Theorem dia2dimlem1 41324

Description: Lemma for dia2dim 41337. 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 494	. . . 4 ⊢ (𝜑 → 𝐾 ∈ HL)
4		dia2dimlem1.p	. . . . 5 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
5	4	simpld 494	. . . 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 40429	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
13	2, 4, 6, 12	syl3anc 1373	. . . 4 ⊢ (𝜑 → (𝑅‘𝐹) ∈ 𝐴)
14		dia2dimlem1.u	. . . . 5 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
15	14	simpld 494	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
16	6	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑇)
17	7, 8, 9, 10	ltrnel 40399	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
18	2, 16, 4, 17	syl3anc 1373	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
19	18	simpld 494	. . . 4 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴)
20		dia2dimlem1.v	. . . . 5 ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
21	20	simpld 494	. . . 4 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
22	4	simprd 495	. . . . . 6 ⊢ (𝜑 → ¬ 𝑃 ≤ 𝑊)
23	7, 9, 10, 11	trlle 40444	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ≤ 𝑊)
24	2, 16, 23	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑅‘𝐹) ≤ 𝑊)
25	14	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
26	3	hllatd 39624	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Lat)
27		eqid 2736	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
28	27, 8	atbase 39549	. . . . . . . . . 10 ⊢ ((𝑅‘𝐹) ∈ 𝐴 → (𝑅‘𝐹) ∈ (Base‘𝐾))
29	13, 28	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝐹) ∈ (Base‘𝐾))
30	27, 8	atbase 39549	. . . . . . . . . 10 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
31	15, 30	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
32	2	simprd 495	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
33	27, 9	lhpbase 40258	. . . . . . . . . 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 18373	. . . . . . . . 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 1374	. . . . . . . 8 ⊢ (𝜑 → (((𝑅‘𝐹) ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊))
38	24, 25, 37	mpbi2and 712	. . . . . . 7 ⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊)
39	27, 8	atbase 39549	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
40	5, 39	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
41	27, 35, 8	hlatjcl 39627	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑅‘𝐹) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾))
42	3, 13, 15, 41	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾))
43	27, 7	lattr 18367	. . . . . . . 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 1374	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝑅‘𝐹) ∨ 𝑈) ≤ 𝑊) → 𝑃 ≤ 𝑊))
45	38, 44	mpan2d 694	. . . . . 6 ⊢ (𝜑 → (𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) → 𝑃 ≤ 𝑊))
46	22, 45	mtod 198	. . . . 5 ⊢ (𝜑 → ¬ 𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈))
47	20	simprd 495	. . . . . . 7 ⊢ (𝜑 → 𝑉 ≤ 𝑊)
48	18	simprd 495	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐹‘𝑃) ≤ 𝑊)
49		nbrne2 5118	. . . . . . 7 ⊢ ((𝑉 ≤ 𝑊 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊) → 𝑉 ≠ (𝐹‘𝑃))
50	47, 48, 49	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝑉 ≠ (𝐹‘𝑃))
51	50	necomd 2987	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) ≠ 𝑉)
52	46, 51	jca 511	. . . 4 ⊢ (𝜑 → (¬ 𝑃 ≤ ((𝑅‘𝐹) ∨ 𝑈) ∧ (𝐹‘𝑃) ≠ 𝑉))
53	26	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝐾 ∈ Lat)
54	40	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑃 ∈ (Base‘𝐾))
55	27, 35, 8	hlatjcl 39627	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑉 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾))
56	3, 21, 15, 55	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾))
57	56	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾))
58	34	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑊 ∈ (Base‘𝐾))
59	7, 35, 8	hlatlej2 39636	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉))
60	3, 19, 21, 59	syl3anc 1373	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉))
61	60	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉))
62		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉))
63	61, 62	breqtrrd 5126	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑉 ≤ (𝑃 ∨ 𝑈))
64		dia2dimlem1.uv	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
65	64	necomd 2987	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ≠ 𝑈)
66	7, 35, 8	hlatexch2 39656	. . . . . . . . . . 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 1385	. . . . . . . . . 10 ⊢ (𝜑 → (𝑉 ≤ (𝑃 ∨ 𝑈) → 𝑃 ≤ (𝑉 ∨ 𝑈)))
68	67	adantr 480	. . . . . . . . 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 39549	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
71	21, 70	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
72	27, 7, 35	latjle12 18373	. . . . . . . . . . 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 1374	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑉 ≤ 𝑊 ∧ 𝑈 ≤ 𝑊) ↔ (𝑉 ∨ 𝑈) ≤ 𝑊))
74	47, 25, 73	mpbi2and 712	. . . . . . . . 9 ⊢ (𝜑 → (𝑉 ∨ 𝑈) ≤ 𝑊)
75	74	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → (𝑉 ∨ 𝑈) ≤ 𝑊)
76	27, 7, 53, 54, 57, 58, 69, 75	lattrd 18369	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑃 ≤ 𝑊)
77	76	ex 412	. . . . . 6 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉) → 𝑃 ≤ 𝑊))
78	77	necon3bd 2946	. . . . 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 39636	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝐹‘𝑃)))
81	3, 5, 19, 80	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝐹‘𝑃)))
82		dia2dimlem1.m	. . . . . . . . . 10 ⊢ ∧ = (meet‘𝐾)
83	7, 35, 82, 8, 9, 10, 11	trlval2 40423	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
84	2, 16, 4, 83	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
85	84	oveq2d 7374	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)))
86	27, 35, 8	hlatjcl 39627	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
87	3, 5, 19, 86	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
88	7, 35, 8	hlatlej1 39635	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
89	3, 5, 19, 88	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
90	27, 7, 35, 82, 8	atmod3i1 40124	. . . . . . . . 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 1385	. . . . . . . 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 40281	. . . . . . . . . . 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 7374	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾)))
96		hlol 39621	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
97	3, 96	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ OL)
98	27, 82, 92	olm11 39487	. . . . . . . . . 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 2771	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)) = (𝑃 ∨ (𝐹‘𝑃)))
101	91, 100	eqtrd 2771	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) = (𝑃 ∨ (𝐹‘𝑃)))
102	85, 101	eqtrd 2771	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃)))
103	81, 102	breqtrrd 5126	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)))
104		dia2dimlem1.rf	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
105	35, 8	hlatjcom 39628	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈))
106	3, 15, 21, 105	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈))
107	104, 106	breqtrd 5124	. . . . . 6 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈))
108		dia2dimlem1.ru	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)
109	7, 35, 8	hlatexch2 39656	. . . . . . 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 1385	. . . . . 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 511	. . . 4 ⊢ (𝜑 → ((𝐹‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)) ∧ 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈)))
113	7, 35, 82, 8	ps-2c 39788	. . . 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 1404	. . 3 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∈ 𝐴)
115	1, 114	eqeltrid 2840	. 2 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
116	27, 35, 8	hlatjcl 39627	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
117	3, 5, 15, 116	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
118	27, 35, 8	hlatjcl 39627	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾))
119	3, 19, 21, 118	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾))
120	27, 7, 82	latmle1 18387	. . . . . . . . . . . 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 1373	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ (𝑃 ∨ 𝑈))
122	1, 121	eqbrtrid 5133	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ≤ (𝑃 ∨ 𝑈))
123	27, 8	atbase 39549	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
124	115, 123	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
125	27, 7, 82	latlem12 18389	. . . . . . . . . . . 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 1374	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 ≤ (𝑃 ∨ 𝑈) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊)))
127	126	biimpd 229	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 ≤ (𝑃 ∨ 𝑈) ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊)))
128	122, 127	mpand 695	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊)))
129	128	imp 406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ ((𝑃 ∨ 𝑈) ∧ 𝑊))
130		eqid 2736	. . . . . . . . . . . . 13 ⊢ (0.‘𝐾) = (0.‘𝐾)
131	7, 82, 130, 8, 9	lhpmat 40290	. . . . . . . . . . . 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 7373	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((0.‘𝐾) ∨ 𝑈))
134	27, 7, 35, 82, 8	atmod4i1 40126	. . . . . . . . . . 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 1385	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((𝑃 ∨ 𝑈) ∧ 𝑊))
136	27, 35, 130	olj02 39486	. . . . . . . . . . 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 2779	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ 𝑊) = 𝑈)
139	138	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → ((𝑃 ∨ 𝑈) ∧ 𝑊) = 𝑈)
140	129, 139	breqtrd 5124	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ 𝑈)
141		hlatl 39620	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
142	3, 141	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ AtLat)
143	142	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝐾 ∈ AtLat)
144	115	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ∈ 𝐴)
145	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑈 ∈ 𝐴)
146	7, 8	atcmp 39571	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑄 ≤ 𝑈 ↔ 𝑄 = 𝑈))
147	143, 144, 145, 146	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (𝑄 ≤ 𝑈 ↔ 𝑄 = 𝑈))
148	140, 147	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 = 𝑈)
149	27, 7, 82	latmle2 18388	. . . . . . . . . . . 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 1373	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ ((𝐹‘𝑃) ∨ 𝑉))
151	1, 150	eqbrtrid 5133	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉))
152	27, 7, 82	latlem12 18389	. . . . . . . . . . . 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 1374	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑄 ≤ 𝑊) ↔ 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)))
154	153	biimpd 229	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)))
155	151, 154	mpand 695	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊)))
156	155	imp 406	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊))
157	7, 82, 130, 8, 9	lhpmat 40290	. . . . . . . . . . . 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 7373	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = ((0.‘𝐾) ∨ 𝑉))
160	27, 8	atbase 39549	. . . . . . . . . . . 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 40126	. . . . . . . . . . 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 1385	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊))
164	27, 35, 130	olj02 39486	. . . . . . . . . . 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 2779	. . . . . . . . 9 ⊢ (𝜑 → (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊) = 𝑉)
167	166	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊) = 𝑉)
168	156, 167	breqtrd 5124	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ 𝑉)
169	21	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑉 ∈ 𝐴)
170	7, 8	atcmp 39571	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑄 ≤ 𝑉 ↔ 𝑄 = 𝑉))
171	143, 144, 169, 170	syl3anc 1373	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (𝑄 ≤ 𝑉 ↔ 𝑄 = 𝑉))
172	168, 171	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 = 𝑉)
173	148, 172	eqtr3d 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑈 = 𝑉)
174	173	ex 412	. . . 4 ⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑈 = 𝑉))
175	174	necon3ad 2945	. . 3 ⊢ (𝜑 → (𝑈 ≠ 𝑉 → ¬ 𝑄 ≤ 𝑊))
176	64, 175	mpd 15	. 2 ⊢ (𝜑 → ¬ 𝑄 ≤ 𝑊)
177	115, 176	jca 511	1 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  lecple 17184  joincjn 18234  meetcmee 18235  0.cp0 18344  1.cp1 18345  Latclat 18354  OLcol 39434  Atomscatm 39523  AtLatcal 39524  HLchlt 39610  LHypclh 40244  LTrncltrn 40361  trLctrl 40418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765  df-proset 18217  df-poset 18236  df-plt 18251  df-lub 18267  df-glb 18268  df-join 18269  df-meet 18270  df-p0 18346  df-p1 18347  df-lat 18355  df-clat 18422  df-oposet 39436  df-ol 39438  df-oml 39439  df-covers 39526  df-ats 39527  df-atl 39558  df-cvlat 39582  df-hlat 39611  df-llines 39758  df-psubsp 39763  df-pmap 39764  df-padd 40056  df-lhyp 40248  df-laut 40249  df-ldil 40364  df-ltrn 40365  df-trl 40419

This theorem is referenced by:  dia2dimlem3  41326  dia2dimlem6  41329