Theorem 4atexlemc 41126

Description: Lemma for 4atexlem7 41132. (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 41114	. . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat)
4		4thatlem0.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
5		4thatlem0.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6	2, 4, 5	4atexlemqtb 41118	. . . 4 ⊢ (𝜑 → (𝑄 ∨ 𝑇) ∈ (Base‘𝐾))
7	2, 4, 5	4atexlempsb 41117	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
8		eqid 2761	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
9		4thatlem0.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
10	8, 9	latmcom 18637	. . . 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 1398	. . 3 ⊢ (𝜑 → ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆)) = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)))
12	1, 11	eqtrid 2808	. 2 ⊢ (𝜑 → 𝐶 = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)))
13	2	4atexlemk 41104	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
14	2	4atexlemp 41107	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
15	2	4atexlems 41109	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
16	2	4atexlemq 41108	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
17	2	4atexlemt 41110	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
18		4thatlem0.l	. . . 4 ⊢ ≤ = (le‘𝐾)
19	2, 18, 4, 5	4atexlempns 41119	. . 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 41125	. . . 4 ⊢ (𝜑 → ¬ 𝑇 ≤ (𝑃 ∨ 𝑄))
24	18, 4, 5	atnlej2 40437	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑇 ≠ 𝑄)
25	24	necomd 3011	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑇 ≤ (𝑃 ∨ 𝑄)) → 𝑄 ≠ 𝑇)
26	13, 17, 14, 16, 23, 25	syl131anc 1410	. . 3 ⊢ (𝜑 → 𝑄 ≠ 𝑇)
27	2	4atexlempnq 41112	. . . 4 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
28	2	4atexlemnslpq 41113	. . . 4 ⊢ (𝜑 → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))
29	18, 4, 5	4atlem0ae 40651	. . . 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 1415	. . 3 ⊢ (𝜑 → ¬ 𝑄 ≤ (𝑃 ∨ 𝑆))
31	8, 5	atbase 40346	. . . . 5 ⊢ (𝑇 ∈ 𝐴 → 𝑇 ∈ (Base‘𝐾))
32	17, 31	syl 18	. . . 4 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾))
33	2, 18, 4, 9, 5, 20, 21	4atexlemu 41121	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
34	2, 18, 4, 9, 5, 20, 21, 22	4atexlemv 41122	. . . . 5 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
35	8, 4, 5	hlatjcl 40424	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) ∈ (Base‘𝐾))
36	13, 33, 34, 35	syl3anc 1398	. . . 4 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ∈ (Base‘𝐾))
37	8, 5	atbase 40346	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
38	16, 37	syl 18	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
39	8, 4	latjcl 18613	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∨ 𝑄) ∈ (Base‘𝐾))
40	3, 7, 38, 39	syl3anc 1398	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∨ 𝑄) ∈ (Base‘𝐾))
41	2	4atexlemkc 41115	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ CvLat)
42	2, 18, 4, 9, 5, 20, 21, 22	4atexlemunv 41123	. . . . 5 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
43	2	4atexlemutvt 41111	. . . . 5 ⊢ (𝜑 → (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))
44	5, 18, 4	cvlsupr4 40402	. . . . 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 1415	. . . 4 ⊢ (𝜑 → 𝑇 ≤ (𝑈 ∨ 𝑉))
46	8, 4, 5	hlatjcl 40424	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
47	13, 14, 16, 46	syl3anc 1398	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
48	2, 20	4atexlemwb 41116	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
49	8, 18, 9	latmle1 18638	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄))
50	3, 47, 48, 49	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∧ 𝑊) ≤ (𝑃 ∨ 𝑄))
51	21, 50	eqbrtrid 5140	. . . . . 6 ⊢ (𝜑 → 𝑈 ≤ (𝑃 ∨ 𝑄))
52	8, 18, 9	latmle1 18638	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆))
53	3, 7, 48, 52	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ 𝑊) ≤ (𝑃 ∨ 𝑆))
54	22, 53	eqbrtrid 5140	. . . . . 6 ⊢ (𝜑 → 𝑉 ≤ (𝑃 ∨ 𝑆))
55	8, 5	atbase 40346	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
56	33, 55	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
57	8, 5	atbase 40346	. . . . . . . 8 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
58	34, 57	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
59	8, 18, 4	latjlej12 18629	. . . . . . 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 1406	. . . . . 6 ⊢ (𝜑 → ((𝑈 ≤ (𝑃 ∨ 𝑄) ∧ 𝑉 ≤ (𝑃 ∨ 𝑆)) → (𝑈 ∨ 𝑉) ≤ ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆))))
61	51, 54, 60	mp2and 712	. . . . 5 ⊢ (𝜑 → (𝑈 ∨ 𝑉) ≤ ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)))
62	4, 5	hlatjass 40427	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
63	13, 14, 16, 15, 62	syl13anc 1399	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ 𝑆) = (𝑃 ∨ (𝑄 ∨ 𝑆)))
64	8, 5	atbase 40346	. . . . . . . 8 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
65	14, 64	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
66	8, 5	atbase 40346	. . . . . . . 8 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾))
67	15, 66	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾))
68	8, 4	latj32 18659	. . . . . . 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 1399	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ 𝑆) = ((𝑃 ∨ 𝑆) ∨ 𝑄))
70	8, 4	latjjdi 18665	. . . . . . 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 1399	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ (𝑄 ∨ 𝑆)) = ((𝑃 ∨ 𝑄) ∨ (𝑃 ∨ 𝑆)))
72	63, 69, 71	3eqtr3rd 2805	. . . . 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 18620	. . 3 ⊢ (𝜑 → 𝑇 ≤ ((𝑃 ∨ 𝑆) ∨ 𝑄))
75	18, 4, 9, 5	2atmat 40618	. . 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 1429	. 2 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ∈ 𝐴)
77	12, 76	eqeltrd 2861	1 ⊢ (𝜑 → 𝐶 ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956   class class class wbr 5103  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  lecple 17435  joincjn 18485  meetcmee 18486  Latclat 18605  Atomscatm 40320  CvLatclc 40322  HLchlt 40407  LHypclh 41041

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-proset 18468  df-poset 18487  df-plt 18502  df-lub 18518  df-glb 18519  df-join 18520  df-meet 18521  df-p0 18597  df-p1 18598  df-lat 18606  df-clat 18673  df-oposet 40233  df-ol 40235  df-oml 40236  df-covers 40323  df-ats 40324  df-atl 40355  df-cvlat 40379  df-hlat 40408  df-llines 40555  df-lplanes 40556  df-lhyp 41045

This theorem is used by:  4atexlemnclw  41127  4atexlemex2  41128  4atexlemcnd  41129