Theorem 4atexlemc 34373

Description: Lemma for 4atexlem7 34379. (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 34361	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat)
4		4thatlem0.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		4thatlem0.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6	2, 4, 5	4atexlemqtb 34365	. . . 4 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
7	2, 4, 5	4atexlempsb 34364	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
8		eqid 2610	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
9		4thatlem0.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
10	8, 9	latmcom 16898	. . . 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 1318	. . 3 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)))
12	1, 11	syl5eq 2656	. 2 ⊢ (𝜑 → 𝐶 = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)))
13	2	4atexlemk 34351	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
14	2	4atexlemp 34354	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
15	2	4atexlems 34356	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
16	2	4atexlemq 34355	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
17	2	4atexlemt 34357	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
18		4thatlem0.l	. . . 4 ⊢ ≤ = (le‘𝐾)
19	2, 18, 4, 5	4atexlempns 34366	. . 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 34372	. . . 4 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
24	18, 4, 5	atnlej2 33684	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ≠ 𝑄)
25	24	necomd 2837	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑄 ≠ 𝑇)
26	13, 17, 14, 16, 23, 25	syl131anc 1331	. . 3 ⊢ (𝜑 → 𝑄 ≠ 𝑇)
27	2	4atexlempnq 34359	. . . 4 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
28	2	4atexlemnslpq 34360	. . . 4 ⊢ (𝜑 → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
29	18, 4, 5	4atlem0ae 33898	. . . 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 1336	. . 3 ⊢ (𝜑 → ¬ 𝑄 ≤ (𝑃 ∨ 𝑆))
31	8, 5	atbase 33594	. . . . 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 34368	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
34	2, 18, 4, 9, 5, 20, 21, 22	4atexlemv 34369	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
35	8, 4, 5	hlatjcl 33671	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) ∈ (Base‘𝐾))
36	13, 33, 34, 35	syl3anc 1318	. . . 4 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ∈ (Base‘𝐾))
37	8, 5	atbase 33594	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
38	16, 37	syl 17	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
39	8, 4	latjcl 16874	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∨ 𝑄) ∈ (Base‘𝐾))
40	3, 7, 38, 39	syl3anc 1318	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∨ 𝑄) ∈ (Base‘𝐾))
41	2	4atexlemkc 34362	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ CvLat)
42	2, 18, 4, 9, 5, 20, 21, 22	4atexlemunv 34370	. . . . 5 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
43	2	4atexlemutvt 34358	. . . . 5 ⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))
44	5, 18, 4	cvlsupr4 33650	. . . . 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 1336	. . . 4 ⊢ (𝜑 → 𝑇 ≤ (𝑈 ∨ 𝑉))
46	8, 4, 5	hlatjcl 33671	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
47	13, 14, 16, 46	syl3anc 1318	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
48	2, 20	4atexlemwb 34363	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
49	8, 18, 9	latmle1 16899	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄))
50	3, 47, 48, 49	syl3anc 1318	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄))
51	21, 50	syl5eqbr 4618	. . . . . 6 ⊢ (𝜑 → 𝑈 ≤ (𝑃 ∨ 𝑄))
52	8, 18, 9	latmle1 16899	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆))
53	3, 7, 48, 52	syl3anc 1318	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆))
54	22, 53	syl5eqbr 4618	. . . . . 6 ⊢ (𝜑 → 𝑉 ≤ (𝑃 ∨ 𝑆))
55	8, 5	atbase 33594	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
56	33, 55	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
57	8, 5	atbase 33594	. . . . . . . 8 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
58	34, 57	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
59	8, 18, 4	latjlej12 16890	. . . . . . 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 1327	. . . . . 6 ⊢ (𝜑 → ((𝑈 ≤ (𝑃 ∨ 𝑄) ∧ 𝑉 ≤ (𝑃 ∨ 𝑆)) → (𝑈 ∨ 𝑉) ≤ ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆))))
61	51, 54, 60	mp2and 711	. . . . 5 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ≤ ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)))
62	4, 5	hlatjass 33674	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
63	13, 14, 16, 15, 62	syl13anc 1320	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
64	8, 5	atbase 33594	. . . . . . . 8 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
65	14, 64	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
66	8, 5	atbase 33594	. . . . . . . 8 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾))
67	15, 66	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾))
68	8, 4	latj32 16920	. . . . . . 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 1320	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ 𝑆) = ((𝑃 ∨ 𝑆) ∨ 𝑄))
70	8, 4	latjjdi 16926	. . . . . . 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 1320	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ (𝑄 ∨ 𝑆)) = ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)))
72	63, 69, 71	3eqtr3rd 2653	. . . . 5 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)) = ((𝑃 ∨ 𝑆) ∨ 𝑄))
73	61, 72	breqtrd 4609	. . . 4 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ≤ ((𝑃 ∨ 𝑆) ∨ 𝑄))
74	8, 18, 3, 32, 36, 40, 45, 73	lattrd 16881	. . 3 ⊢ (𝜑 → 𝑇 ≤ ((𝑃 ∨ 𝑆) ∨ 𝑄))
75	18, 4, 9, 5	2atmat 33865	. . 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 1350	. 2 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ∈ 𝐴)
77	12, 76	eqeltrd 2688	1 ⊢ (𝜑 → 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  Latclat 16868  Atomscatm 33568  CvLatclc 33570  HLchlt 33655  LHypclh 34288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lhyp 34292

This theorem is referenced by:  4atexlemnclw  34374  4atexlemex2  34375  4atexlemcnd  34376