Theorem 4atexlemc 38010

Description: Lemma for 4atexlem7 38016. (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 37998	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat)
4		4thatlem0.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		4thatlem0.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6	2, 4, 5	4atexlemqtb 38002	. . . 4 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
7	2, 4, 5	4atexlempsb 38001	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
8		eqid 2738	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
9		4thatlem0.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
10	8, 9	latmcom 18096	. . . 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 1369	. . 3 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)))
12	1, 11	syl5eq 2791	. 2 ⊢ (𝜑 → 𝐶 = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)))
13	2	4atexlemk 37988	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
14	2	4atexlemp 37991	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
15	2	4atexlems 37993	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
16	2	4atexlemq 37992	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
17	2	4atexlemt 37994	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
18		4thatlem0.l	. . . 4 ⊢ ≤ = (le‘𝐾)
19	2, 18, 4, 5	4atexlempns 38003	. . 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 38009	. . . 4 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
24	18, 4, 5	atnlej2 37321	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ≠ 𝑄)
25	24	necomd 2998	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑄 ≠ 𝑇)
26	13, 17, 14, 16, 23, 25	syl131anc 1381	. . 3 ⊢ (𝜑 → 𝑄 ≠ 𝑇)
27	2	4atexlempnq 37996	. . . 4 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
28	2	4atexlemnslpq 37997	. . . 4 ⊢ (𝜑 → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
29	18, 4, 5	4atlem0ae 37535	. . . 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 1386	. . 3 ⊢ (𝜑 → ¬ 𝑄 ≤ (𝑃 ∨ 𝑆))
31	8, 5	atbase 37230	. . . . 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 38005	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
34	2, 18, 4, 9, 5, 20, 21, 22	4atexlemv 38006	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
35	8, 4, 5	hlatjcl 37308	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) ∈ (Base‘𝐾))
36	13, 33, 34, 35	syl3anc 1369	. . . 4 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ∈ (Base‘𝐾))
37	8, 5	atbase 37230	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
38	16, 37	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
39	8, 4	latjcl 18072	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∨ 𝑄) ∈ (Base‘𝐾))
40	3, 7, 38, 39	syl3anc 1369	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∨ 𝑄) ∈ (Base‘𝐾))
41	2	4atexlemkc 37999	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ CvLat)
42	2, 18, 4, 9, 5, 20, 21, 22	4atexlemunv 38007	. . . . 5 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
43	2	4atexlemutvt 37995	. . . . 5 ⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))
44	5, 18, 4	cvlsupr4 37286	. . . . 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 1386	. . . 4 ⊢ (𝜑 → 𝑇 ≤ (𝑈 ∨ 𝑉))
46	8, 4, 5	hlatjcl 37308	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
47	13, 14, 16, 46	syl3anc 1369	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
48	2, 20	4atexlemwb 38000	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
49	8, 18, 9	latmle1 18097	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄))
50	3, 47, 48, 49	syl3anc 1369	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄))
51	21, 50	eqbrtrid 5105	. . . . . 6 ⊢ (𝜑 → 𝑈 ≤ (𝑃 ∨ 𝑄))
52	8, 18, 9	latmle1 18097	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆))
53	3, 7, 48, 52	syl3anc 1369	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆))
54	22, 53	eqbrtrid 5105	. . . . . 6 ⊢ (𝜑 → 𝑉 ≤ (𝑃 ∨ 𝑆))
55	8, 5	atbase 37230	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
56	33, 55	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
57	8, 5	atbase 37230	. . . . . . . 8 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
58	34, 57	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
59	8, 18, 4	latjlej12 18088	. . . . . . 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 1377	. . . . . 6 ⊢ (𝜑 → ((𝑈 ≤ (𝑃 ∨ 𝑄) ∧ 𝑉 ≤ (𝑃 ∨ 𝑆)) → (𝑈 ∨ 𝑉) ≤ ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆))))
61	51, 54, 60	mp2and 695	. . . . 5 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ≤ ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)))
62	4, 5	hlatjass 37311	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
63	13, 14, 16, 15, 62	syl13anc 1370	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
64	8, 5	atbase 37230	. . . . . . . 8 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
65	14, 64	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
66	8, 5	atbase 37230	. . . . . . . 8 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾))
67	15, 66	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾))
68	8, 4	latj32 18118	. . . . . . 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 1370	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ 𝑆) = ((𝑃 ∨ 𝑆) ∨ 𝑄))
70	8, 4	latjjdi 18124	. . . . . . 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 1370	. . . . . 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 5096	. . . 4 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ≤ ((𝑃 ∨ 𝑆) ∨ 𝑄))
74	8, 18, 3, 32, 36, 40, 45, 73	lattrd 18079	. . 3 ⊢ (𝜑 → 𝑇 ≤ ((𝑃 ∨ 𝑆) ∨ 𝑄))
75	18, 4, 9, 5	2atmat 37502	. . 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 1400	. 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 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  lecple 16895  joincjn 17944  meetcmee 17945  Latclat 18064  Atomscatm 37204  CvLatclc 37206  HLchlt 37291  LHypclh 37925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-proset 17928  df-poset 17946  df-plt 17963  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p0 18058  df-p1 18059  df-lat 18065  df-clat 18132  df-oposet 37117  df-ol 37119  df-oml 37120  df-covers 37207  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292  df-llines 37439  df-lplanes 37440  df-lhyp 37929

This theorem is referenced by:  4atexlemnclw  38011  4atexlemex2  38012  4atexlemcnd  38013