in Lat$
11		simp21	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A) \land S \in A) \to P \in A$
12		simp22	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A) \land S \in A) \to Q \in A$
13		eqid	$⊢ {Base}_{K} = {Base}_{K}$
14	13 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
15	9 11 12 14	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A) \land S \in A) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
16		simp1r	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A) \land S \in A) \to W \in H$
17		simp3	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A) \land S \in A) \to S \in A$
18	1 2 3 4 5 6 7 13	cdleme1b	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land S \in A)) \to F \in {Base}_{K}$
19	9 16 11 12 17 18	syl23anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A) \land S \in A) \to F \in {Base}_{K}$
20		simp23	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A) \land S \in A) \to R \in A$
21	13 2 4	hlatjcl	$⊢ (K \in HL \land R \in A \land S \in A) \to R \underset{˙}{\lor} S \in {Base}_{K}$
22	9 20 17 21	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A) \land S \in A) \to R \underset{˙}{\lor} S \in {Base}_{K}$
23	13 5	lhpbase	$⊢ W \in H \to W \in {Base}_{K}$
24	16 23	syl	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A) \land S \in A) \to W \in {Base}_{K}$
25	13 3	latmcl	$⊢ (K \in Lat \land R \underset{˙}{\lor} S \in {Base}_{K} \land W \in {Base}_{K}) \to (R \underset{˙}{\lor} S) \underset{˙}{\land} W \in {Base}_{K}$
26	10 22 24 25	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A) \land S \in A) \to (R \underset{˙}{\lor} S) \underset{˙}{\land} W \in {Base}_{K}$
27	13 2	latjcl	$⊢ (K \in Lat \land F \in {Base}_{K} \land (R \underset{˙}{\lor} S) \underset{˙}{\land} W \in {Base}_{K}) \to F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \in {Base}_{K}$
28	10 19 26 27	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A) \land S \in A) \to F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \in {Base}_{K}$
29	13 1 3	latmle1	$⊢ (K \in Lat \land P \underset{˙}{\lor} Q \in {Base}_{K} \land F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \in {Base}_{K}) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
30	10 15 28 29	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A) \land S \in A) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\land} (F \underset{˙}{\lor} ((R \underset{˙}{\lor} S) \underset{˙}{\land} W))) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
31	8 30	eqbrtrid	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A) \land S \in A) \to G \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$

Metamath Proof Explorer

Theorem cdleme4a

Proof