Theorem dia2dimlem1 39527

Description: Lemma for dia2dim 39540. 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 495	. . . 4 ⊢ (𝜑 → 𝐾 ∈ HL)
4		dia2dimlem1.p	. . . . 5 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
5	4	simpld 495	. . . 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 38632	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
13	2, 4, 6, 12	syl3anc 1371	. . . 4 ⊢ (𝜑 → (𝑅‘𝐹) ∈ 𝐴)
14		dia2dimlem1.u	. . . . 5 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
15	14	simpld 495	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
16	6	simpld 495	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑇)
17	7, 8, 9, 10	ltrnel 38602	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
18	2, 16, 4, 17	syl3anc 1371	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
19	18	simpld 495	. . . 4 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴)
20		dia2dimlem1.v	. . . . 5 ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
21	20	simpld 495	. . . 4 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
22	4	simprd 496	. . . . . 6 ⊢ (𝜑 → ¬ 𝑃 ≤ 𝑊)
23	7, 9, 10, 11	trlle 38647	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊)
24	2, 16, 23	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑅‘𝐹) ≤ 𝑊)
25	14	simprd 496	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
26	3	hllatd 37826	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Lat)
27		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
28	27, 8	atbase 37751	. . . . . . . . . 10 ⊢ ((𝑅‘𝐹) ∈ 𝐴 → (𝑅‘𝐹) ∈ (Base‘𝐾))
29	13, 28	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝐹) ∈ (Base‘𝐾))
30	27, 8	atbase 37751	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
31	15, 30	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
32	2	simprd 496	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
33	27, 9	lhpbase 38461	. . . . . . . . . 10 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
34	32, 33	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
35		dia2dimlem1.j	. . . . . . . . . 10 ⊢ ∨ = (join‘𝐾)
36	27, 7, 35	latjle12 18339	. . . . . . . . 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 1372	. . . . . . . 8 ⊢ (𝜑 → (((𝑅‘𝐹) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊))
38	24, 25, 37	mpbi2and 710	. . . . . . 7 ⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊)
39	27, 8	atbase 37751	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
40	5, 39	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
41	27, 35, 8	hlatjcl 37829	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑅‘𝐹) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾))
42	3, 13, 15, 41	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾))
43	27, 7	lattr 18333	. . . . . . . 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 1372	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊) → 𝑃 ≤ 𝑊))
45	38, 44	mpan2d 692	. . . . . 6 ⊢ (𝜑 → (𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) → 𝑃 ≤ 𝑊))
46	22, 45	mtod 197	. . . . 5 ⊢ (𝜑 → ¬ 𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈))
47	20	simprd 496	. . . . . . 7 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
48	18	simprd 496	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐹‘𝑃) ≤ 𝑊)
49		nbrne2 5125	. . . . . . 7 ⊢ ((𝑉 ≤ 𝑊 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊) → 𝑉 ≠ (𝐹‘𝑃))
50	47, 48, 49	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝑉 ≠ (𝐹‘𝑃))
51	50	necomd 2999	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) ≠ 𝑉)
52	46, 51	jca 512	. . . 4 ⊢ (𝜑 → (¬ 𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) ∧ (𝐹‘𝑃) ≠ 𝑉))
53	26	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝐾 ∈ Lat)
54	40	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑃 ∈ (Base‘𝐾))
55	27, 35, 8	hlatjcl 37829	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑉 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾))
56	3, 21, 15, 55	syl3anc 1371	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾))
57	56	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾))
58	34	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑊 ∈ (Base‘𝐾))
59	7, 35, 8	hlatlej2 37838	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉))
60	3, 19, 21, 59	syl3anc 1371	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉))
61	60	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉))
62		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉))
63	61, 62	breqtrrd 5133	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑉 ≤ (𝑃 ∨ 𝑈))
64		dia2dimlem1.uv	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
65	64	necomd 2999	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ≠ 𝑈)
66	7, 35, 8	hlatexch2 37859	. . . . . . . . . . 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 1383	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉 ≤ (𝑃 ∨ 𝑈) → 𝑃 ≤ (𝑉 ∨ 𝑈)))
68	67	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑉 ≤ (𝑃 ∨ 𝑈) → 𝑃 ≤ (𝑉 ∨ 𝑈)))
69	63, 68	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑃 ≤ (𝑉 ∨ 𝑈))
70	27, 8	atbase 37751	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
71	21, 70	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
72	27, 7, 35	latjle12 18339	. . . . . . . . . . 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 1372	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑉 ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ (𝑉 ∨ 𝑈) ≤ 𝑊))
74	47, 25, 73	mpbi2and 710	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∨ 𝑈) ≤ 𝑊)
75	74	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑉 ∨ 𝑈) ≤ 𝑊)
76	27, 7, 53, 54, 57, 58, 69, 75	lattrd 18335	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑃 ≤ 𝑊)
77	76	ex 413	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉) → 𝑃 ≤ 𝑊))
78	77	necon3bd 2957	. . . . 5 ⊢ (𝜑 → (¬ 𝑃 ≤ 𝑊 → (𝑃 ∨ 𝑈) ≠ ((𝐹‘𝑃) ∨ 𝑉)))
79	22, 78	mpd 15	. . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑈) ≠ ((𝐹‘𝑃) ∨ 𝑉))
80	7, 35, 8	hlatlej2 37838	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝐹‘𝑃)))
81	3, 5, 19, 80	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝐹‘𝑃)))
82		dia2dimlem1.m	. . . . . . . . . 10 ⊢ ∧ = (meet‘𝐾)
83	7, 35, 82, 8, 9, 10, 11	trlval2 38626	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
84	2, 16, 4, 83	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
85	84	oveq2d 7373	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)))
86	27, 35, 8	hlatjcl 37829	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
87	3, 5, 19, 86	syl3anc 1371	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
88	7, 35, 8	hlatlej1 37837	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
89	3, 5, 19, 88	syl3anc 1371	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
90	27, 7, 35, 82, 8	atmod3i1 38327	. . . . . . . . 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 1383	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)))
92		eqid 2736	. . . . . . . . . . . 12 ⊢ (1.‘𝐾) = (1.‘𝐾)
93	7, 35, 92, 8, 9	lhpjat2 38484	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∨ 𝑊) = (1.‘𝐾))
94	2, 4, 93	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∨ 𝑊) = (1.‘𝐾))
95	94	oveq2d 7373	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾)))
96		hlol 37823	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
97	3, 96	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ OL)
98	27, 82, 92	olm11 37689	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OL ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾)) = (𝑃 ∨ (𝐹‘𝑃)))
99	97, 87, 98	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾)) = (𝑃 ∨ (𝐹‘𝑃)))
100	95, 99	eqtrd 2776	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)) = (𝑃 ∨ (𝐹‘𝑃)))
101	91, 100	eqtrd 2776	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) = (𝑃 ∨ (𝐹‘𝑃)))
102	85, 101	eqtrd 2776	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃)))
103	81, 102	breqtrrd 5133	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)))
104		dia2dimlem1.rf	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
105	35, 8	hlatjcom 37830	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈))
106	3, 15, 21, 105	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈))
107	104, 106	breqtrd 5131	. . . . . 6 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈))
108		dia2dimlem1.ru	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)
109	7, 35, 8	hlatexch2 37859	. . . . . . 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 1383	. . . . . 6 ⊢ (𝜑 → ((𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈) → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈)))
111	107, 110	mpd 15	. . . . 5 ⊢ (𝜑 → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))
112	103, 111	jca 512	. . . 4 ⊢ (𝜑 → ((𝐹‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈)))
113	7, 35, 82, 8	ps-2c 37991	. . . 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 1402	. . 3 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∈ 𝐴)
115	1, 114	eqeltrid 2842	. 2 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
116	27, 35, 8	hlatjcl 37829	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
117	3, 5, 15, 116	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
118	27, 35, 8	hlatjcl 37829	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾))
119	3, 19, 21, 118	syl3anc 1371	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾))
120	27, 7, 82	latmle1 18353	. . . . . . . . . . . 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 1371	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ (𝑃 ∨ 𝑈))
122	1, 121	eqbrtrid 5140	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ≤ (𝑃 ∨ 𝑈))
123	27, 8	atbase 37751	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
124	115, 123	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
125	27, 7, 82	latlem12 18355	. . . . . . . . . . . 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 1372	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 ≤ (𝑃 ∨ 𝑈) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊)))
127	126	biimpd 228	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 ≤ (𝑃 ∨ 𝑈) ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊)))
128	122, 127	mpand 693	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊)))
129	128	imp 407	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊))
130		eqid 2736	. . . . . . . . . . . . 13 ⊢ (0.‘𝐾) = (0.‘𝐾)
131	7, 82, 130, 8, 9	lhpmat 38493	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∧ 𝑊) = (0.‘𝐾))
132	2, 4, 131	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∧ 𝑊) = (0.‘𝐾))
133	132	oveq1d 7372	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((0.‘𝐾) ∨ 𝑈))
134	27, 7, 35, 82, 8	atmod4i1 38329	. . . . . . . . . . 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 1383	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((𝑃 ∨ 𝑈) ∧ 𝑊))
136	27, 35, 130	olj02 37688	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OL ∧ 𝑈 ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ 𝑈) = 𝑈)
137	97, 31, 136	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((0.‘𝐾) ∨ 𝑈) = 𝑈)
138	133, 135, 137	3eqtr3d 2784	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ 𝑊) = 𝑈)
139	138	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → ((𝑃 ∨ 𝑈) ∧ 𝑊) = 𝑈)
140	129, 139	breqtrd 5131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ 𝑈)
141		hlatl 37822	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
142	3, 141	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ AtLat)
143	142	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝐾 ∈ AtLat)
144	115	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ∈ 𝐴)
145	15	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑈 ∈ 𝐴)
146	7, 8	atcmp 37773	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑄 ≤ 𝑈 ↔ 𝑄 = 𝑈))
147	143, 144, 145, 146	syl3anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (𝑄 ≤ 𝑈 ↔ 𝑄 = 𝑈))
148	140, 147	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 = 𝑈)
149	27, 7, 82	latmle2 18354	. . . . . . . . . . . 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 1371	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ ((𝐹‘𝑃) ∨ 𝑉))
151	1, 150	eqbrtrid 5140	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉))
152	27, 7, 82	latlem12 18355	. . . . . . . . . . . 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 1372	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)))
154	153	biimpd 228	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)))
155	151, 154	mpand 693	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)))
156	155	imp 407	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊))
157	7, 82, 130, 8, 9	lhpmat 38493	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) → ((𝐹‘𝑃) ∧ 𝑊) = (0.‘𝐾))
158	2, 18, 157	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐹‘𝑃) ∧ 𝑊) = (0.‘𝐾))
159	158	oveq1d 7372	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = ((0.‘𝐾) ∨ 𝑉))
160	27, 8	atbase 37751	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑃) ∈ 𝐴 → (𝐹‘𝑃) ∈ (Base‘𝐾))
161	19, 160	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘𝑃) ∈ (Base‘𝐾))
162	27, 7, 35, 82, 8	atmod4i1 38329	. . . . . . . . . . 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 1383	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊))
164	27, 35, 130	olj02 37688	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ OL ∧ 𝑉 ∈ (Base‘𝐾)) → ((0.‘𝐾) ∨ 𝑉) = 𝑉)
165	97, 71, 164	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((0.‘𝐾) ∨ 𝑉) = 𝑉)
166	159, 163, 165	3eqtr3d 2784	. . . . . . . . 9 ⊢ (𝜑 → (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊) = 𝑉)
167	166	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊) = 𝑉)
168	156, 167	breqtrd 5131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ 𝑉)
169	21	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑉 ∈ 𝐴)
170	7, 8	atcmp 37773	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑄 ≤ 𝑉 ↔ 𝑄 = 𝑉))
171	143, 144, 169, 170	syl3anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (𝑄 ≤ 𝑉 ↔ 𝑄 = 𝑉))
172	168, 171	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 = 𝑉)
173	148, 172	eqtr3d 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑈 = 𝑉)
174	173	ex 413	. . . 4 ⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑈 = 𝑉))
175	174	necon3ad 2956	. . 3 ⊢ (𝜑 → (𝑈 ≠ 𝑉 → ¬ 𝑄 ≤ 𝑊))
176	64, 175	mpd 15	. 2 ⊢ (𝜑 → ¬ 𝑄 ≤ 𝑊)
177	115, 176	jca 512	1 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = 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  0.cp0 18312  1.cp1 18313  Latclat 18320  OLcol 37636  Atomscatm 37725  AtLatcal 37726  HLchlt 37812  LHypclh 38447  LTrncltrn 38564  trLctrl 38621

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-iin 4957  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-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  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-psubsp 37966  df-pmap 37967  df-padd 38259  df-lhyp 38451  df-laut 38452  df-ldil 38567  df-ltrn 38568  df-trl 38622

This theorem is referenced by:  dia2dimlem3  39529  dia2dimlem6  39532