Theorem dia2dimlem3 41900

Description: Lemma for dia2dim 41911. 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 40973	. . . . . . . 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 40210	. . . . . 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 40198	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat)
19		eqid 2765	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
20	19, 7	atbase 40123	. . . . . . 7 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
21	14, 20	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
22	19, 15, 7	hlatjcl 40201	. . . . . . 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 41003	. . . . . . . 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 40201	. . . . . . 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 18550	. . . . . 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 40202	. . . . . . . 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 5139	. . . . . 6 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈))
39		dia2dimlem3.ru	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)
40	6, 15, 7	hlatexch2 40230	. . . . . . 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 18548	. . . . . 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 41898	. . . . . 6 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
50	6, 15, 31, 7, 8, 9, 24	trlval2 40997	. . . . . 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 7437	. . . . . . . 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 40201	. . . . . . . . . 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 40209	. . . . . . . . . 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 40700	. . . . . . . . 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 40206	. . . . . . . . . 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 7434	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
65	54, 61, 64	3eqtrd 2804	. . . . . . 7 ⊢ (𝜑 → (𝑄 ∨ (𝐷‘𝑄)) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
66	65	oveq1d 7434	. . . . . 6 ⊢ (𝜑 → ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊) = ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))
67		hlol 40195	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
68	2, 67	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ OL)
69	19, 15, 7	hlatjcl 40201	. . . . . . . . 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 40123	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
72	28, 71	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
73	19, 15	latjcl 18519	. . . . . . . 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 40832	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
77	75, 76	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
78	19, 31	latm32 40065	. . . . . . 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 40997	. . . . . . . . . 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 7434	. . . . . . . 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 40700	. . . . . . . . 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 2801	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) = ((𝑅‘𝐹) ∨ 𝑈))
87	86	oveq1d 7434	. . . . . 6 ⊢ (𝜑 → ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
88	66, 79, 87	3eqtrd 2804	. . . . 5 ⊢ (𝜑 → ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊) = (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
89	51, 88	eqtr2d 2801	. . . 4 ⊢ (𝜑 → (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (𝑅‘𝐷))
90	34, 45, 89	3brtr3d 5144	. . 3 ⊢ (𝜑 → 𝑉 ≤ (𝑅‘𝐷))
91		hlatl 40194	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
92	2, 91	syl 18	. . . 4 ⊢ (𝜑 → 𝐾 ∈ AtLat)
93		hlop 40196	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
94	2, 93	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ OP)
95		eqid 2765	. . . . . . . . . 10 ⊢ (0.‘𝐾) = (0.‘𝐾)
96		eqid 2765	. . . . . . . . . 10 ⊢ (lt‘𝐾) = (lt‘𝐾)
97	95, 96, 7	0ltat 40125	. . . . . . . . 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 40200	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
100	2, 99	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Poset)
101	19, 95	op0cl 40018	. . . . . . . . . 10 ⊢ (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
102	94, 101	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾))
103	19, 8, 9, 24	trlcl 40998	. . . . . . . . . 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 18421	. . . . . . . . 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 40024	. . . . . . . 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 2965	. . . . 5 ⊢ (𝜑 → ¬ (𝑅‘𝐷) = (0.‘𝐾))
112	95, 7, 8, 9, 24	trlator0 41005	. . . . . . . 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 40145	. . . 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 2771	1 ⊢ (𝜑 → (𝑅‘𝐷) = 𝑉)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146   ≠ wne 2960   class class class wbr 5111  ‘cfv 6540  (class class class)co 7419  Basecbs 17293  lecple 17341  Posetcpo 18387  ltcplt 18388  joincjn 18391  meetcmee 18392  0.cp0 18501  Latclat 18511  OPcops 40006  OLcol 40008  Atomscatm 40097  AtLatcal 40098  HLchlt 40184  LHypclh 40818  LTrncltrn 40935  trLctrl 40992

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-map 8832  df-proset 18374  df-poset 18393  df-plt 18408  df-lub 18424  df-glb 18425  df-join 18426  df-meet 18427  df-p0 18503  df-p1 18504  df-lat 18512  df-clat 18579  df-oposet 40010  df-ol 40012  df-oml 40013  df-covers 40100  df-ats 40101  df-atl 40132  df-cvlat 40156  df-hlat 40185  df-llines 40332  df-psubsp 40337  df-pmap 40338  df-padd 40630  df-lhyp 40822  df-laut 40823  df-ldil 40938  df-ltrn 40939  df-trl 40993

This theorem is used by:  dia2dimlem5  41902