cdleme3b

Description: Part of proof of Lemma E in Crawley p. 113. Lemma leading to cdleme3fa and cdleme3 . (Contributed by NM, 6-Jun-2012)

	Ref	Expression
Hypotheses	cdleme1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	cdleme1.j	$⊢ \underset{˙}{\lor} = join (K)$
	cdleme1.m	$⊢ \underset{˙}{\land} = meet (K)$
	cdleme1.a	$⊢ A = Atoms (K)$
	cdleme1.h	$⊢ H = LHyp (K)$
	cdleme1.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
	cdleme1.f	$⊢ F = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
Assertion	cdleme3b	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to F \neq R$

Proof

Step	Hyp	Ref	Expression
1		cdleme1.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		cdleme1.j	$⊢ \underset{˙}{\lor} = join (K)$
3		cdleme1.m	$⊢ \underset{˙}{\land} = meet (K)$
4		cdleme1.a	$⊢ A = Atoms (K)$
5		cdleme1.h	$⊢ H = LHyp (K)$
6		cdleme1.u	$⊢ U = (P \underset{˙}{\lor} Q) \underset{˙}{\land} W$
7		cdleme1.f	$⊢ F = (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W))$
8		simpll	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to K \in HL$
9		simpr3l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to R \in A$
10		eqid	$⊢ {Base}_{K} = {Base}_{K}$
11	10 4	atbase	$⊢ R \in A \to R \in {Base}_{K}$
12	9 11	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to R \in {Base}_{K}$
13		hllat	$⊢ K \in HL \to K \in Lat$
14	13	ad2antrr	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to K \in Lat$
15	1 2 3 4 5 6	lhpat2	$⊢ ((K \in HL \land W \in H) \land (P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q)) \to U \in A$
16	15	3adant3r3	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to U \in A$
17	10 4	atbase	$⊢ U \in A \to U \in {Base}_{K}$
18	16 17	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to U \in {Base}_{K}$
19	10 2	latjcl	$⊢ (K \in Lat \land R \in {Base}_{K} \land U \in {Base}_{K}) \to R \underset{˙}{\lor} U \in {Base}_{K}$
20	14 12 18 19	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to R \underset{˙}{\lor} U \in {Base}_{K}$
21		simpr2l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to Q \in A$
22	10 4	atbase	$⊢ Q \in A \to Q \in {Base}_{K}$
23	21 22	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to Q \in {Base}_{K}$
24		simpr1l	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to P \in A$
25	10 4	atbase	$⊢ P \in A \to P \in {Base}_{K}$
26	24 25	syl	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to P \in {Base}_{K}$
27	10 2	latjcl	$⊢ (K \in Lat \land P \in {Base}_{K} \land R \in {Base}_{K}) \to P \underset{˙}{\lor} R \in {Base}_{K}$
28	14 26 12 27	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to P \underset{˙}{\lor} R \in {Base}_{K}$
29	10 5	lhpbase	$⊢ W \in H \to W \in {Base}_{K}$
30	29	ad2antlr	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to W \in {Base}_{K}$
31	10 3	latmcl	$⊢ (K \in Lat \land P \underset{˙}{\lor} R \in {Base}_{K} \land W \in {Base}_{K}) \to (P \underset{˙}{\lor} R) \underset{˙}{\land} W \in {Base}_{K}$
32	14 28 30 31	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to (P \underset{˙}{\lor} R) \underset{˙}{\land} W \in {Base}_{K}$
33	10 2	latjcl	$⊢ (K \in Lat \land Q \in {Base}_{K} \land (P \underset{˙}{\lor} R) \underset{˙}{\land} W \in {Base}_{K}) \to Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) \in {Base}_{K}$
34	14 23 32 33	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) \in {Base}_{K}$
35	10 3	latmcl	$⊢ (K \in Lat \land R \underset{˙}{\lor} U \in {Base}_{K} \land Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W) \in {Base}_{K}) \to (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)) \in {Base}_{K}$
36	14 20 34 35	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to (R \underset{˙}{\lor} U) \underset{˙}{\land} (Q \underset{˙}{\lor} ((P \underset{˙}{\lor} R) \underset{˙}{\land} W)) \in {Base}_{K}$
37	7 36	eqeltrid	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to F \in {Base}_{K}$
38	10 2	latjcl	$⊢ (K \in Lat \land R \in {Base}_{K} \land F \in {Base}_{K}) \to R \underset{˙}{\lor} F \in {Base}_{K}$
39	14 12 37 38	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to R \underset{˙}{\lor} F \in {Base}_{K}$
40	10 2	latjcl	$⊢ (K \in Lat \land P \in {Base}_{K} \land Q \in {Base}_{K}) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
41	14 26 23 40	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to P \underset{˙}{\lor} Q \in {Base}_{K}$
42	10 1 3	latmle2	$⊢ (K \in Lat \land P \underset{˙}{\lor} Q \in {Base}_{K} \land W \in {Base}_{K}) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W) \underset{˙}{\leq} W$
43	14 41 30 42	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to ((P \underset{˙}{\lor} Q) \underset{˙}{\land} W) \underset{˙}{\leq} W$
44	6 43	eqbrtrid	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to U \underset{˙}{\leq} W$
45		simpr3r	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to \neg R \underset{˙}{\leq} W$
46		nbrne2	$⊢ (U \underset{˙}{\leq} W \land \neg R \underset{˙}{\leq} W) \to U \neq R$
47	44 45 46	syl2anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to U \neq R$
48	47	necomd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to R \neq U$
49		eqid	$⊢ ⋖_{K} = ⋖_{K}$
50	2 49 4	atcvr1	$⊢ (K \in HL \land R \in A \land U \in A) \to (R \neq U \leftrightarrow R ⋖_{K} (R \underset{˙}{\lor} U))$
51	8 9 16 50	syl3anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to (R \neq U \leftrightarrow R ⋖_{K} (R \underset{˙}{\lor} U))$
52	48 51	mpbid	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to R ⋖_{K} (R \underset{˙}{\lor} U)$
53		simpr3	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to (R \in A \land \neg R \underset{˙}{\leq} W)$
54	24 21 53	3jca	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W))$
55	1 2 3 4 5 6 7	cdleme1	$⊢ ((K \in HL \land W \in H) \land (P \in A \land Q \in A \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to R \underset{˙}{\lor} F = R \underset{˙}{\lor} U$
56	54 55	syldan	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to R \underset{˙}{\lor} F = R \underset{˙}{\lor} U$
57	52 56	breqtrrd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to R ⋖_{K} (R \underset{˙}{\lor} F)$
58	10 49	cvrne	$⊢ ((K \in HL \land R \in {Base}_{K} \land R \underset{˙}{\lor} F \in {Base}_{K}) \land R ⋖_{K} (R \underset{˙}{\lor} F)) \to R \neq R \underset{˙}{\lor} F$
59	8 12 39 57 58	syl31anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to R \neq R \underset{˙}{\lor} F$
60		oveq2	$⊢ F = R \to R \underset{˙}{\lor} F = R \underset{˙}{\lor} R$
61	60	adantl	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \land F = R) \to R \underset{˙}{\lor} F = R \underset{˙}{\lor} R$
62	2 4	hlatjidm	$⊢ (K \in HL \land R \in A) \to R \underset{˙}{\lor} R = R$
63	8 9 62	syl2anc	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to R \underset{˙}{\lor} R = R$
64	63	adantr	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \land F = R) \to R \underset{˙}{\lor} R = R$
65	61 64	eqtr2d	$⊢ (((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \land F = R) \to R = R \underset{˙}{\lor} F$
66	65	ex	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to (F = R \to R = R \underset{˙}{\lor} F)$
67	66	necon3d	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to (R \neq R \underset{˙}{\lor} F \to F \neq R)$
68	59 67	mpd	$⊢ ((K \in HL \land W \in H) \land ((P \in A \land \neg P \underset{˙}{\leq} W) \land (Q \in A \land P \neq Q) \land (R \in A \land \neg R \underset{˙}{\leq} W))) \to F \neq R$

Metamath Proof Explorer

Theorem cdleme3b

Proof