Theorem dalem38 37336

Description: Lemma for dath 37362. 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 37326	. . . . . 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 37331	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ≤ (𝐻 ∨ 𝑐))
14	2	dalemkelat 37250	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
15	14	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
16	2, 5	dalempeb 37265	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
17	16	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ (Base‘𝐾))
18	2	dalemkehl 37249	. . . . . . . . 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 37322	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
21	6	dalemccea 37309	. . . . . . . . 9 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
22	21	3ad2ant3 1136	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
23		eqid 2738	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
24	23, 4, 5	hlatjcl 36993	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝐺 ∨ 𝑐) ∈ (Base‘𝐾))
25	19, 20, 22, 24	syl3anc 1372	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝑐) ∈ (Base‘𝐾))
26	2, 5	dalemqeb 37266	. . . . . . . 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 37327	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)
29	23, 4, 5	hlatjcl 36993	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝐻 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝐻 ∨ 𝑐) ∈ (Base‘𝐾))
30	19, 28, 22, 29	syl3anc 1372	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐻 ∨ 𝑐) ∈ (Base‘𝐾))
31	23, 3, 4	latjlej12 17786	. . . . . . 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 1380	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ≤ (𝐺 ∨ 𝑐) ∧ 𝑄 ≤ (𝐻 ∨ 𝑐)) → (𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐))))
33	11, 13, 32	mp2and 699	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐)))
34	23, 5	atbase 36915	. . . . . . 7 ⊢ (𝐺 ∈ 𝐴 → 𝐺 ∈ (Base‘𝐾))
35	20, 34	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ (Base‘𝐾))
36	23, 5	atbase 36915	. . . . . . 7 ⊢ (𝐻 ∈ 𝐴 → 𝐻 ∈ (Base‘𝐾))
37	28, 36	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ (Base‘𝐾))
38	6, 5	dalemcceb 37315	. . . . . . 7 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
39	38	3ad2ant3 1136	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
40	23, 4	latjjdir 17823	. . . . . 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 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝑐) = ((𝐺 ∨ 𝑐) ∨ (𝐻 ∨ 𝑐)))
42	33, 41	breqtrrd 5055	. . . 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 37335	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ≤ (𝐼 ∨ 𝑐))
45	2, 4, 5	dalempjqeb 37271	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
46	45	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
47	23, 4, 5	hlatjcl 36993	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
48	19, 20, 28, 47	syl3anc 1372	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
49	23, 4	latjcl 17770	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∨ 𝑐) ∈ (Base‘𝐾))
50	15, 48, 39, 49	syl3anc 1372	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝑐) ∈ (Base‘𝐾))
51	2, 5	dalemreb 37267	. . . . . 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 37332	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴)
54	23, 4, 5	hlatjcl 36993	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝐼 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) → (𝐼 ∨ 𝑐) ∈ (Base‘𝐾))
55	19, 53, 22, 54	syl3anc 1372	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐼 ∨ 𝑐) ∈ (Base‘𝐾))
56	23, 3, 4	latjlej12 17786	. . . . 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 1380	. . . 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 699	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐)))
59	23, 5	atbase 36915	. . . . 5 ⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾))
60	53, 59	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾))
61	23, 4	latjjdir 17823	. . . 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 1373	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) = (((𝐺 ∨ 𝐻) ∨ 𝑐) ∨ (𝐼 ∨ 𝑐)))
63	58, 62	breqtrrd 5055	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))
64	1, 63	eqbrtrid 5062	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2113   ≠ wne 2934   class class class wbr 5027  ‘cfv 6333  (class class class)co 7164  Basecbs 16579  lecple 16668  joincjn 17663  meetcmee 17664  Latclat 17764  Atomscatm 36889  HLchlt 36976  LPlanesclpl 37118

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 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-proset 17647  df-poset 17665  df-plt 17677  df-lub 17693  df-glb 17694  df-join 17695  df-meet 17696  df-p0 17758  df-lat 17765  df-clat 17827  df-oposet 36802  df-ol 36804  df-oml 36805  df-covers 36892  df-ats 36893  df-atl 36924  df-cvlat 36948  df-hlat 36977  df-llines 37124  df-lplanes 37125

This theorem is referenced by:  dalem39  37337