Theorem dia2dimlem3 41840

Description: Lemma for dia2dim 41851. Define a translation 𝐷 whose trace is atom 𝑉. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)

Hypotheses

Ref	Expression
dia2dimlem3.l	⊢ ≤ = (le‘𝐾)
dia2dimlem3.j	⊢ ∨ = (join‘𝐾)
dia2dimlem3.m	⊢ ∧ = (meet‘𝐾)
dia2dimlem3.a	⊢ 𝐴 = (Atoms‘𝐾)
dia2dimlem3.h	⊢ 𝐻 = (LHyp‘𝐾)
dia2dimlem3.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dia2dimlem3.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dia2dimlem3.q	⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))
dia2dimlem3.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dia2dimlem3.u	⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
dia2dimlem3.v	⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
dia2dimlem3.p	⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
dia2dimlem3.f	⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))
dia2dimlem3.rf	⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
dia2dimlem3.uv	⊢ (𝜑 → 𝑈 ≠ 𝑉)
dia2dimlem3.ru	⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)
dia2dimlem3.rv	⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉)
dia2dimlem3.d	⊢ (𝜑 → 𝐷 ∈ 𝑇)
dia2dimlem3.dv	⊢ (𝜑 → (𝐷‘𝑄) = (𝐹‘𝑃))

Assertion

Ref	Expression
dia2dimlem3	⊢ (𝜑 → (𝑅‘𝐷) = 𝑉)

Proof of Theorem dia2dimlem3

Step	Hyp	Ref	Expression
1		dia2dimlem3.k	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2	1	simpld 499	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL)
3		dia2dimlem3.f	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))
4	3	simpld 499	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ 𝑇)
5		dia2dimlem3.p	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
6		dia2dimlem3.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
7		dia2dimlem3.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
8		dia2dimlem3.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
9		dia2dimlem3.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
10	6, 7, 8, 9	ltrnel 40913	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
11	1, 4, 5, 10	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
12	11	simpld 499	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴)
13		dia2dimlem3.v	. . . . . . 7 ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
14	13	simpld 499	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
15		dia2dimlem3.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
16	6, 15, 7	hlatlej2 40150	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉))
17	2, 12, 14, 16	syl3anc 1398	. . . . 5 ⊢ (𝜑 → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉))
18	2	hllatd 40138	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat)
19		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
20	19, 7	atbase 40063	. . . . . . 7 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
21	14, 20	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
22	19, 15, 7	hlatjcl 40141	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾))
23	2, 12, 14, 22	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾))
24		dia2dimlem3.r	. . . . . . . . 9 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
25	6, 7, 8, 9, 24	trlat 40943	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
26	1, 5, 3, 25	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ∈ 𝐴)
27		dia2dimlem3.u	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
28	27	simpld 499	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
29	19, 15, 7	hlatjcl 40141	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑅‘𝐹) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾))
30	2, 26, 28, 29	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾))
31		dia2dimlem3.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
32	19, 6, 31	latmlem2 18521	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑉 ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾) ∧ ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾))) → (𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉) → (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) ≤ (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))))
33	18, 21, 23, 30, 32	syl13anc 1399	. . . . 5 ⊢ (𝜑 → (𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉) → (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) ≤ (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))))
34	17, 33	mpd 16	. . . 4 ⊢ (𝜑 → (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) ≤ (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
35		dia2dimlem3.rf	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
36	15, 7	hlatjcom 40142	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈))
37	2, 28, 14, 36	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈))
38	35, 37	breqtrd 5137	. . . . . 6 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈))
39		dia2dimlem3.ru	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)
40	6, 15, 7	hlatexch2 40170	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ ((𝑅‘𝐹) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) ∧ (𝑅‘𝐹) ≠ 𝑈) → ((𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈) → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈)))
41	2, 26, 14, 28, 39, 40	syl131anc 1410	. . . . . 6 ⊢ (𝜑 → ((𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈) → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈)))
42	38, 41	mpd 16	. . . . 5 ⊢ (𝜑 → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))
43	19, 6, 31	latleeqm2 18519	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑉 ∈ (Base‘𝐾) ∧ ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾)) → (𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈) ↔ (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) = 𝑉))
44	18, 21, 30, 43	syl3anc 1398	. . . . 5 ⊢ (𝜑 → (𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈) ↔ (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) = 𝑉))
45	42, 44	mpbid 235	. . . 4 ⊢ (𝜑 → (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) = 𝑉)
46		dia2dimlem3.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑇)
47		dia2dimlem3.q	. . . . . . 7 ⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))
48		dia2dimlem3.uv	. . . . . . 7 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
49	6, 15, 31, 7, 8, 9, 24, 47, 1, 27, 13, 5, 3, 35, 48, 39	dia2dimlem1 41838	. . . . . 6 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
50	6, 15, 31, 7, 8, 9, 24	trlval2 40937	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑅‘𝐷) = ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊))
51	1, 46, 49, 50	syl3anc 1398	. . . . 5 ⊢ (𝜑 → (𝑅‘𝐷) = ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊))
52	47	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
53		dia2dimlem3.dv	. . . . . . . . 9 ⊢ (𝜑 → (𝐷‘𝑄) = (𝐹‘𝑃))
54	52, 53	oveq12d 7428	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∨ (𝐷‘𝑄)) = (((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∨ (𝐹‘𝑃)))
55	5	simpld 499	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
56	19, 15, 7	hlatjcl 40141	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
57	2, 55, 28, 56	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
58	6, 15, 7	hlatlej1 40149	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ 𝑉))
59	2, 12, 14, 58	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ 𝑉))
60	19, 6, 15, 31, 7	atmod4i1 40640	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ (𝑃 ∨ 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾)) ∧ (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ 𝑉)) → (((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∨ (𝐹‘𝑃)) = (((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
61	2, 12, 57, 23, 59, 60	syl131anc 1410	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∨ (𝐹‘𝑃)) = (((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
62	15, 7	hlatj32 40146	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴)) → ((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) = ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈))
63	2, 55, 28, 12, 62	syl13anc 1399	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) = ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈))
64	63	oveq1d 7425	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
65	54, 61, 64	3eqtrd 2802	. . . . . . 7 ⊢ (𝜑 → (𝑄 ∨ (𝐷‘𝑄)) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
66	65	oveq1d 7425	. . . . . 6 ⊢ (𝜑 → ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊) = ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))
67		hlol 40135	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
68	2, 67	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ OL)
69	19, 15, 7	hlatjcl 40141	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
70	2, 55, 12, 69	syl3anc 1398	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
71	19, 7	atbase 40063	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
72	28, 71	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
73	19, 15	latjcl 18490	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∈ (Base‘𝐾))
74	18, 70, 72, 73	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∈ (Base‘𝐾))
75	1	simprd 500	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
76	19, 8	lhpbase 40772	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
77	75, 76	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
78	19, 31	latm32 40005	. . . . . . 7 ⊢ ((𝐾 ∈ OL ∧ (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∈ (Base‘𝐾) ∧ ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊) = ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
79	68, 74, 23, 77, 78	syl13anc 1399	. . . . . 6 ⊢ (𝜑 → ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊) = ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
80	6, 15, 31, 7, 8, 9, 24	trlval2 40937	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
81	1, 4, 5, 80	syl3anc 1398	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
82	81	oveq1d 7425	. . . . . . . 8 ⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) = (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑈))
83	27	simprd 500	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
84	19, 6, 15, 31, 7	atmod4i1 40640	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑈 ∈ 𝐴 ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑈 ≤ 𝑊) → (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑈) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊))
85	2, 28, 70, 77, 83, 84	syl131anc 1410	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑈) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊))
86	82, 85	eqtr2d 2799	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) = ((𝑅‘𝐹) ∨ 𝑈))
87	86	oveq1d 7425	. . . . . 6 ⊢ (𝜑 → ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
88	66, 79, 87	3eqtrd 2802	. . . . 5 ⊢ (𝜑 → ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊) = (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
89	51, 88	eqtr2d 2799	. . . 4 ⊢ (𝜑 → (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (𝑅‘𝐷))
90	34, 45, 89	3brtr3d 5142	. . 3 ⊢ (𝜑 → 𝑉 ≤ (𝑅‘𝐷))
91		hlatl 40134	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
92	2, 91	syl 18	. . . 4 ⊢ (𝜑 → 𝐾 ∈ AtLat)
93		hlop 40136	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
94	2, 93	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ OP)
95		eqid 2763	. . . . . . . . . 10 ⊢ (0.‘𝐾) = (0.‘𝐾)
96		eqid 2763	. . . . . . . . . 10 ⊢ (lt‘𝐾) = (lt‘𝐾)
97	95, 96, 7	0ltat 40065	. . . . . . . . 9 ⊢ ((𝐾 ∈ OP ∧ 𝑉 ∈ 𝐴) → (0.‘𝐾)(lt‘𝐾)𝑉)
98	94, 14, 97	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)𝑉)
99		hlpos 40140	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
100	2, 99	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Poset)
101	19, 95	op0cl 39958	. . . . . . . . . 10 ⊢ (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
102	94, 101	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾))
103	19, 8, 9, 24	trlcl 40938	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → (𝑅‘𝐷) ∈ (Base‘𝐾))
104	1, 46, 103	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝐷) ∈ (Base‘𝐾))
105	19, 6, 96	pltletr 18392	. . . . . . . . 9 ⊢ ((𝐾 ∈ Poset ∧ ((0.‘𝐾) ∈ (Base‘𝐾) ∧ 𝑉 ∈ (Base‘𝐾) ∧ (𝑅‘𝐷) ∈ (Base‘𝐾))) → (((0.‘𝐾)(lt‘𝐾)𝑉 ∧ 𝑉 ≤ (𝑅‘𝐷)) → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷)))
106	100, 102, 21, 104, 105	syl13anc 1399	. . . . . . . 8 ⊢ (𝜑 → (((0.‘𝐾)(lt‘𝐾)𝑉 ∧ 𝑉 ≤ (𝑅‘𝐷)) → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷)))
107	98, 90, 106	mp2and 711	. . . . . . 7 ⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷))
108	19, 96, 95	opltn0 39964	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ (𝑅‘𝐷) ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷) ↔ (𝑅‘𝐷) ≠ (0.‘𝐾)))
109	94, 104, 108	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷) ↔ (𝑅‘𝐷) ≠ (0.‘𝐾)))
110	107, 109	mpbid 235	. . . . . 6 ⊢ (𝜑 → (𝑅‘𝐷) ≠ (0.‘𝐾))
111	110	neneqd 2963	. . . . 5 ⊢ (𝜑 → ¬ (𝑅‘𝐷) = (0.‘𝐾))
112	95, 7, 8, 9, 24	trlator0 40945	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ((𝑅‘𝐷) ∈ 𝐴 ∨ (𝑅‘𝐷) = (0.‘𝐾)))
113	1, 46, 112	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → ((𝑅‘𝐷) ∈ 𝐴 ∨ (𝑅‘𝐷) = (0.‘𝐾)))
114	113	orcomd 884	. . . . . 6 ⊢ (𝜑 → ((𝑅‘𝐷) = (0.‘𝐾) ∨ (𝑅‘𝐷) ∈ 𝐴))
115	114	ord 877	. . . . 5 ⊢ (𝜑 → (¬ (𝑅‘𝐷) = (0.‘𝐾) → (𝑅‘𝐷) ∈ 𝐴))
116	111, 115	mpd 16	. . . 4 ⊢ (𝜑 → (𝑅‘𝐷) ∈ 𝐴)
117	6, 7	atcmp 40085	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑉 ∈ 𝐴 ∧ (𝑅‘𝐷) ∈ 𝐴) → (𝑉 ≤ (𝑅‘𝐷) ↔ 𝑉 = (𝑅‘𝐷)))
118	92, 14, 116, 117	syl3anc 1398	. . 3 ⊢ (𝜑 → (𝑉 ≤ (𝑅‘𝐷) ↔ 𝑉 = (𝑅‘𝐷)))
119	90, 118	mpbid 235	. 2 ⊢ (𝜑 → 𝑉 = (𝑅‘𝐷))
120	119	eqcomd 2769	1 ⊢ (𝜑 → (𝑅‘𝐷) = 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  lecple 17312  Posetcpo 18358  ltcplt 18359  joincjn 18362  meetcmee 18363  0.cp0 18472  Latclat 18482  OPcops 39946  OLcol 39948  Atomscatm 40037  AtLatcal 40038  HLchlt 40124  LHypclh 40758  LTrncltrn 40875  trLctrl 40932

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 39950  df-ol 39952  df-oml 39953  df-covers 40040  df-ats 40041  df-atl 40072  df-cvlat 40096  df-hlat 40125  df-llines 40272  df-psubsp 40277  df-pmap 40278  df-padd 40570  df-lhyp 40762  df-laut 40763  df-ldil 40878  df-ltrn 40879  df-trl 40933

This theorem is referenced by:  dia2dimlem5  41842