Theorem dia2dimlem3 41558

Description: Lemma for dia2dim 41569. 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 495	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL)
3		dia2dimlem3.f	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))
4	3	simpld 495	. . . . . . . 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 40631	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
11	1, 4, 5, 10	syl3anc 1379	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
12	11	simpld 495	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴)
13		dia2dimlem3.v	. . . . . . 7 ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
14	13	simpld 495	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
15		dia2dimlem3.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
16	6, 15, 7	hlatlej2 39868	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉))
17	2, 12, 14, 16	syl3anc 1379	. . . . 5 ⊢ (𝜑 → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉))
18	2	hllatd 39856	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat)
19		eqid 2739	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
20	19, 7	atbase 39781	. . . . . . 7 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
21	14, 20	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
22	19, 15, 7	hlatjcl 39859	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾))
23	2, 12, 14, 22	syl3anc 1379	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾))
24		dia2dimlem3.r	. . . . . . . . 9 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
25	6, 7, 8, 9, 24	trlat 40661	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
26	1, 5, 3, 25	syl3anc 1379	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ∈ 𝐴)
27		dia2dimlem3.u	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
28	27	simpld 495	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
29	19, 15, 7	hlatjcl 39859	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑅‘𝐹) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾))
30	2, 26, 28, 29	syl3anc 1379	. . . . . 6 ⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾))
31		dia2dimlem3.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
32	19, 6, 31	latmlem2 18427	. . . . . 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 1380	. . . . 5 ⊢ (𝜑 → (𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉) → (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) ≤ (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))))
34	17, 33	mpd 15	. . . 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 39860	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈))
37	2, 28, 14, 36	syl3anc 1379	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈))
38	35, 37	breqtrd 5098	. . . . . 6 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈))
39		dia2dimlem3.ru	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)
40	6, 15, 7	hlatexch2 39888	. . . . . . 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 1391	. . . . . 6 ⊢ (𝜑 → ((𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈) → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈)))
42	38, 41	mpd 15	. . . . 5 ⊢ (𝜑 → 𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈))
43	19, 6, 31	latleeqm2 18425	. . . . . 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 1379	. . . . 5 ⊢ (𝜑 → (𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈) ↔ (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) = 𝑉))
45	42, 44	mpbid 233	. . . 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 41556	. . . . . 6 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
50	6, 15, 31, 7, 8, 9, 24	trlval2 40655	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑅‘𝐷) = ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊))
51	1, 46, 49, 50	syl3anc 1379	. . . . 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 7374	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∨ (𝐷‘𝑄)) = (((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∨ (𝐹‘𝑃)))
55	5	simpld 495	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
56	19, 15, 7	hlatjcl 39859	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
57	2, 55, 28, 56	syl3anc 1379	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
58	6, 15, 7	hlatlej1 39867	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ 𝑉))
59	2, 12, 14, 58	syl3anc 1379	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ 𝑉))
60	19, 6, 15, 31, 7	atmod4i1 40358	. . . . . . . . 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 1391	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∨ (𝐹‘𝑃)) = (((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
62	15, 7	hlatj32 39864	. . . . . . . . . 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 1380	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) = ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈))
64	63	oveq1d 7371	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
65	54, 61, 64	3eqtrd 2778	. . . . . . 7 ⊢ (𝜑 → (𝑄 ∨ (𝐷‘𝑄)) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
66	65	oveq1d 7371	. . . . . 6 ⊢ (𝜑 → ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊) = ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))
67		hlol 39853	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
68	2, 67	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ OL)
69	19, 15, 7	hlatjcl 39859	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
70	2, 55, 12, 69	syl3anc 1379	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
71	19, 7	atbase 39781	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
72	28, 71	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
73	19, 15	latjcl 18396	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∈ (Base‘𝐾))
74	18, 70, 72, 73	syl3anc 1379	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∈ (Base‘𝐾))
75	1	simprd 496	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
76	19, 8	lhpbase 40490	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
77	75, 76	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
78	19, 31	latm32 39723	. . . . . . 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 1380	. . . . . 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 40655	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
81	1, 4, 5, 80	syl3anc 1379	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
82	81	oveq1d 7371	. . . . . . . 8 ⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) = (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑈))
83	27	simprd 496	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
84	19, 6, 15, 31, 7	atmod4i1 40358	. . . . . . . . 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 1391	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑈) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊))
86	82, 85	eqtr2d 2775	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) = ((𝑅‘𝐹) ∨ 𝑈))
87	86	oveq1d 7371	. . . . . 6 ⊢ (𝜑 → ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
88	66, 79, 87	3eqtrd 2778	. . . . 5 ⊢ (𝜑 → ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊) = (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
89	51, 88	eqtr2d 2775	. . . 4 ⊢ (𝜑 → (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (𝑅‘𝐷))
90	34, 45, 89	3brtr3d 5103	. . 3 ⊢ (𝜑 → 𝑉 ≤ (𝑅‘𝐷))
91		hlatl 39852	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
92	2, 91	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ AtLat)
93		hlop 39854	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
94	2, 93	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ OP)
95		eqid 2739	. . . . . . . . . 10 ⊢ (0.‘𝐾) = (0.‘𝐾)
96		eqid 2739	. . . . . . . . . 10 ⊢ (lt‘𝐾) = (lt‘𝐾)
97	95, 96, 7	0ltat 39783	. . . . . . . . 9 ⊢ ((𝐾 ∈ OP ∧ 𝑉 ∈ 𝐴) → (0.‘𝐾)(lt‘𝐾)𝑉)
98	94, 14, 97	syl2anc 590	. . . . . . . 8 ⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)𝑉)
99		hlpos 39858	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
100	2, 99	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Poset)
101	19, 95	op0cl 39676	. . . . . . . . . 10 ⊢ (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
102	94, 101	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾))
103	19, 8, 9, 24	trlcl 40656	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → (𝑅‘𝐷) ∈ (Base‘𝐾))
104	1, 46, 103	syl2anc 590	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝐷) ∈ (Base‘𝐾))
105	19, 6, 96	pltletr 18298	. . . . . . . . 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 1380	. . . . . . . 8 ⊢ (𝜑 → (((0.‘𝐾)(lt‘𝐾)𝑉 ∧ 𝑉 ≤ (𝑅‘𝐷)) → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷)))
107	98, 90, 106	mp2and 705	. . . . . . 7 ⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷))
108	19, 96, 95	opltn0 39682	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ (𝑅‘𝐷) ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷) ↔ (𝑅‘𝐷) ≠ (0.‘𝐾)))
109	94, 104, 108	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷) ↔ (𝑅‘𝐷) ≠ (0.‘𝐾)))
110	107, 109	mpbid 233	. . . . . 6 ⊢ (𝜑 → (𝑅‘𝐷) ≠ (0.‘𝐾))
111	110	neneqd 2939	. . . . 5 ⊢ (𝜑 → ¬ (𝑅‘𝐷) = (0.‘𝐾))
112	95, 7, 8, 9, 24	trlator0 40663	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ((𝑅‘𝐷) ∈ 𝐴 ∨ (𝑅‘𝐷) = (0.‘𝐾)))
113	1, 46, 112	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → ((𝑅‘𝐷) ∈ 𝐴 ∨ (𝑅‘𝐷) = (0.‘𝐾)))
114	113	orcomd 877	. . . . . 6 ⊢ (𝜑 → ((𝑅‘𝐷) = (0.‘𝐾) ∨ (𝑅‘𝐷) ∈ 𝐴))
115	114	ord 870	. . . . 5 ⊢ (𝜑 → (¬ (𝑅‘𝐷) = (0.‘𝐾) → (𝑅‘𝐷) ∈ 𝐴))
116	111, 115	mpd 15	. . . 4 ⊢ (𝜑 → (𝑅‘𝐷) ∈ 𝐴)
117	6, 7	atcmp 39803	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑉 ∈ 𝐴 ∧ (𝑅‘𝐷) ∈ 𝐴) → (𝑉 ≤ (𝑅‘𝐷) ↔ 𝑉 = (𝑅‘𝐷)))
118	92, 14, 116, 117	syl3anc 1379	. . 3 ⊢ (𝜑 → (𝑉 ≤ (𝑅‘𝐷) ↔ 𝑉 = (𝑅‘𝐷)))
119	90, 118	mpbid 233	. 2 ⊢ (𝜑 → 𝑉 = (𝑅‘𝐷))
120	119	eqcomd 2745	1 ⊢ (𝜑 → (𝑅‘𝐷) = 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ≠ wne 2934   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  lecple 17218  Posetcpo 18264  ltcplt 18265  joincjn 18268  meetcmee 18269  0.cp0 18378  Latclat 18388  OPcops 39664  OLcol 39666  Atomscatm 39755  AtLatcal 39756  HLchlt 39842  LHypclh 40476  LTrncltrn 40593  trLctrl 40650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8765  df-proset 18251  df-poset 18270  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-p1 18381  df-lat 18389  df-clat 18456  df-oposet 39668  df-ol 39670  df-oml 39671  df-covers 39758  df-ats 39759  df-atl 39790  df-cvlat 39814  df-hlat 39843  df-llines 39990  df-psubsp 39995  df-pmap 39996  df-padd 40288  df-lhyp 40480  df-laut 40481  df-ldil 40596  df-ltrn 40597  df-trl 40651

This theorem is referenced by:  dia2dimlem5  41560