Theorem 4atexlemc 38083

Description: Lemma for 4atexlem7 38089. (Contributed by NM, 24-Nov-2012.)

Hypotheses

Ref	Expression
4thatlem.ph	⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))))
4thatlem0.l	⊢ ≤ = (le‘𝐾)
4thatlem0.j	⊢ ∨ = (join‘𝐾)
4thatlem0.m	⊢ ∧ = (meet‘𝐾)
4thatlem0.a	⊢ 𝐴 = (Atoms‘𝐾)
4thatlem0.h	⊢ 𝐻 = (LHyp‘𝐾)
4thatlem0.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
4thatlem0.v	⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
4thatlem0.c	⊢ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))

Assertion

Ref	Expression
4atexlemc	⊢ (𝜑 → 𝐶 ∈ 𝐴)

Proof of Theorem 4atexlemc

Step	Hyp	Ref	Expression
1		4thatlem0.c	. . 3 ⊢ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
2		4thatlem.ph	. . . . 5 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))))
3	2	4atexlemkl 38071	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat)
4		4thatlem0.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		4thatlem0.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6	2, 4, 5	4atexlemqtb 38075	. . . 4 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
7	2, 4, 5	4atexlempsb 38074	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
8		eqid 2738	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
9		4thatlem0.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
10	8, 9	latmcom 18181	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)))
11	3, 6, 7, 10	syl3anc 1370	. . 3 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)))
12	1, 11	eqtrid 2790	. 2 ⊢ (𝜑 → 𝐶 = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)))
13	2	4atexlemk 38061	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
14	2	4atexlemp 38064	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
15	2	4atexlems 38066	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
16	2	4atexlemq 38065	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
17	2	4atexlemt 38067	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
18		4thatlem0.l	. . . 4 ⊢ ≤ = (le‘𝐾)
19	2, 18, 4, 5	4atexlempns 38076	. . 3 ⊢ (𝜑 → 𝑃 ≠ 𝑆)
20		4thatlem0.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
21		4thatlem0.u	. . . . 5 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
22		4thatlem0.v	. . . . 5 ⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
23	2, 18, 4, 9, 5, 20, 21, 22	4atexlemntlpq 38082	. . . 4 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
24	18, 4, 5	atnlej2 37394	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ≠ 𝑄)
25	24	necomd 2999	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑄 ≠ 𝑇)
26	13, 17, 14, 16, 23, 25	syl131anc 1382	. . 3 ⊢ (𝜑 → 𝑄 ≠ 𝑇)
27	2	4atexlempnq 38069	. . . 4 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
28	2	4atexlemnslpq 38070	. . . 4 ⊢ (𝜑 → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
29	18, 4, 5	4atlem0ae 37608	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))) → ¬ 𝑄 ≤ (𝑃 ∨ 𝑆))
30	13, 14, 16, 15, 27, 28, 29	syl132anc 1387	. . 3 ⊢ (𝜑 → ¬ 𝑄 ≤ (𝑃 ∨ 𝑆))
31	8, 5	atbase 37303	. . . . 5 ⊢ (𝑇 ∈ 𝐴 → 𝑇 ∈ (Base‘𝐾))
32	17, 31	syl 17	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾))
33	2, 18, 4, 9, 5, 20, 21	4atexlemu 38078	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
34	2, 18, 4, 9, 5, 20, 21, 22	4atexlemv 38079	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
35	8, 4, 5	hlatjcl 37381	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) ∈ (Base‘𝐾))
36	13, 33, 34, 35	syl3anc 1370	. . . 4 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ∈ (Base‘𝐾))
37	8, 5	atbase 37303	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
38	16, 37	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
39	8, 4	latjcl 18157	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∨ 𝑄) ∈ (Base‘𝐾))
40	3, 7, 38, 39	syl3anc 1370	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∨ 𝑄) ∈ (Base‘𝐾))
41	2	4atexlemkc 38072	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ CvLat)
42	2, 18, 4, 9, 5, 20, 21, 22	4atexlemunv 38080	. . . . 5 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
43	2	4atexlemutvt 38068	. . . . 5 ⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))
44	5, 18, 4	cvlsupr4 37359	. . . . 5 ⊢ ((𝐾 ∈ CvLat ∧ (𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) ∧ (𝑈 ≠ 𝑉 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) → 𝑇 ≤ (𝑈 ∨ 𝑉))
45	41, 33, 34, 17, 42, 43, 44	syl132anc 1387	. . . 4 ⊢ (𝜑 → 𝑇 ≤ (𝑈 ∨ 𝑉))
46	8, 4, 5	hlatjcl 37381	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
47	13, 14, 16, 46	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
48	2, 20	4atexlemwb 38073	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
49	8, 18, 9	latmle1 18182	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄))
50	3, 47, 48, 49	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄))
51	21, 50	eqbrtrid 5109	. . . . . 6 ⊢ (𝜑 → 𝑈 ≤ (𝑃 ∨ 𝑄))
52	8, 18, 9	latmle1 18182	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆))
53	3, 7, 48, 52	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆))
54	22, 53	eqbrtrid 5109	. . . . . 6 ⊢ (𝜑 → 𝑉 ≤ (𝑃 ∨ 𝑆))
55	8, 5	atbase 37303	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
56	33, 55	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
57	8, 5	atbase 37303	. . . . . . . 8 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
58	34, 57	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
59	8, 18, 4	latjlej12 18173	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑈 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) ∧ (𝑉 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))) → ((𝑈 ≤ (𝑃 ∨ 𝑄) ∧ 𝑉 ≤ (𝑃 ∨ 𝑆)) → (𝑈 ∨ 𝑉) ≤ ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆))))
60	3, 56, 47, 58, 7, 59	syl122anc 1378	. . . . . 6 ⊢ (𝜑 → ((𝑈 ≤ (𝑃 ∨ 𝑄) ∧ 𝑉 ≤ (𝑃 ∨ 𝑆)) → (𝑈 ∨ 𝑉) ≤ ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆))))
61	51, 54, 60	mp2and 696	. . . . 5 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ≤ ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)))
62	4, 5	hlatjass 37384	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
63	13, 14, 16, 15, 62	syl13anc 1371	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
64	8, 5	atbase 37303	. . . . . . . 8 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
65	14, 64	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
66	8, 5	atbase 37303	. . . . . . . 8 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾))
67	15, 66	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾))
68	8, 4	latj32 18203	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → ((𝑃 ∨ 𝑄) ∨ 𝑆) = ((𝑃 ∨ 𝑆) ∨ 𝑄))
69	3, 65, 38, 67, 68	syl13anc 1371	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ 𝑆) = ((𝑃 ∨ 𝑆) ∨ 𝑄))
70	8, 4	latjjdi 18209	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (𝑃 ∨ (𝑄 ∨ 𝑆)) = ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)))
71	3, 65, 38, 67, 70	syl13anc 1371	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ (𝑄 ∨ 𝑆)) = ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)))
72	63, 69, 71	3eqtr3rd 2787	. . . . 5 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)) = ((𝑃 ∨ 𝑆) ∨ 𝑄))
73	61, 72	breqtrd 5100	. . . 4 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ≤ ((𝑃 ∨ 𝑆) ∨ 𝑄))
74	8, 18, 3, 32, 36, 40, 45, 73	lattrd 18164	. . 3 ⊢ (𝜑 → 𝑇 ≤ ((𝑃 ∨ 𝑆) ∨ 𝑄))
75	18, 4, 9, 5	2atmat 37575	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑆) ∧ (𝑄 ≠ 𝑇 ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑆) ∧ 𝑇 ≤ ((𝑃 ∨ 𝑆) ∨ 𝑄))) → ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ∈ 𝐴)
76	13, 14, 15, 16, 17, 19, 26, 30, 74, 75	syl333anc 1401	. 2 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ∈ 𝐴)
77	12, 76	eqeltrd 2839	1 ⊢ (𝜑 → 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  lecple 16969  joincjn 18029  meetcmee 18030  Latclat 18149  Atomscatm 37277  CvLatclc 37279  HLchlt 37364  LHypclh 37998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-p1 18144  df-lat 18150  df-clat 18217  df-oposet 37190  df-ol 37192  df-oml 37193  df-covers 37280  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-llines 37512  df-lplanes 37513  df-lhyp 38002

This theorem is referenced by:  4atexlemnclw  38084  4atexlemex2  38085  4atexlemcnd  38086