Theorem dia2dimlem1 41047

Description: Lemma for dia2dim 41060. 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 40152	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
13	2, 4, 6, 12	syl3anc 1370	. . . 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 40122	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
18	2, 16, 4, 17	syl3anc 1370	. . . . 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 40167	. . . . . . . . 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 39346	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Lat)
27		eqid 2735	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
28	27, 8	atbase 39271	. . . . . . . . . 10 ⊢ ((𝑅‘𝐹) ∈ 𝐴 → (𝑅‘𝐹) ∈ (Base‘𝐾))
29	13, 28	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝐹) ∈ (Base‘𝐾))
30	27, 8	atbase 39271	. . . . . . . . . 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 39981	. . . . . . . . . 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 18508	. . . . . . . . 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 1371	. . . . . . . 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 39271	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
40	5, 39	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
41	27, 35, 8	hlatjcl 39349	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑅‘𝐹) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾))
42	3, 13, 15, 41	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾))
43	27, 7	lattr 18502	. . . . . . . 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 1371	. . . . . . 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 5168	. . . . . . 7 ⊢ ((𝑉 ≤ 𝑊 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊) → 𝑉 ≠ (𝐹‘𝑃))
50	47, 48, 49	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝑉 ≠ (𝐹‘𝑃))
51	50	necomd 2994	. . . . 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 39349	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑉 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑉 ∨ 𝑈) ∈ (Base‘𝐾))
56	3, 21, 15, 55	syl3anc 1370	. . . . . . . . 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 39358	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉))
60	3, 19, 21, 59	syl3anc 1370	. . . . . . . . . . 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 5176	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑉 ≤ (𝑃 ∨ 𝑈))
64		dia2dimlem1.uv	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
65	64	necomd 2994	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ≠ 𝑈)
66	7, 35, 8	hlatexch2 39379	. . . . . . . . . . 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 1382	. . . . . . . . . 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 39271	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
71	21, 70	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
72	27, 7, 35	latjle12 18508	. . . . . . . . . . 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 1371	. . . . . . . . . 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 18504	. . . . . . 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 2952	. . . . 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 39358	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝐹‘𝑃)))
81	3, 5, 19, 80	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝐹‘𝑃)))
82		dia2dimlem1.m	. . . . . . . . . 10 ⊢ ∧ = (meet‘𝐾)
83	7, 35, 82, 8, 9, 10, 11	trlval2 40146	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
84	2, 16, 4, 83	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
85	84	oveq2d 7447	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)))
86	27, 35, 8	hlatjcl 39349	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
87	3, 5, 19, 86	syl3anc 1370	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
88	7, 35, 8	hlatlej1 39357	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
89	3, 5, 19, 88	syl3anc 1370	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ (𝐹‘𝑃)))
90	27, 7, 35, 82, 8	atmod3i1 39847	. . . . . . . . 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 1382	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)))
92		eqid 2735	. . . . . . . . . . . 12 ⊢ (1.‘𝐾) = (1.‘𝐾)
93	7, 35, 92, 8, 9	lhpjat2 40004	. . . . . . . . . . 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 7447	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾)))
96		hlol 39343	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
97	3, 96	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ OL)
98	27, 82, 92	olm11 39209	. . . . . . . . . 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 2775	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)) = (𝑃 ∨ (𝐹‘𝑃)))
101	91, 100	eqtrd 2775	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) = (𝑃 ∨ (𝐹‘𝑃)))
102	85, 101	eqtrd 2775	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃)))
103	81, 102	breqtrrd 5176	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)))
104		dia2dimlem1.rf	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
105	35, 8	hlatjcom 39350	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈))
106	3, 15, 21, 105	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈))
107	104, 106	breqtrd 5174	. . . . . 6 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈))
108		dia2dimlem1.ru	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)
109	7, 35, 8	hlatexch2 39379	. . . . . . 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 1382	. . . . . 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 39511	. . . 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 1401	. . 3 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∈ 𝐴)
115	1, 114	eqeltrid 2843	. 2 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
116	27, 35, 8	hlatjcl 39349	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
117	3, 5, 15, 116	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
118	27, 35, 8	hlatjcl 39349	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾))
119	3, 19, 21, 118	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾))
120	27, 7, 82	latmle1 18522	. . . . . . . . . . . 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 1370	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ (𝑃 ∨ 𝑈))
122	1, 121	eqbrtrid 5183	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ≤ (𝑃 ∨ 𝑈))
123	27, 8	atbase 39271	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
124	115, 123	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
125	27, 7, 82	latlem12 18524	. . . . . . . . . . . 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 1371	. . . . . . . . . . 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 2735	. . . . . . . . . . . . 13 ⊢ (0.‘𝐾) = (0.‘𝐾)
131	7, 82, 130, 8, 9	lhpmat 40013	. . . . . . . . . . . 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 7446	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((0.‘𝐾) ∨ 𝑈))
134	27, 7, 35, 82, 8	atmod4i1 39849	. . . . . . . . . . 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 1382	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((𝑃 ∨ 𝑈) ∧ 𝑊))
136	27, 35, 130	olj02 39208	. . . . . . . . . . 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 2783	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ 𝑊) = 𝑈)
139	138	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → ((𝑃 ∨ 𝑈) ∧ 𝑊) = 𝑈)
140	129, 139	breqtrd 5174	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ 𝑈)
141		hlatl 39342	. . . . . . . . . 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 39293	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑄 ≤ 𝑈 ↔ 𝑄 = 𝑈))
147	143, 144, 145, 146	syl3anc 1370	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (𝑄 ≤ 𝑈 ↔ 𝑄 = 𝑈))
148	140, 147	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 = 𝑈)
149	27, 7, 82	latmle2 18523	. . . . . . . . . . . 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 1370	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ≤ ((𝐹‘𝑃) ∨ 𝑉))
151	1, 150	eqbrtrid 5183	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉))
152	27, 7, 82	latlem12 18524	. . . . . . . . . . . 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 1371	. . . . . . . . . . 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 40013	. . . . . . . . . . . 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 7446	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = ((0.‘𝐾) ∨ 𝑉))
160	27, 8	atbase 39271	. . . . . . . . . . . 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 39849	. . . . . . . . . . 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 1382	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊))
164	27, 35, 130	olj02 39208	. . . . . . . . . . 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 2783	. . . . . . . . 9 ⊢ (𝜑 → (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊) = 𝑉)
167	166	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊) = 𝑉)
168	156, 167	breqtrd 5174	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ 𝑉)
169	21	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑉 ∈ 𝐴)
170	7, 8	atcmp 39293	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑄 ≤ 𝑉 ↔ 𝑄 = 𝑉))
171	143, 144, 169, 170	syl3anc 1370	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (𝑄 ≤ 𝑉 ↔ 𝑄 = 𝑉))
172	168, 171	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 = 𝑉)
173	148, 172	eqtr3d 2777	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑈 = 𝑉)
174	173	ex 412	. . . 4 ⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑈 = 𝑉))
175	174	necon3ad 2951	. . 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 1537   ∈ wcel 2106   ≠ wne 2938   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  0.cp0 18481  1.cp1 18482  Latclat 18489  OLcol 39156  Atomscatm 39245  AtLatcal 39246  HLchlt 39332  LHypclh 39967  LTrncltrn 40084  trLctrl 40141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142

This theorem is referenced by:  dia2dimlem3  41049  dia2dimlem6  41052