cdleme35d

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013)

	Ref	Expression
Hypotheses	cdleme35.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme35.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme35.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme35.a	$⊢ A = Atoms (K)$
	cdleme35.h	$⊢ H = LHyp (K)$
	cdleme35.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme35.f	$⊢ F = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
Assertion	cdleme35d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (Q \underset{˙}{\lor} F) \underset{˙}{\land} W = (P \underset{˙}{\lor} R) \underset{˙}{\land} W$

Proof

Step	Hyp	Ref	Expression
1		cdleme35.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme35.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme35.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme35.a	$⊢ A = Atoms (K)$
5		cdleme35.h	$⊢ H = LHyp (K)$
6		cdleme35.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme35.f	$⊢ F = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
8	1 2 3 4 5 6 7	cdleme35c	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to Q \underset{˙}{\lor} F = Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)$
9	8	oveq1d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (Q \underset{˙}{\lor} F) \underset{˙}{\land} W = (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)) \underset{˙}{\land} W$
10		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to K \in HL$
11		simp13l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to Q \in A$
12	10	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to K \in Lat$
13		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \in A$
14		simp2rl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to R \in A$
15		eqid	$⊢ {Base}_{K} = {Base}_{K}$
16	15 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land R \in A) \to P \underset{˙}{\lor} R \in {Base}_{K}$
17	10 13 14 16	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} R \in {Base}_{K}$
18		simp11r	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to W \in H$
19	15 5	lhpbase	$⊢ W \in H \to W \in {Base}_{K}$
20	18 19	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to W \in {Base}_{K}$
21	15 3	latmcl	$⊢ (K \in Lat \land P \underset{˙}{\lor} R \in {Base}_{K} \land W \in {Base}_{K}) \to (P \underset{˙}{\lor} R) \underset{˙}{\land} W \in {Base}_{K}$
22	12 17 20 21	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (P \underset{˙}{\lor} R) \underset{˙}{\land} W \in {Base}_{K}$
23	15 1 3	latmle2	$⊢ (K \in Lat \land P \underset{˙}{\lor} R \in {Base}_{K} \land W \in {Base}_{K}) \to ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) \underset{˙}{\leq} W$
24	12 17 20 23	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) \underset{˙}{\leq} W$
25	15 1 2 3 4	atmod4i2	$⊢ (K \in HL \land (Q \in A \land (P \underset{˙}{\lor} R) \underset{˙}{\land} W \in {Base}_{K} \land W \in {Base}_{K}) \land ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) \underset{˙}{\leq} W) \to (Q \underset{˙}{\land} W) \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) = (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)) \underset{˙}{\land} W$
26	10 11 22 20 24 25	syl131anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (Q \underset{˙}{\land} W) \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) = (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)) \underset{˙}{\land} W$
27		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (K \in HL \land W \in H)$
28		simp13	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (Q \in A \land \neg Q \underset{˙}{\leq} W)$
29		eqid	$⊢ 0. (K) = 0. (K)$
30	1 3 29 4 5	lhpmat	$⊢ ((K \in HL \land W \in H) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \to Q \underset{˙}{\land} W = 0. (K)$
31	27 28 30	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to Q \underset{˙}{\land} W = 0. (K)$
32	31	oveq1d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (Q \underset{˙}{\land} W) \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) = 0. (K) \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)$
33		hlol	$⊢ K \in HL \to K \in OL$
34	10 33	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to K \in OL$
35	15 2 29	olj02	$⊢ (K \in OL \land (P \underset{˙}{\lor} R) \underset{˙}{\land} W \in {Base}_{K}) \to 0. (K) \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) = (P \underset{˙}{\lor} R) \underset{˙}{\land} W$
36	34 22 35	syl2anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to 0. (K) \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) = (P \underset{˙}{\lor} R) \underset{˙}{\land} W$
37	32 36	eqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (Q \underset{˙}{\land} W) \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) = (P \underset{˙}{\lor} R) \underset{˙}{\land} W$
38	9 26 37	3eqtr2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (Q \underset{˙}{\lor} F) \underset{˙}{\land} W = (P \underset{˙}{\lor} R) \underset{˙}{\land} W$

Metamath Proof Explorer

Theorem cdleme35d

Proof