Theorem 4atexlemc 38532

Description: Lemma for 4atexlem7 38538. (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 38520	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat)
4		4thatlem0.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		4thatlem0.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6	2, 4, 5	4atexlemqtb 38524	. . . 4 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
7	2, 4, 5	4atexlempsb 38523	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
8		eqid 2736	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
9		4thatlem0.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
10	8, 9	latmcom 18352	. . . 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 1371	. . 3 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)))
12	1, 11	eqtrid 2788	. 2 ⊢ (𝜑 → 𝐶 = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)))
13	2	4atexlemk 38510	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
14	2	4atexlemp 38513	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
15	2	4atexlems 38515	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
16	2	4atexlemq 38514	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
17	2	4atexlemt 38516	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
18		4thatlem0.l	. . . 4 ⊢ ≤ = (le‘𝐾)
19	2, 18, 4, 5	4atexlempns 38525	. . 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 38531	. . . 4 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
24	18, 4, 5	atnlej2 37843	. . . . 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 1383	. . 3 ⊢ (𝜑 → 𝑄 ≠ 𝑇)
27	2	4atexlempnq 38518	. . . 4 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
28	2	4atexlemnslpq 38519	. . . 4 ⊢ (𝜑 → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
29	18, 4, 5	4atlem0ae 38057	. . . 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 1388	. . 3 ⊢ (𝜑 → ¬ 𝑄 ≤ (𝑃 ∨ 𝑆))
31	8, 5	atbase 37751	. . . . 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 38527	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
34	2, 18, 4, 9, 5, 20, 21, 22	4atexlemv 38528	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
35	8, 4, 5	hlatjcl 37829	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) ∈ (Base‘𝐾))
36	13, 33, 34, 35	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ∈ (Base‘𝐾))
37	8, 5	atbase 37751	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
38	16, 37	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
39	8, 4	latjcl 18328	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∨ 𝑄) ∈ (Base‘𝐾))
40	3, 7, 38, 39	syl3anc 1371	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∨ 𝑄) ∈ (Base‘𝐾))
41	2	4atexlemkc 38521	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ CvLat)
42	2, 18, 4, 9, 5, 20, 21, 22	4atexlemunv 38529	. . . . 5 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
43	2	4atexlemutvt 38517	. . . . 5 ⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))
44	5, 18, 4	cvlsupr4 37807	. . . . 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 1388	. . . 4 ⊢ (𝜑 → 𝑇 ≤ (𝑈 ∨ 𝑉))
46	8, 4, 5	hlatjcl 37829	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
47	13, 14, 16, 46	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
48	2, 20	4atexlemwb 38522	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
49	8, 18, 9	latmle1 18353	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄))
50	3, 47, 48, 49	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄))
51	21, 50	eqbrtrid 5140	. . . . . 6 ⊢ (𝜑 → 𝑈 ≤ (𝑃 ∨ 𝑄))
52	8, 18, 9	latmle1 18353	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆))
53	3, 7, 48, 52	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆))
54	22, 53	eqbrtrid 5140	. . . . . 6 ⊢ (𝜑 → 𝑉 ≤ (𝑃 ∨ 𝑆))
55	8, 5	atbase 37751	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
56	33, 55	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
57	8, 5	atbase 37751	. . . . . . . 8 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
58	34, 57	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
59	8, 18, 4	latjlej12 18344	. . . . . . 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 1379	. . . . . 6 ⊢ (𝜑 → ((𝑈 ≤ (𝑃 ∨ 𝑄) ∧ 𝑉 ≤ (𝑃 ∨ 𝑆)) → (𝑈 ∨ 𝑉) ≤ ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆))))
61	51, 54, 60	mp2and 697	. . . . 5 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ≤ ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)))
62	4, 5	hlatjass 37832	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
63	13, 14, 16, 15, 62	syl13anc 1372	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
64	8, 5	atbase 37751	. . . . . . . 8 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
65	14, 64	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
66	8, 5	atbase 37751	. . . . . . . 8 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾))
67	15, 66	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾))
68	8, 4	latj32 18374	. . . . . . 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 1372	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ 𝑆) = ((𝑃 ∨ 𝑆) ∨ 𝑄))
70	8, 4	latjjdi 18380	. . . . . . 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 1372	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ (𝑄 ∨ 𝑆)) = ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)))
72	63, 69, 71	3eqtr3rd 2785	. . . . 5 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)) = ((𝑃 ∨ 𝑆) ∨ 𝑄))
73	61, 72	breqtrd 5131	. . . 4 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ≤ ((𝑃 ∨ 𝑆) ∨ 𝑄))
74	8, 18, 3, 32, 36, 40, 45, 73	lattrd 18335	. . 3 ⊢ (𝜑 → 𝑇 ≤ ((𝑃 ∨ 𝑆) ∨ 𝑄))
75	18, 4, 9, 5	2atmat 38024	. . 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 1402	. 2 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ∈ 𝐴)
77	12, 76	eqeltrd 2838	1 ⊢ (𝜑 → 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  meetcmee 18201  Latclat 18320  Atomscatm 37725  CvLatclc 37727  HLchlt 37812  LHypclh 38447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-p1 18315  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962  df-lhyp 38451

This theorem is referenced by:  4atexlemnclw  38533  4atexlemex2  38534  4atexlemcnd  38535