Theorem dia2dimlem3 41033

Description: Lemma for dia2dim 41044. 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 494	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL)
3		dia2dimlem3.f	. . . . . . . . 9 ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))
4	3	simpld 494	. . . . . . . 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 40106	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
11	1, 4, 5, 10	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
12	11	simpld 494	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐴)
13		dia2dimlem3.v	. . . . . . 7 ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
14	13	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ 𝐴)
15		dia2dimlem3.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
16	6, 15, 7	hlatlej2 39342	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉))
17	2, 12, 14, 16	syl3anc 1373	. . . . 5 ⊢ (𝜑 → 𝑉 ≤ ((𝐹‘𝑃) ∨ 𝑉))
18	2	hllatd 39330	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat)
19		eqid 2729	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
20	19, 7	atbase 39255	. . . . . . 7 ⊢ (𝑉 ∈ 𝐴 → 𝑉 ∈ (Base‘𝐾))
21	14, 20	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑉 ∈ (Base‘𝐾))
22	19, 15, 7	hlatjcl 39333	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾))
23	2, 12, 14, 22	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝑃) ∨ 𝑉) ∈ (Base‘𝐾))
24		dia2dimlem3.r	. . . . . . . . 9 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
25	6, 7, 8, 9, 24	trlat 40136	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑅‘𝐹) ∈ 𝐴)
26	1, 5, 3, 25	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ∈ 𝐴)
27		dia2dimlem3.u	. . . . . . . 8 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
28	27	simpld 494	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
29	19, 15, 7	hlatjcl 39333	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑅‘𝐹) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾))
30	2, 26, 28, 29	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) ∈ (Base‘𝐾))
31		dia2dimlem3.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
32	19, 6, 31	latmlem2 18405	. . . . . 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 1374	. . . . 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 39334	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈))
37	2, 28, 14, 36	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝑈 ∨ 𝑉) = (𝑉 ∨ 𝑈))
38	35, 37	breqtrd 5128	. . . . . 6 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑉 ∨ 𝑈))
39		dia2dimlem3.ru	. . . . . . 7 ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)
40	6, 15, 7	hlatexch2 39363	. . . . . . 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 1385	. . . . . 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 18403	. . . . . 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 1373	. . . . 5 ⊢ (𝜑 → (𝑉 ≤ ((𝑅‘𝐹) ∨ 𝑈) ↔ (((𝑅‘𝐹) ∨ 𝑈) ∧ 𝑉) = 𝑉))
45	42, 44	mpbid 232	. . . 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 41031	. . . . . 6 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
50	6, 15, 31, 7, 8, 9, 24	trlval2 40130	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑅‘𝐷) = ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊))
51	1, 46, 49, 50	syl3anc 1373	. . . . 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 7387	. . . . . . . 8 ⊢ (𝜑 → (𝑄 ∨ (𝐷‘𝑄)) = (((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∨ (𝐹‘𝑃)))
55	5	simpld 494	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
56	19, 15, 7	hlatjcl 39333	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴) → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
57	2, 55, 28, 56	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝑃 ∨ 𝑈) ∈ (Base‘𝐾))
58	6, 15, 7	hlatlej1 39341	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝐹‘𝑃) ∈ 𝐴 ∧ 𝑉 ∈ 𝐴) → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ 𝑉))
59	2, 12, 14, 58	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘𝑃) ≤ ((𝐹‘𝑃) ∨ 𝑉))
60	19, 6, 15, 31, 7	atmod4i1 39833	. . . . . . . . 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 1385	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∨ (𝐹‘𝑃)) = (((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
62	15, 7	hlatj32 39338	. . . . . . . . . 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 1374	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) = ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈))
64	63	oveq1d 7384	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 ∨ 𝑈) ∨ (𝐹‘𝑃)) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
65	54, 61, 64	3eqtrd 2768	. . . . . . 7 ⊢ (𝜑 → (𝑄 ∨ (𝐷‘𝑄)) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
66	65	oveq1d 7384	. . . . . 6 ⊢ (𝜑 → ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊) = ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) ∧ 𝑊))
67		hlol 39327	. . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
68	2, 67	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ OL)
69	19, 15, 7	hlatjcl 39333	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ (𝐹‘𝑃) ∈ 𝐴) → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
70	2, 55, 12, 69	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾))
71	19, 7	atbase 39255	. . . . . . . . 9 ⊢ (𝑈 ∈ 𝐴 → 𝑈 ∈ (Base‘𝐾))
72	28, 71	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
73	19, 15	latjcl 18374	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ (𝐹‘𝑃)) ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∈ (Base‘𝐾))
74	18, 70, 72, 73	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∈ (Base‘𝐾))
75	1	simprd 495	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐻)
76	19, 8	lhpbase 39965	. . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾))
77	75, 76	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾))
78	19, 31	latm32 39197	. . . . . . 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 1374	. . . . . 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 40130	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
81	1, 4, 5, 80	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝐹) = ((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊))
82	81	oveq1d 7384	. . . . . . . 8 ⊢ (𝜑 → ((𝑅‘𝐹) ∨ 𝑈) = (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑈))
83	27	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ≤ 𝑊)
84	19, 6, 15, 31, 7	atmod4i1 39833	. . . . . . . . 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 1385	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∧ 𝑊) ∨ 𝑈) = (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊))
86	82, 85	eqtr2d 2765	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) = ((𝑅‘𝐹) ∨ 𝑈))
87	86	oveq1d 7384	. . . . . 6 ⊢ (𝜑 → ((((𝑃 ∨ (𝐹‘𝑃)) ∨ 𝑈) ∧ 𝑊) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
88	66, 79, 87	3eqtrd 2768	. . . . 5 ⊢ (𝜑 → ((𝑄 ∨ (𝐷‘𝑄)) ∧ 𝑊) = (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)))
89	51, 88	eqtr2d 2765	. . . 4 ⊢ (𝜑 → (((𝑅‘𝐹) ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉)) = (𝑅‘𝐷))
90	34, 45, 89	3brtr3d 5133	. . 3 ⊢ (𝜑 → 𝑉 ≤ (𝑅‘𝐷))
91		hlatl 39326	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
92	2, 91	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ AtLat)
93		hlop 39328	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
94	2, 93	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ OP)
95		eqid 2729	. . . . . . . . . 10 ⊢ (0.‘𝐾) = (0.‘𝐾)
96		eqid 2729	. . . . . . . . . 10 ⊢ (lt‘𝐾) = (lt‘𝐾)
97	95, 96, 7	0ltat 39257	. . . . . . . . 9 ⊢ ((𝐾 ∈ OP ∧ 𝑉 ∈ 𝐴) → (0.‘𝐾)(lt‘𝐾)𝑉)
98	94, 14, 97	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)𝑉)
99		hlpos 39332	. . . . . . . . . 10 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
100	2, 99	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ Poset)
101	19, 95	op0cl 39150	. . . . . . . . . 10 ⊢ (𝐾 ∈ OP → (0.‘𝐾) ∈ (Base‘𝐾))
102	94, 101	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (0.‘𝐾) ∈ (Base‘𝐾))
103	19, 8, 9, 24	trlcl 40131	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → (𝑅‘𝐷) ∈ (Base‘𝐾))
104	1, 46, 103	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝐷) ∈ (Base‘𝐾))
105	19, 6, 96	pltletr 18278	. . . . . . . . 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 1374	. . . . . . . 8 ⊢ (𝜑 → (((0.‘𝐾)(lt‘𝐾)𝑉 ∧ 𝑉 ≤ (𝑅‘𝐷)) → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷)))
107	98, 90, 106	mp2and 699	. . . . . . 7 ⊢ (𝜑 → (0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷))
108	19, 96, 95	opltn0 39156	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ (𝑅‘𝐷) ∈ (Base‘𝐾)) → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷) ↔ (𝑅‘𝐷) ≠ (0.‘𝐾)))
109	94, 104, 108	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((0.‘𝐾)(lt‘𝐾)(𝑅‘𝐷) ↔ (𝑅‘𝐷) ≠ (0.‘𝐾)))
110	107, 109	mpbid 232	. . . . . 6 ⊢ (𝜑 → (𝑅‘𝐷) ≠ (0.‘𝐾))
111	110	neneqd 2930	. . . . 5 ⊢ (𝜑 → ¬ (𝑅‘𝐷) = (0.‘𝐾))
112	95, 7, 8, 9, 24	trlator0 40138	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝑇) → ((𝑅‘𝐷) ∈ 𝐴 ∨ (𝑅‘𝐷) = (0.‘𝐾)))
113	1, 46, 112	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ((𝑅‘𝐷) ∈ 𝐴 ∨ (𝑅‘𝐷) = (0.‘𝐾)))
114	113	orcomd 871	. . . . . 6 ⊢ (𝜑 → ((𝑅‘𝐷) = (0.‘𝐾) ∨ (𝑅‘𝐷) ∈ 𝐴))
115	114	ord 864	. . . . 5 ⊢ (𝜑 → (¬ (𝑅‘𝐷) = (0.‘𝐾) → (𝑅‘𝐷) ∈ 𝐴))
116	111, 115	mpd 15	. . . 4 ⊢ (𝜑 → (𝑅‘𝐷) ∈ 𝐴)
117	6, 7	atcmp 39277	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑉 ∈ 𝐴 ∧ (𝑅‘𝐷) ∈ 𝐴) → (𝑉 ≤ (𝑅‘𝐷) ↔ 𝑉 = (𝑅‘𝐷)))
118	92, 14, 116, 117	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝑉 ≤ (𝑅‘𝐷) ↔ 𝑉 = (𝑅‘𝐷)))
119	90, 118	mpbid 232	. 2 ⊢ (𝜑 → 𝑉 = (𝑅‘𝐷))
120	119	eqcomd 2735	1 ⊢ (𝜑 → (𝑅‘𝐷) = 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5102  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  lecple 17203  Posetcpo 18244  ltcplt 18245  joincjn 18248  meetcmee 18249  0.cp0 18358  Latclat 18366  OPcops 39138  OLcol 39140  Atomscatm 39229  AtLatcal 39230  HLchlt 39316  LHypclh 39951  LTrncltrn 40068  trLctrl 40125

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-map 8778  df-proset 18231  df-poset 18250  df-plt 18265  df-lub 18281  df-glb 18282  df-join 18283  df-meet 18284  df-p0 18360  df-p1 18361  df-lat 18367  df-clat 18434  df-oposet 39142  df-ol 39144  df-oml 39145  df-covers 39232  df-ats 39233  df-atl 39264  df-cvlat 39288  df-hlat 39317  df-llines 39465  df-psubsp 39470  df-pmap 39471  df-padd 39763  df-lhyp 39955  df-laut 39956  df-ldil 40071  df-ltrn 40072  df-trl 40126

This theorem is referenced by:  dia2dimlem5  41035