cdleme35fnpq

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. (Contributed by NM, 19-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	cdleme35fnpq	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to \neg F \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$

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		simp3	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
9		simp11	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (K \in HL \land W \in H)$
10		simp12l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \in A$
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	1 2 3 4 5 6	cdlemeulpq	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A)) \to U \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
13	9 10 11 12	syl12anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to U \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
14		simp11l	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to K \in HL$
15	14	hllatd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to K \in Lat$
16		simp2rl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to R \in A$
17		eqid	$⊢ {Base}_{K} = {Base}_{K}$
18	1 2 3 4 5 6 7 17	cdleme1b	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land R \in A)) \to F \in {Base}_{K}$
19	9 10 11 16 18	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to F \in {Base}_{K}$
20	1 2 3 4 5 6 17	cdleme0aa	$⊢ ((K \in HL \land W \in H) \land P \in A \land Q \in A) \to U \in {Base}_{K}$
21	9 10 11 20	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to U \in {Base}_{K}$
22	17 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
23	14 10 11 22	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
24	17 1 2	latjle12	$⊢ (K \in Lat \land (F \in {Base}_{K} \land U \in {Base}_{K} \land P \underset{˙}{\lor} Q \in {Base}_{K})) \to ((F \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \leftrightarrow (F \underset{˙}{\lor} U) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
25	15 19 21 23 24	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ((F \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \leftrightarrow (F \underset{˙}{\lor} U) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
26	25	biimpd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ((F \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land U \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (F \underset{˙}{\lor} U) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
27	13 26	mpan2d	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (F \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to (F \underset{˙}{\lor} U) \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
28	17 4	atbase	$⊢ R \in A \to R \in {Base}_{K}$
29	16 28	syl	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to R \in {Base}_{K}$
30	17 1 2	latlej1	$⊢ (K \in Lat \land R \in {Base}_{K} \land U \in {Base}_{K}) \to R \underset{˙}{\leq} (R \underset{˙}{\lor} U)$
31	15 29 21 30	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to R \underset{˙}{\leq} (R \underset{˙}{\lor} U)$
32	1 2 3 4 5 6 7	cdleme35a	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to F \underset{˙}{\lor} U = R \underset{˙}{\lor} U$
33	31 32	breqtrrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to R \underset{˙}{\leq} (F \underset{˙}{\lor} U)$
34	17 2	latjcl	$⊢ (K \in Lat \land F \in {Base}_{K} \land U \in {Base}_{K}) \to F \underset{˙}{\lor} U \in {Base}_{K}$
35	15 19 21 34	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to F \underset{˙}{\lor} U \in {Base}_{K}$
36	17 1	lattr	$⊢ (K \in Lat \land (R \in {Base}_{K} \land F \underset{˙}{\lor} U \in {Base}_{K} \land P \underset{˙}{\lor} Q \in {Base}_{K})) \to ((R \underset{˙}{\leq} (F \underset{˙}{\lor} U) \land (F \underset{˙}{\lor} U) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
37	15 29 35 23 36	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ((R \underset{˙}{\leq} (F \underset{˙}{\lor} U) \land (F \underset{˙}{\lor} U) \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
38	33 37	mpand	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ((F \underset{˙}{\lor} U) \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
39	27 38	syld	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (F \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to R \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
40	8 39	mtod	$⊢ (((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land \neg Q \underset{˙}{\leq} W)) \land (P \neq Q \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land \neg R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to \neg F \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$

Metamath Proof Explorer

Theorem cdleme35fnpq

Proof