Theorem dalem39 35519

Description: Lemma for dath 35544. Auxiliary atoms 𝐺, 𝐻, and 𝐼 are not colinear. (Contributed by NM, 4-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
dalem39	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝐻 ≤ (𝐼 ∨ 𝐺))

Proof of Theorem dalem39

Step	Hyp	Ref	Expression
1		dalem.ph	. . . . 5 ⊢ (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)))))
2	1	dalemkehl 35431	. . . 4 ⊢ (𝜑 → 𝐾 ∈ HL)
3	2	3ad2ant1 1127	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
4	1	dalemyeo 35440	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
5	4	3ad2ant1 1127	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ 𝑂)
6		dalem.ps	. . . . . 6 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
7	6	dalemccea 35491	. . . . 5 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
8	7	3ad2ant3 1129	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
9	6	dalem-ccly 35493	. . . . 5 ⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌)
10	9	3ad2ant3 1129	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑐 ≤ 𝑌)
11		dalem.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
12		dalem.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
13		dalem.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
14		dalem38.o	. . . . 5 ⊢ 𝑂 = (LPlanes‘𝐾)
15		eqid 2771	. . . . 5 ⊢ (LVols‘𝐾) = (LVols‘𝐾)
16	11, 12, 13, 14, 15	lvoli3 35385	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑂 ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌) → (𝑌 ∨ 𝑐) ∈ (LVols‘𝐾))
17	3, 5, 8, 10, 16	syl31anc 1479	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑌 ∨ 𝑐) ∈ (LVols‘𝐾))
18		dalem38.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
19		dalem38.y	. . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
20		dalem38.z	. . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
21		dalem38.i	. . . 4 ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))
22	1, 11, 12, 13, 6, 18, 14, 19, 20, 21	dalem34 35514	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴)
23		dalem38.g	. . . 4 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
24	1, 11, 12, 13, 6, 18, 14, 19, 20, 23	dalem23 35504	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
25	11, 12, 13, 15	lvolnle3at 35390	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑌 ∨ 𝑐) ∈ (LVols‘𝐾)) ∧ (𝐼 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴)) → ¬ (𝑌 ∨ 𝑐) ≤ ((𝐼 ∨ 𝐺) ∨ 𝑐))
26	3, 17, 22, 24, 8, 25	syl23anc 1483	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ (𝑌 ∨ 𝑐) ≤ ((𝐼 ∨ 𝐺) ∨ 𝑐))
27		dalem38.h	. . . . . . 7 ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
28	1, 11, 12, 13, 6, 18, 14, 19, 20, 23, 27, 21	dalem38 35518	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))
29	1	dalemkelat 35432	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
30	29	3ad2ant1 1127	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
31	1, 11, 12, 13, 6, 18, 14, 19, 20, 27	dalem29 35509	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)
32		eqid 2771	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
33	32, 12, 13	hlatjcl 35175	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
34	3, 24, 31, 33	syl3anc 1476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
35	32, 13	atbase 35098	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾))
36	22, 35	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾))
37	32, 12	latjcl 17259	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾))
38	30, 34, 36, 37	syl3anc 1476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾))
39	6, 13	dalemcceb 35497	. . . . . . . 8 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
40	39	3ad2ant3 1129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
41	32, 11, 12	latlej2 17269	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → 𝑐 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))
42	30, 38, 40, 41	syl3anc 1476	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))
43	1, 14	dalemyeb 35457	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
44	43	3ad2ant1 1127	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾))
45	32, 12	latjcl 17259	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) ∈ (Base‘𝐾))
46	30, 38, 40, 45	syl3anc 1476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) ∈ (Base‘𝐾))
47	32, 11, 12	latjle12 17270	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑌 ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) ∈ (Base‘𝐾))) → ((𝑌 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) ∧ 𝑐 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐)) ↔ (𝑌 ∨ 𝑐) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐)))
48	30, 44, 40, 46, 47	syl13anc 1478	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑌 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) ∧ 𝑐 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐)) ↔ (𝑌 ∨ 𝑐) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐)))
49	28, 42, 48	mpbi2and 691	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑌 ∨ 𝑐) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))
50	12, 13	hlatjrot 35181	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴)) → ((𝐺 ∨ 𝐻) ∨ 𝐼) = ((𝐼 ∨ 𝐺) ∨ 𝐻))
51	3, 24, 31, 22, 50	syl13anc 1478	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) = ((𝐼 ∨ 𝐺) ∨ 𝐻))
52	51	oveq1d 6808	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) = (((𝐼 ∨ 𝐺) ∨ 𝐻) ∨ 𝑐))
53	49, 52	breqtrd 4812	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑌 ∨ 𝑐) ≤ (((𝐼 ∨ 𝐺) ∨ 𝐻) ∨ 𝑐))
54	53	adantr 466	. . 3 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝐻 ≤ (𝐼 ∨ 𝐺)) → (𝑌 ∨ 𝑐) ≤ (((𝐼 ∨ 𝐺) ∨ 𝐻) ∨ 𝑐))
55	32, 13	atbase 35098	. . . . . . 7 ⊢ (𝐻 ∈ 𝐴 → 𝐻 ∈ (Base‘𝐾))
56	31, 55	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ (Base‘𝐾))
57	32, 12, 13	hlatjcl 35175	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝐼 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝐼 ∨ 𝐺) ∈ (Base‘𝐾))
58	3, 22, 24, 57	syl3anc 1476	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐼 ∨ 𝐺) ∈ (Base‘𝐾))
59	32, 11, 12	latleeqj2 17272	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝐻 ∈ (Base‘𝐾) ∧ (𝐼 ∨ 𝐺) ∈ (Base‘𝐾)) → (𝐻 ≤ (𝐼 ∨ 𝐺) ↔ ((𝐼 ∨ 𝐺) ∨ 𝐻) = (𝐼 ∨ 𝐺)))
60	30, 56, 58, 59	syl3anc 1476	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐻 ≤ (𝐼 ∨ 𝐺) ↔ ((𝐼 ∨ 𝐺) ∨ 𝐻) = (𝐼 ∨ 𝐺)))
61	60	biimpa 462	. . . 4 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝐻 ≤ (𝐼 ∨ 𝐺)) → ((𝐼 ∨ 𝐺) ∨ 𝐻) = (𝐼 ∨ 𝐺))
62	61	oveq1d 6808	. . 3 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝐻 ≤ (𝐼 ∨ 𝐺)) → (((𝐼 ∨ 𝐺) ∨ 𝐻) ∨ 𝑐) = ((𝐼 ∨ 𝐺) ∨ 𝑐))
63	54, 62	breqtrd 4812	. 2 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝐻 ≤ (𝐼 ∨ 𝐺)) → (𝑌 ∨ 𝑐) ≤ ((𝐼 ∨ 𝐺) ∨ 𝑐))
64	26, 63	mtand 817	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝐻 ≤ (𝐼 ∨ 𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145   ≠ wne 2943   class class class wbr 4786  ‘cfv 6031  (class class class)co 6793  Basecbs 16064  lecple 16156  joincjn 17152  meetcmee 17153  Latclat 17253  Atomscatm 35072  HLchlt 35159  LPlanesclpl 35300  LVolsclvol 35301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-preset 17136  df-poset 17154  df-plt 17166  df-lub 17182  df-glb 17183  df-join 17184  df-meet 17185  df-p0 17247  df-lat 17254  df-clat 17316  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35306  df-lplanes 35307  df-lvols 35308

This theorem is referenced by:  dalem40  35520  dalem41  35521