Theorem dia2dimlem1 41858

Description: Lemma for dia2dim 41871. Show properties of the auxiliary atom 𝑄. Part of proof of Lemma M in [Crawley] p. 121 line 3. (Contributed by NM, 8-Sep-2014.)

Hypotheses

Ref	Expression
dia2dimlem1.l	⊢ ≤ = (le‘𝐾)
dia2dimlem1.j	⊢ ∨ = (join‘𝐾)
dia2dimlem1.m	⊢ ∧ = (meet‘𝐾)
dia2dimlem1.a	⊢ 𝐴 = (Atoms‘𝐾)
dia2dimlem1.h	⊢ 𝐻 = (LHyp‘𝐾)
dia2dimlem1.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dia2dimlem1.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dia2dimlem1.q	⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))
dia2dimlem1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dia2dimlem1.u	⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
dia2dimlem1.v	⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
dia2dimlem1.p	⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
dia2dimlem1.f	⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))
dia2dimlem1.rf	⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
dia2dimlem1.uv	⊢ (𝜑 → 𝑈 ≠ 𝑉)
dia2dimlem1.ru	⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)

Assertion

Ref	Expression
dia2dimlem1	⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))

Proof of Theorem dia2dimlem1

Step	Hyp	Ref	Expression
1		dia2dimlem1.q	. . 3 ⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))
2		dia2dimlem1.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3	2	simpld 499	. . . 4 ⊢ (𝜑 → 𝐾 ∈ HL)
4		dia2dimlem1.p	. . . . 5 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
5	4	simpld 499	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
6		dia2dimlem1.f	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))
7		dia2dimlem1.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
8		dia2dimlem1.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
9		dia2dimlem1.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
10		dia2dimlem1.t	. . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11		dia2dimlem1.r	. . . . . 6 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
12	7, 8, 9, 10, 11	trlat 40963	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
13	2, 4, 6, 12	syl3anc 1398	. . . 4 ⊢ (𝜑 → (𝑅‘𝐹) ∈ 𝐴)
14		dia2dimlem1.u	. . . . 5 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
15	14	simpld 499	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
16	6	simpld 499	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑇)
17	7, 8, 9, 10	ltrnel 40933	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
18	2, 16, 4, 17	syl3anc 1398	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
19	18	simpld 499	. . . 4 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴)
20		dia2dimlem1.v	. . . . 5 ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
21	20	simpld 499	. . . 4 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
22	4	simprd 500	. . . . . 6 ⊢ (𝜑 → ¬ 𝑃 ≤ 𝑊)
23	7, 9, 10, 11	trlle 40978	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊)
24	2, 16, 23	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → (𝑅‘𝐹) ≤ 𝑊)
25	14	simprd 500	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
26	3	hllatd 40158	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Lat)
27		eqid 2763	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
28	27, 8	atbase 40083	. . . . . . . . . 10 ⊢ ((𝑅‘𝐹) ∈ 𝐴 → (𝑅‘𝐹) ∈ (Base‘𝐾))
29	13, 28	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝐹) ∈ (Base‘𝐾))
30	27, 8	atbase 40083	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
31	15, 30	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
32	2	simprd 500	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
33	27, 9	lhpbase 40792	. . . . . . . . . 10 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
34	32, 33	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
35		dia2dimlem1.j	. . . . . . . . . 10 ⊢ ∨ = (join‘𝐾)
36	27, 7, 35	latjle12 18501	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ ((𝑅‘𝐹) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → (((𝑅‘𝐹) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊))
37	26, 29, 31, 34, 36	syl13anc 1399	. . . . . . . 8 ⊢ (𝜑 → (((𝑅‘𝐹) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊))
38	24, 25, 37	mpbi2and 724	. . . . . . 7 ⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊)
39	27, 8	atbase 40083	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
40	5, 39	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
41	27, 35, 8	hlatjcl 40161	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑅‘𝐹) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾))
42	3, 13, 15, 41	syl3anc 1398	. . . . . . . 8 ⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾))
43	27, 7	lattr 18495	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊) → 𝑃 ≤ 𝑊))
44	26, 40, 42, 34, 43	syl13anc 1399	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊) → 𝑃 ≤ 𝑊))
45	38, 44	mpan2d 706	. . . . . 6 ⊢ (𝜑 → (𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) → 𝑃 ≤ 𝑊))
46	22, 45	mtod 201	. . . . 5 ⊢ (𝜑 → ¬ 𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈))
47	20	simprd 500	. . . . . . 7 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
48	18	simprd 500	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐹‘𝑃) ≤ 𝑊)
49		nbrne2 5131	. . . . . . 7 ⊢ ((𝑉 ≤ 𝑊 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊) → 𝑉 ≠ (𝐹‘𝑃))
50	47, 48, 49	syl2anc 595	. . . . . 6 ⊢ (𝜑 → 𝑉 ≠ (𝐹‘𝑃))
51	50	necomd 3013	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) ≠ 𝑉)
52	46, 51	jca 520	. . . 4 ⊢ (𝜑 → (¬ 𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) ∧ (𝐹‘𝑃) ≠ 𝑉))
53	26	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝐾 ∈ Lat)
54	40	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑃 ∈ (Base‘𝐾))
55	27, 35, 8	hlatjcl 40161	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑉 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾))
56	3, 21, 15, 55	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾))
57	56	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾))
58	34	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑊 ∈ (Base‘𝐾))
59	7, 35, 8	hlatlej2 40170	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉))
60	3, 19, 21, 59	syl3anc 1398	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉))
61	60	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉))
62		simpr 489	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉))
63	61, 62	breqtrrd 5139	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑉 ≤ (𝑃 ∨ 𝑈))
64		dia2dimlem1.uv	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
65	64	necomd 3013	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ≠ 𝑈)
66	7, 35, 8	hlatexch2 40190	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑉 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ 𝑉 ≠ 𝑈) → (𝑉 ≤ (𝑃 ∨ 𝑈) → 𝑃 ≤ (𝑉 ∨ 𝑈)))
67	3, 21, 5, 15, 65, 66	syl131anc 1410	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉 ≤ (𝑃 ∨ 𝑈) → 𝑃 ≤ (𝑉 ∨ 𝑈)))
68	67	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑉 ≤ (𝑃 ∨ 𝑈) → 𝑃 ≤ (𝑉 ∨ 𝑈)))
69	63, 68	mpd 16	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑃 ≤ (𝑉 ∨ 𝑈))
70	27, 8	atbase 40083	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
71	21, 70	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
72	27, 7, 35	latjle12 18501	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ (𝑉 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑉 ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ (𝑉 ∨ 𝑈) ≤ 𝑊))
73	26, 71, 31, 34, 72	syl13anc 1399	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑉 ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ (𝑉 ∨ 𝑈) ≤ 𝑊))
74	47, 25, 73	mpbi2and 724	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∨ 𝑈) ≤ 𝑊)
75	74	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑉 ∨ 𝑈) ≤ 𝑊)
76	27, 7, 53, 54, 57, 58, 69, 75	lattrd 18497	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑃 ≤ 𝑊)
77	76	ex 417	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉) → 𝑃 ≤ 𝑊))
78	77	necon3bd 2972	. . . . 5 ⊢ (𝜑 → (¬ 𝑃 ≤ 𝑊 → (𝑃 ∨ 𝑈) ≠ ((𝐹‘𝑃) ∨ 𝑉)))
79	22, 78	mpd 16	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑈) ≠ ((𝐹‘𝑃) ∨ 𝑉))
80	7, 35, 8	hlatlej2 40170	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝐹‘𝑃)))
81	3, 5, 19, 80	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝐹‘𝑃)))
82		dia2dimlem1.m	. . . . . . . . . 10 ⊢ ∧ = (meet‘𝐾)
83	7, 35, 82, 8, 9, 10, 11	trlval2 40957	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
84	2, 16, 4, 83	syl3anc 1398	. . . . . . . 8 ⊢ (𝜑 → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
85	84	oveq2d 7426	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)))
86	27, 35, 8	hlatjcl 40161	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
87	3, 5, 19, 86	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
88	7, 35, 8	hlatlej1 40169	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
89	3, 5, 19, 88	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
90	27, 7, 35, 82, 8	atmod3i1 40658	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃))) → (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)))
91	3, 5, 87, 34, 89, 90	syl131anc 1410	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)))
92		eqid 2763	. . . . . . . . . . . 12 ⊢ (1.‘𝐾) = (1.‘𝐾)
93	7, 35, 92, 8, 9	lhpjat2 40815	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ 𝑊) = (1.‘𝐾))
94	2, 4, 93	syl2anc 595	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∨ 𝑊) = (1.‘𝐾))
95	94	oveq2d 7426	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾)))
96		hlol 40155	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
97	3, 96	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ OL)
98	27, 82, 92	olm11 40021	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OL ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾)) = (𝑃 ∨ (𝐹‘𝑃)))
99	97, 87, 98	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾)) = (𝑃 ∨ (𝐹‘𝑃)))
100	95, 99	eqtrd 2798	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)) = (𝑃 ∨ (𝐹‘𝑃)))
101	91, 100	eqtrd 2798	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) = (𝑃 ∨ (𝐹‘𝑃)))
102	85, 101	eqtrd 2798	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃)))
103	81, 102	breqtrrd 5139	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)))
104		dia2dimlem1.rf	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
105	35, 8	hlatjcom 40162	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈))
106	3, 15, 21, 105	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈))
107	104, 106	breqtrd 5137	. . . . . 6 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈))
108		dia2dimlem1.ru	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)
109	7, 35, 8	hlatexch2 40190	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ ((𝑅‘𝐹) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ (𝑅‘𝐹) ≠ 𝑈) → ((𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈) → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈)))
110	3, 13, 21, 15, 108, 109	syl131anc 1410	. . . . . 6 ⊢ (𝜑 → ((𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈) → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈)))
111	107, 110	mpd 16	. . . . 5 ⊢ (𝜑 → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))
112	103, 111	jca 520	. . . 4 ⊢ (𝜑 → ((𝐹‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈)))
113	7, 35, 82, 8	ps-2c 40322	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝑅‘𝐹) ∈ 𝐴) ∧ (𝑈 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) ∧ ((¬ 𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) ∧ (𝐹‘𝑃) ≠ 𝑉) ∧ (𝑃 ∨ 𝑈) ≠ ((𝐹‘𝑃) ∨ 𝑉) ∧ ((𝐹‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈)))) → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∈ 𝐴)
114	3, 5, 13, 15, 19, 21, 52, 79, 112, 113	syl333anc 1429	. . 3 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∈ 𝐴)
115	1, 114	eqeltrid 2867	. 2 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
116	27, 35, 8	hlatjcl 40161	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
117	3, 5, 15, 116	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
118	27, 35, 8	hlatjcl 40161	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾))
119	3, 19, 21, 118	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾))
120	27, 7, 82	latmle1 18515	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ (𝑃 ∨ 𝑈))
121	26, 117, 119, 120	syl3anc 1398	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ (𝑃 ∨ 𝑈))
122	1, 121	eqbrtrid 5146	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ≤ (𝑃 ∨ 𝑈))
123	27, 8	atbase 40083	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
124	115, 123	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
125	27, 7, 82	latlem12 18517	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑄 ≤ (𝑃 ∨ 𝑈) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊)))
126	26, 124, 117, 34, 125	syl13anc 1399	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 ≤ (𝑃 ∨ 𝑈) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊)))
127	126	biimpd 232	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 ≤ (𝑃 ∨ 𝑈) ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊)))
128	122, 127	mpand 707	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊)))
129	128	imp 411	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊))
130		eqid 2763	. . . . . . . . . . . . 13 ⊢ (0.‘𝐾) = (0.‘𝐾)
131	7, 82, 130, 8, 9	lhpmat 40824	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∧ 𝑊) = (0.‘𝐾))
132	2, 4, 131	syl2anc 595	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∧ 𝑊) = (0.‘𝐾))
133	132	oveq1d 7425	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((0.‘𝐾) ∨ 𝑈))
134	27, 7, 35, 82, 8	atmod4i1 40660	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑈 ∈ 𝐴 ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑈 ≤ 𝑊) → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((𝑃 ∨ 𝑈) ∧ 𝑊))
135	3, 15, 40, 34, 25, 134	syl131anc 1410	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((𝑃 ∨ 𝑈) ∧ 𝑊))
136	27, 35, 130	olj02 40020	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OL ∧ 𝑈 ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ 𝑈) = 𝑈)
137	97, 31, 136	syl2anc 595	. . . . . . . . . 10 ⊢ (𝜑 → ((0.‘𝐾) ∨ 𝑈) = 𝑈)
138	133, 135, 137	3eqtr3d 2806	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ 𝑊) = 𝑈)
139	138	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → ((𝑃 ∨ 𝑈) ∧ 𝑊) = 𝑈)
140	129, 139	breqtrd 5137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ 𝑈)
141		hlatl 40154	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
142	3, 141	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ AtLat)
143	142	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝐾 ∈ AtLat)
144	115	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ∈ 𝐴)
145	15	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑈 ∈ 𝐴)
146	7, 8	atcmp 40105	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑄 ≤ 𝑈 ↔ 𝑄 = 𝑈))
147	143, 144, 145, 146	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (𝑄 ≤ 𝑈 ↔ 𝑄 = 𝑈))
148	140, 147	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 = 𝑈)
149	27, 7, 82	latmle2 18516	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ ((𝐹‘𝑃) ∨ 𝑉))
150	26, 117, 119, 149	syl3anc 1398	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ ((𝐹‘𝑃) ∨ 𝑉))
151	1, 150	eqbrtrid 5146	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉))
152	27, 7, 82	latlem12 18517	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)))
153	26, 124, 119, 34, 152	syl13anc 1399	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)))
154	153	biimpd 232	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)))
155	151, 154	mpand 707	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)))
156	155	imp 411	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊))
157	7, 82, 130, 8, 9	lhpmat 40824	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) → ((𝐹‘𝑃) ∧ 𝑊) = (0.‘𝐾))
158	2, 18, 157	syl2anc 595	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹‘𝑃) ∧ 𝑊) = (0.‘𝐾))
159	158	oveq1d 7425	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = ((0.‘𝐾) ∨ 𝑉))
160	27, 8	atbase 40083	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑃) ∈ 𝐴 → (𝐹‘𝑃) ∈ (Base‘𝐾))
161	19, 160	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘𝑃) ∈ (Base‘𝐾))
162	27, 7, 35, 82, 8	atmod4i1 40660	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑉 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑉 ≤ 𝑊) → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊))
163	3, 21, 161, 34, 47, 162	syl131anc 1410	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊))
164	27, 35, 130	olj02 40020	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OL ∧ 𝑉 ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ 𝑉) = 𝑉)
165	97, 71, 164	syl2anc 595	. . . . . . . . . 10 ⊢ (𝜑 → ((0.‘𝐾) ∨ 𝑉) = 𝑉)
166	159, 163, 165	3eqtr3d 2806	. . . . . . . . 9 ⊢ (𝜑 → (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊) = 𝑉)
167	166	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊) = 𝑉)
168	156, 167	breqtrd 5137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ 𝑉)
169	21	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑉 ∈ 𝐴)
170	7, 8	atcmp 40105	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑄 ≤ 𝑉 ↔ 𝑄 = 𝑉))
171	143, 144, 169, 170	syl3anc 1398	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (𝑄 ≤ 𝑉 ↔ 𝑄 = 𝑉))
172	168, 171	mpbid 235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 = 𝑉)
173	148, 172	eqtr3d 2800	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑈 = 𝑉)
174	173	ex 417	. . . 4 ⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑈 = 𝑉))
175	174	necon3ad 2971	. . 3 ⊢ (𝜑 → (𝑈 ≠ 𝑉 → ¬ 𝑄 ≤ 𝑊))
176	64, 175	mpd 16	. 2 ⊢ (𝜑 → ¬ 𝑄 ≤ 𝑊)
177	115, 176	jca 520	1 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ≠ wne 2958   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  lecple 17312  joincjn 18362  meetcmee 18363  0.cp0 18472  1.cp1 18473  Latclat 18482  OLcol 39968  Atomscatm 40057  AtLatcal 40058  HLchlt 40144  LHypclh 40778  LTrncltrn 40895  trLctrl 40952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822  df-proset 18345  df-poset 18364  df-plt 18379  df-lub 18395  df-glb 18396  df-join 18397  df-meet 18398  df-p0 18474  df-p1 18475  df-lat 18483  df-clat 18550  df-oposet 39970  df-ol 39972  df-oml 39973  df-covers 40060  df-ats 40061  df-atl 40092  df-cvlat 40116  df-hlat 40145  df-llines 40292  df-psubsp 40297  df-pmap 40298  df-padd 40590  df-lhyp 40782  df-laut 40783  df-ldil 40898  df-ltrn 40899  df-trl 40953

This theorem is referenced by:  dia2dimlem3  41860  dia2dimlem6  41863