Theorem dia2dimlem1 41065

Description: Lemma for dia2dim 41078. 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 40170	. . . . 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 40140	. . . . . 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 40185	. . . . . . . . 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 39364	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Lat)
27		eqid 2730	. . . . . . . . . . 11 ⊢ (Base‘𝐾) = (Base‘𝐾)
28	27, 8	atbase 39289	. . . . . . . . . 10 ⊢ ((𝑅‘𝐹) ∈ 𝐴 → (𝑅‘𝐹) ∈ (Base‘𝐾))
29	13, 28	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝐹) ∈ (Base‘𝐾))
30	27, 8	atbase 39289	. . . . . . . . . 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 39999	. . . . . . . . . 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 18416	. . . . . . . . 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 39289	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
40	5, 39	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
41	27, 35, 8	hlatjcl 39367	. . . . . . . . 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 18410	. . . . . . . 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 5130	. . . . . . 7 ⊢ ((𝑉 ≤ 𝑊 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊) → 𝑉 ≠ (𝐹‘𝑃))
50	47, 48, 49	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝑉 ≠ (𝐹‘𝑃))
51	50	necomd 2981	. . . . 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 39367	. . . . . . . . . 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 39376	. . . . . . . . . . . 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 5138	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑈) = ((𝐹‘𝑃) ∨ 𝑉)) → 𝑉 ≤ (𝑃 ∨ 𝑈))
64		dia2dimlem1.uv	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
65	64	necomd 2981	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ≠ 𝑈)
66	7, 35, 8	hlatexch2 39397	. . . . . . . . . . 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 39289	. . . . . . . . . . . 12 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
71	21, 70	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
72	27, 7, 35	latjle12 18416	. . . . . . . . . . 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 18412	. . . . . . 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 2940	. . . . 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 39376	. . . . . . 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 40164	. . . . . . . . 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 7406	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)))
86	27, 35, 8	hlatjcl 39367	. . . . . . . . . 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 39375	. . . . . . . . . 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 39865	. . . . . . . . 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 2730	. . . . . . . . . . . 12 ⊢ (1.‘𝐾) = (1.‘𝐾)
93	7, 35, 92, 8, 9	lhpjat2 40022	. . . . . . . . . . 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 7406	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ (1.‘𝐾)))
96		hlol 39361	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
97	3, 96	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ OL)
98	27, 82, 92	olm11 39227	. . . . . . . . . 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 2765	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∧ (𝑃 ∨ 𝑊)) = (𝑃 ∨ (𝐹‘𝑃)))
101	91, 100	eqtrd 2765	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊)) = (𝑃 ∨ (𝐹‘𝑃)))
102	85, 101	eqtrd 2765	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ (𝑅‘𝐹)) = (𝑃 ∨ (𝐹‘𝑃)))
103	81, 102	breqtrrd 5138	. . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) ≤ (𝑃 ∨ (𝑅‘𝐹)))
104		dia2dimlem1.rf	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
105	35, 8	hlatjcom 39368	. . . . . . . 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 5136	. . . . . 6 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈))
108		dia2dimlem1.ru	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)
109	7, 35, 8	hlatexch2 39397	. . . . . . 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 39529	. . . 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 2833	. 2 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
116	27, 35, 8	hlatjcl 39367	. . . . . . . . . . . . 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 39367	. . . . . . . . . . . . 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 18430	. . . . . . . . . . . 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 5145	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ≤ (𝑃 ∨ 𝑈))
123	27, 8	atbase 39289	. . . . . . . . . . . . 13 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
124	115, 123	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
125	27, 7, 82	latlem12 18432	. . . . . . . . . . . 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 2730	. . . . . . . . . . . . 13 ⊢ (0.‘𝐾) = (0.‘𝐾)
131	7, 82, 130, 8, 9	lhpmat 40031	. . . . . . . . . . . 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 7405	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 ∧ 𝑊) ∨ 𝑈) = ((0.‘𝐾) ∨ 𝑈))
134	27, 7, 35, 82, 8	atmod4i1 39867	. . . . . . . . . . 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 39226	. . . . . . . . . . 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 2773	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∧ 𝑊) = 𝑈)
139	138	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → ((𝑃 ∨ 𝑈) ∧ 𝑊) = 𝑈)
140	129, 139	breqtrd 5136	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ 𝑈)
141		hlatl 39360	. . . . . . . . . 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 39311	. . . . . . . 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 18431	. . . . . . . . . . . 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 5145	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ≤ ((𝐹‘𝑃) ∨ 𝑉))
152	27, 7, 82	latlem12 18432	. . . . . . . . . . . 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 40031	. . . . . . . . . . . 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 7405	. . . . . . . . . 10 ⊢ (𝜑 → (((𝐹‘𝑃) ∧ 𝑊) ∨ 𝑉) = ((0.‘𝐾) ∨ 𝑉))
160	27, 8	atbase 39289	. . . . . . . . . . . 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 39867	. . . . . . . . . . 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 39226	. . . . . . . . . . 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 2773	. . . . . . . . 9 ⊢ (𝜑 → (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊) = 𝑉)
167	166	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → (((𝐹‘𝑃) ∨ 𝑉) ∧ 𝑊) = 𝑉)
168	156, 167	breqtrd 5136	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑄 ≤ 𝑉)
169	21	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑉 ∈ 𝐴)
170	7, 8	atcmp 39311	. . . . . . . 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 2767	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 ≤ 𝑊) → 𝑈 = 𝑉)
174	173	ex 412	. . . 4 ⊢ (𝜑 → (𝑄 ≤ 𝑊 → 𝑈 = 𝑉))
175	174	necon3ad 2939	. . 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 1540   ∈ wcel 2109   ≠ wne 2926   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  lecple 17234  joincjn 18279  meetcmee 18280  0.cp0 18389  1.cp1 18390  Latclat 18397  OLcol 39174  Atomscatm 39263  AtLatcal 39264  HLchlt 39350  LHypclh 39985  LTrncltrn 40102  trLctrl 40159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804  df-proset 18262  df-poset 18281  df-plt 18296  df-lub 18312  df-glb 18313  df-join 18314  df-meet 18315  df-p0 18391  df-p1 18392  df-lat 18398  df-clat 18465  df-oposet 39176  df-ol 39178  df-oml 39179  df-covers 39266  df-ats 39267  df-atl 39298  df-cvlat 39322  df-hlat 39351  df-llines 39499  df-psubsp 39504  df-pmap 39505  df-padd 39797  df-lhyp 39989  df-laut 39990  df-ldil 40105  df-ltrn 40106  df-trl 40160

This theorem is referenced by:  dia2dimlem3  41067  dia2dimlem6  41070