cdlemednpq

Description: Part of proof of Lemma E in Crawley p. 113. Utility lemma. D represents s_2. (Contributed by NM, 18-Nov-2012)

	Ref	Expression
Hypotheses	cdlemeda.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdlemeda.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdlemeda.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdlemeda.a	$⊢ A = Atoms (K)$
	cdlemeda.h	$⊢ H = LHyp (K)$
	cdlemeda.d	$⊢ D = (R \underset{˙}{\lor} S) \underset{˙}{\land} W$
Assertion	cdlemednpq	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$

Proof

Step	Hyp	Ref	Expression
1		cdlemeda.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdlemeda.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdlemeda.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdlemeda.a	$⊢ A = Atoms (K)$
5		cdlemeda.h	$⊢ H = LHyp (K)$
6		cdlemeda.d	$⊢ D = (R \underset{˙}{\lor} S) \underset{˙}{\land} W$
7		simp1l	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in HL$
8	7	hllatd	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in Lat$
9		simp23l	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \in A$
10		simp31l	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to S \in A$
11		eqid	$⊢ {Base}_{K} = {Base}_{K}$
12	11 2 4	hlatjcl	$⊢ (K \in HL \land R \in A \land S \in A) \to R \underset{˙}{\lor} S \in {Base}_{K}$
13	7 9 10 12	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \underset{˙}{\lor} S \in {Base}_{K}$
14		simp1r	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to W \in H$
15	11 5	lhpbase	$⊢ W \in H \to W \in {Base}_{K}$
16	14 15	syl	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to W \in {Base}_{K}$
17	11 1 3	latmle2	$⊢ (K \in Lat \land R \underset{˙}{\lor} S \in {Base}_{K} \land W \in {Base}_{K}) \to ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} W$
18	8 13 16 17	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} W$
19	6 18	eqbrtrid	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to D \underset{˙}{\leq} W$
20		simp23r	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg R \underset{˙}{\leq} W$
21		nbrne2	$⊢ (D \underset{˙}{\leq} W \land \neg R \underset{˙}{\leq} W) \to D \neq R$
22	19 20 21	syl2anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to D \neq R$
23	8	adantr	$⊢ (((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to K \in Lat$
24	13	adantr	$⊢ (((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to R \underset{˙}{\lor} S \in {Base}_{K}$
25	16	adantr	$⊢ (((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to W \in {Base}_{K}$
26	11 1 3	latmle1	$⊢ (K \in Lat \land R \underset{˙}{\lor} S \in {Base}_{K} \land W \in {Base}_{K}) \to ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} (R \underset{˙}{\lor} S)$
27	23 24 25 26	syl3anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ((R \underset{˙}{\lor} S) \underset{˙}{\land} W) \underset{˙}{\leq} (R \underset{˙}{\lor} S)$
28	6 27	eqbrtrid	$⊢ (((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to D \underset{˙}{\leq} (R \underset{˙}{\lor} S)$
29		simpr	$⊢ (((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
30		simp31r	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg S \underset{˙}{\leq} W$
31		simp32	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
32		simp33	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$
33	1 2 3 4 5 6	cdlemeda	$⊢ ((K \in HL \land W \in H) \land (S \in A \land \neg S \underset{˙}{\leq} W) \land (R \in A \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to D \in A$
34	7 14 10 30 9 31 32 33	syl223anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to D \in A$
35	11 4	atbase	$⊢ D \in A \to D \in {Base}_{K}$
36	34 35	syl	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to D \in {Base}_{K}$
37	36	adantr	$⊢ (((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to D \in {Base}_{K}$
38		simp21	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \in A$
39		simp22	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to Q \in A$
40	11 2 4	hlatjcl	$⊢ (K \in HL \land P \in A \land Q \in A) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
41	7 38 39 40	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
42	41	adantr	$⊢ (((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
43	11 1 3	latlem12	$⊢ (K \in Lat \land (D \in {Base}_{K} \land R \underset{˙}{\lor} S \in {Base}_{K} \land P \underset{˙}{\lor} Q \in {Base}_{K})) \to ((D \underset{˙}{\leq} (R \underset{˙}{\lor} S) \land D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \leftrightarrow D \underset{˙}{\leq} ((R \underset{˙}{\lor} S) \underset{˙}{\land} (P \underset{˙}{\lor} Q)))$
44	23 37 24 42 43	syl13anc	$⊢ (((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to ((D \underset{˙}{\leq} (R \underset{˙}{\lor} S) \land D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \leftrightarrow D \underset{˙}{\leq} ((R \underset{˙}{\lor} S) \underset{˙}{\land} (P \underset{˙}{\lor} Q)))$
45	28 29 44	mpbi2and	$⊢ (((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to D \underset{˙}{\leq} ((R \underset{˙}{\lor} S) \underset{˙}{\land} (P \underset{˙}{\lor} Q))$
46		hlatl	$⊢ K \in HL \to K \in AtLat$
47	7 46	syl	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in AtLat$
48		eqid	$⊢ 0. (K) = 0. (K)$
49	11 1 3 48 4	atnle	$⊢ (K \in AtLat \land S \in A \land P \underset{˙}{\lor} Q \in {Base}_{K}) \to (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \leftrightarrow S \underset{˙}{\land} (P \underset{˙}{\lor} Q) = 0. (K))$
50	47 10 41 49	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (\neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \leftrightarrow S \underset{˙}{\land} (P \underset{˙}{\lor} Q) = 0. (K))$
51	32 50	mpbid	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to S \underset{˙}{\land} (P \underset{˙}{\lor} Q) = 0. (K)$
52	51	oveq2d	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \underset{˙}{\lor} (S \underset{˙}{\land} (P \underset{˙}{\lor} Q)) = R \underset{˙}{\lor} 0. (K)$
53	11 4	atbase	$⊢ S \in A \to S \in {Base}_{K}$
54	10 53	syl	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to S \in {Base}_{K}$
55	11 1 2 3 4	atmod1i1	$⊢ (K \in HL \land (R \in A \land S \in {Base}_{K} \land P \underset{˙}{\lor} Q \in {Base}_{K}) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to R \underset{˙}{\lor} (S \underset{˙}{\land} (P \underset{˙}{\lor} Q)) = (R \underset{˙}{\lor} S) \underset{˙}{\land} (P \underset{˙}{\lor} Q)$
56	7 9 54 41 31 55	syl131anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \underset{˙}{\lor} (S \underset{˙}{\land} (P \underset{˙}{\lor} Q)) = (R \underset{˙}{\lor} S) \underset{˙}{\land} (P \underset{˙}{\lor} Q)$
57		hlol	$⊢ K \in HL \to K \in OL$
58	7 57	syl	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to K \in OL$
59	11 4	atbase	$⊢ R \in A \to R \in {Base}_{K}$
60	9 59	syl	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \in {Base}_{K}$
61	11 2 48	olj01	$⊢ (K \in OL \land R \in {Base}_{K}) \to R \underset{˙}{\lor} 0. (K) = R$
62	58 60 61	syl2anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to R \underset{˙}{\lor} 0. (K) = R$
63	52 56 62	3eqtr3d	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (R \underset{˙}{\lor} S) \underset{˙}{\land} (P \underset{˙}{\lor} Q) = R$
64	63	adantr	$⊢ (((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to (R \underset{˙}{\lor} S) \underset{˙}{\land} (P \underset{˙}{\lor} Q) = R$
65	45 64	breqtrd	$⊢ (((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \land D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)) \to D \underset{˙}{\leq} R$
66	65	ex	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (D \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to D \underset{˙}{\leq} R)$
67	1 4	atcmp	$⊢ (K \in AtLat \land D \in A \land R \in A) \to (D \underset{˙}{\leq} R \leftrightarrow D = R)$
68	47 34 9 67	syl3anc	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (D \underset{˙}{\leq} R \leftrightarrow D = R)$
69	66 68	sylibd	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (D \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \to D = R)$
70	69	necon3ad	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to (D \neq R \to \neg D \underset{˙}{\leq} (P \underset{˙}{\lor} Q))$
71	22 70	mpd	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W)) \land ((S \in A \land \neg S \underset{˙}{\leq} W) \land R \underset{˙}{\leq} (P \underset{˙}{\lor} Q) \land \neg S \underset{˙}{\leq} (P \underset{˙}{\lor} Q))) \to \neg D \underset{˙}{\leq} (P \underset{˙}{\lor} Q)$

Metamath Proof Explorer

Theorem cdlemednpq

Proof