Theorem dalem38 40086

Description: Lemma for dath 40112. Plane 𝑌 belongs to the 3-dimensional volume 𝐺𝐻𝐼𝑐. (Contributed by NM, 5-Aug-2012.)

Hypotheses

Ref	Expression
dalem.ph	⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))
dalem.l	⊢ ≤ = (le‘𝐾)
dalem.j	⊢ ∨ = (join‘𝐾)
dalem.a	⊢ 𝐴 = (Atoms‘𝐾)
dalem.ps	⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
dalem38.m	⊢ ∧ = (meet‘𝐾)
dalem38.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem38.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem38.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem38.g	⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
dalem38.h	⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
dalem38.i	⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))

Assertion

Ref	Expression
dalem38	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))

Proof of Theorem dalem38

Step	Hyp	Ref	Expression
1		dalem38.y	. 2 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
2		dalem.ph	. . . . . . 7 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))
3		dalem.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
4		dalem.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
5		dalem.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
6		dalem.ps	. . . . . . 7 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
7		dalem38.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
8		dalem38.o	. . . . . . 7 ⊢ 𝑂 = (LPlanes‘𝐾)
9		dalem38.z	. . . . . . 7 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
10		dalem38.g	. . . . . . 7 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
11	2, 3, 4, 5, 6, 7, 8, 1, 9, 10	dalem28 40076	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ (𝐺 ∨ 𝑐))
12		dalem38.h	. . . . . . 7 ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
13	2, 3, 4, 5, 6, 7, 8, 1, 9, 12	dalem33 40081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ≤ (𝐻 ∨ 𝑐))
14	2	dalemkelat 40000	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
15	14	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
16	2, 5	dalempeb 40015	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
17	16	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ (Base‘𝐾))
18	2	dalemkehl 39999	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ HL)
19	18	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
20	2, 3, 4, 5, 6, 7, 8, 1, 9, 10	dalem23 40072	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
21	6	dalemccea 40059	. . . . . . . . 9 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
22	21	3ad2ant3 1136	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
23		eqid 2737	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
24	23, 4, 5	hlatjcl 39743	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝐺 ∨ 𝑐) ∈ (Base‘𝐾))
25	19, 20, 22, 24	syl3anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝑐) ∈ (Base‘𝐾))
26	2, 5	dalemqeb 40016	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
27	26	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ∈ (Base‘𝐾))
28	2, 3, 4, 5, 6, 7, 8, 1, 9, 12	dalem29 40077	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)
29	23, 4, 5	hlatjcl 39743	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝐻 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝐻 ∨ 𝑐) ∈ (Base‘𝐾))
30	19, 28, 22, 29	syl3anc 1374	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐻 ∨ 𝑐) ∈ (Base‘𝐾))
31	23, 3, 4	latjlej12 18390	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ (𝐺 ∨ 𝑐) ∈ (Base‘𝐾)) ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝐻 ∨ 𝑐) ∈ (Base‘𝐾))) → ((𝑃 ≤ (𝐺 ∨ 𝑐) ∧ 𝑄 ≤ (𝐻 ∨ 𝑐)) → (𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐))))
32	15, 17, 25, 27, 30, 31	syl122anc 1382	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ≤ (𝐺 ∨ 𝑐) ∧ 𝑄 ≤ (𝐻 ∨ 𝑐)) → (𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐))))
33	11, 13, 32	mp2and 700	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐)))
34	23, 5	atbase 39665	. . . . . . 7 ⊢ (𝐺 ∈ 𝐴 → 𝐺 ∈ (Base‘𝐾))
35	20, 34	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ (Base‘𝐾))
36	23, 5	atbase 39665	. . . . . . 7 ⊢ (𝐻 ∈ 𝐴 → 𝐻 ∈ (Base‘𝐾))
37	28, 36	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ (Base‘𝐾))
38	6, 5	dalemcceb 40065	. . . . . . 7 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
39	38	3ad2ant3 1136	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
40	23, 4	latjjdir 18427	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾))) → ((𝐺 ∨ 𝐻) ∨ 𝑐) = ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐)))
41	15, 35, 37, 39, 40	syl13anc 1375	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝑐) = ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐)))
42	33, 41	breqtrrd 5128	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝐻) ∨ 𝑐))
43		dalem38.i	. . . . 5 ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))
44	2, 3, 4, 5, 6, 7, 8, 1, 9, 43	dalem37 40085	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ≤ (𝐼 ∨ 𝑐))
45	2, 4, 5	dalempjqeb 40021	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
46	45	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
47	23, 4, 5	hlatjcl 39743	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
48	19, 20, 28, 47	syl3anc 1374	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
49	23, 4	latjcl 18374	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∨ 𝑐) ∈ (Base‘𝐾))
50	15, 48, 39, 49	syl3anc 1374	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝑐) ∈ (Base‘𝐾))
51	2, 5	dalemreb 40017	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
52	51	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ∈ (Base‘𝐾))
53	2, 3, 4, 5, 6, 7, 8, 1, 9, 43	dalem34 40082	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴)
54	23, 4, 5	hlatjcl 39743	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝐼 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝐼 ∨ 𝑐) ∈ (Base‘𝐾))
55	19, 53, 22, 54	syl3anc 1374	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐼 ∨ 𝑐) ∈ (Base‘𝐾))
56	23, 3, 4	latjlej12 18390	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ ((𝐺 ∨ 𝐻) ∨ 𝑐) ∈ (Base‘𝐾)) ∧ (𝑅 ∈ (Base‘𝐾) ∧ (𝐼 ∨ 𝑐) ∈ (Base‘𝐾))) → (((𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝐻) ∨ 𝑐) ∧ 𝑅 ≤ (𝐼 ∨ 𝑐)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐))))
57	15, 46, 50, 52, 55, 56	syl122anc 1382	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝐻) ∨ 𝑐) ∧ 𝑅 ≤ (𝐼 ∨ 𝑐)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐))))
58	42, 44, 57	mp2and 700	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐)))
59	23, 5	atbase 39665	. . . . 5 ⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾))
60	53, 59	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾))
61	23, 4	latjjdir 18427	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ ((𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾))) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) = (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐)))
62	15, 48, 60, 39, 61	syl13anc 1375	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) = (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐)))
63	58, 62	breqtrrd 5128	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))
64	1, 63	eqbrtrid 5135	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  lecple 17196  joincjn 18246  meetcmee 18247  Latclat 18366  Atomscatm 39639  HLchlt 39726  LPlanesclpl 39868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-proset 18229  df-poset 18248  df-plt 18263  df-lub 18279  df-glb 18280  df-join 18281  df-meet 18282  df-p0 18358  df-lat 18367  df-clat 18434  df-oposet 39552  df-ol 39554  df-oml 39555  df-covers 39642  df-ats 39643  df-atl 39674  df-cvlat 39698  df-hlat 39727  df-llines 39874  df-lplanes 39875

This theorem is referenced by:  dalem39  40087