Theorem dia2dimlem3 41940

Description: Lemma for dia2dim 41951. 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 500	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL)
3		dia2dimlem3.f	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))
4	3	simpld 500	. . . . . . . 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 41013	. . . . . . . 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 500	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴)
13		dia2dimlem3.v	. . . . . . 7 ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
14	13	simpld 500	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
15		dia2dimlem3.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
16	6, 15, 7	hlatlej2 40250	. . . . . 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 40238	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat)
19		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
20	19, 7	atbase 40163	. . . . . . 7 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
21	14, 20	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
22	19, 15, 7	hlatjcl 40241	. . . . . . 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 41043	. . . . . . . 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 500	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
29	19, 15, 7	hlatjcl 40241	. . . . . . 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 18559	. . . . . 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 40242	. . . . . . . 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 5131	. . . . . 6 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈))
39		dia2dimlem3.ru	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)
40	6, 15, 7	hlatexch2 40270	. . . . . . 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 18557	. . . . . 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 41938	. . . . . 6 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
50	6, 15, 31, 7, 8, 9, 24	trlval2 41037	. . . . . 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 7432	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∨ (𝐷‘𝑄)) = (((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∨ (𝐹‘𝑃)))
55	5	simpld 500	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
56	19, 15, 7	hlatjcl 40241	. . . . . . . . . 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 40249	. . . . . . . . . 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 40740	. . . . . . . . 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 40246	. . . . . . . . . 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 7429	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
65	54, 61, 64	3eqtrd 2799	. . . . . . 7 ⊢ (𝜑 → (𝑄 ∨ (𝐷‘𝑄)) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
66	65	oveq1d 7429	. . . . . 6 ⊢ (𝜑 → ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊) = ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))
67		hlol 40235	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
68	2, 67	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ OL)
69	19, 15, 7	hlatjcl 40241	. . . . . . . . 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 40163	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
72	28, 71	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
73	19, 15	latjcl 18528	. . . . . . . 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 501	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
76	19, 8	lhpbase 40872	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
77	75, 76	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
78	19, 31	latm32 40105	. . . . . . 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 41037	. . . . . . . . . 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 7429	. . . . . . . 8 ⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) = (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑈))
83	27	simprd 501	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
84	19, 6, 15, 31, 7	atmod4i1 40740	. . . . . . . . 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 2796	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) = ((𝑅‘𝐹) ∨ 𝑈))
87	86	oveq1d 7429	. . . . . 6 ⊢ (𝜑 → ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
88	66, 79, 87	3eqtrd 2799	. . . . 5 ⊢ (𝜑 → ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊) = (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
89	51, 88	eqtr2d 2796	. . . 4 ⊢ (𝜑 → (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (𝑅‘𝐷))
90	34, 45, 89	3brtr3d 5136	. . 3 ⊢ (𝜑 → 𝑉 ≤ (𝑅‘𝐷))
91		hlatl 40234	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
92	2, 91	syl 18	. . . 4 ⊢ (𝜑 → 𝐾 ∈ AtLat)
93		hlop 40236	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
94	2, 93	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ OP)
95		eqid 2760	. . . . . . . . . 10 ⊢ (0.‘𝐾) = (0.‘𝐾)
96		eqid 2760	. . . . . . . . . 10 ⊢ (lt‘𝐾) = (lt‘𝐾)
97	95, 96, 7	0ltat 40165	. . . . . . . . 9 ⊢ ((𝐾 ∈ OP ∧ 𝑉 ∈ 𝐴) → (0.‘𝐾)(lt‘𝐾)𝑉)
98	94, 14, 97	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)𝑉)
99		hlpos 40240	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
100	2, 99	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Poset)
101	19, 95	op0cl 40058	. . . . . . . . . 10 ⊢ (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
102	94, 101	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾))
103	19, 8, 9, 24	trlcl 41038	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → (𝑅‘𝐷) ∈ (Base‘𝐾))
104	1, 46, 103	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝐷) ∈ (Base‘𝐾))
105	19, 6, 96	pltletr 18430	. . . . . . . . 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 712	. . . . . . 7 ⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷))
108	19, 96, 95	opltn0 40064	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ (𝑅‘𝐷) ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷) ↔ (𝑅‘𝐷) ≠ (0.‘𝐾)))
109	94, 104, 108	syl2anc 596	. . . . . . 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 2960	. . . . 5 ⊢ (𝜑 → ¬ (𝑅‘𝐷) = (0.‘𝐾))
112	95, 7, 8, 9, 24	trlator0 41045	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ((𝑅‘𝐷) ∈ 𝐴 ∨ (𝑅‘𝐷) = (0.‘𝐾)))
113	1, 46, 112	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → ((𝑅‘𝐷) ∈ 𝐴 ∨ (𝑅‘𝐷) = (0.‘𝐾)))
114	113	orcomd 885	. . . . . 6 ⊢ (𝜑 → ((𝑅‘𝐷) = (0.‘𝐾) ∨ (𝑅‘𝐷) ∈ 𝐴))
115	114	ord 878	. . . . 5 ⊢ (𝜑 → (¬ (𝑅‘𝐷) = (0.‘𝐾) → (𝑅‘𝐷) ∈ 𝐴))
116	111, 115	mpd 16	. . . 4 ⊢ (𝜑 → (𝑅‘𝐷) ∈ 𝐴)
117	6, 7	atcmp 40185	. . . 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 2766	1 ⊢ (𝜑 → (𝑅‘𝐷) = 𝑉)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ≠ wne 2955   class class class wbr 5103  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  lecple 17350  Posetcpo 18396  ltcplt 18397  joincjn 18400  meetcmee 18401  0.cp0 18510  Latclat 18520  OPcops 40046  OLcol 40048  Atomscatm 40137  AtLatcal 40138  HLchlt 40224  LHypclh 40858  LTrncltrn 40975  trLctrl 41032

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8829  df-proset 18383  df-poset 18402  df-plt 18417  df-lub 18433  df-glb 18434  df-join 18435  df-meet 18436  df-p0 18512  df-p1 18513  df-lat 18521  df-clat 18588  df-oposet 40050  df-ol 40052  df-oml 40053  df-covers 40140  df-ats 40141  df-atl 40172  df-cvlat 40196  df-hlat 40225  df-llines 40372  df-psubsp 40377  df-pmap 40378  df-padd 40670  df-lhyp 40862  df-laut 40863  df-ldil 40978  df-ltrn 40979  df-trl 41033

This theorem is used by:  dia2dimlem5  41942