Theorem dalem39 39693

Description: Lemma for dath 39718. 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 39605	. . . 4 ⊢ (𝜑 → 𝐾 ∈ HL)
3	2	3ad2ant1 1132	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
4	1	dalemyeo 39614	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
5	4	3ad2ant1 1132	. . . 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 39665	. . . . 5 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
8	7	3ad2ant3 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
9	6	dalem-ccly 39667	. . . . 5 ⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌)
10	9	3ad2ant3 1134	. . . 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 2734	. . . . 5 ⊢ (LVols‘𝐾) = (LVols‘𝐾)
16	11, 12, 13, 14, 15	lvoli3 39559	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑂 ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌) → (𝑌 ∨ 𝑐) ∈ (LVols‘𝐾))
17	3, 5, 8, 10, 16	syl31anc 1372	. . 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 39688	. . 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 39678	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
25	11, 12, 13, 15	lvolnle3at 39564	. . 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 1376	. 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 39692	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))
29	1	dalemkelat 39606	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
30	29	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
31	1, 11, 12, 13, 6, 18, 14, 19, 20, 27	dalem29 39683	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)
32		eqid 2734	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
33	32, 12, 13	hlatjcl 39348	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
34	3, 24, 31, 33	syl3anc 1370	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
35	32, 13	atbase 39270	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾))
36	22, 35	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾))
37	32, 12	latjcl 18496	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾))
38	30, 34, 36, 37	syl3anc 1370	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾))
39	6, 13	dalemcceb 39671	. . . . . . . 8 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
40	39	3ad2ant3 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
41	32, 11, 12	latlej2 18506	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → 𝑐 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))
42	30, 38, 40, 41	syl3anc 1370	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))
43	1, 14	dalemyeb 39631	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
44	43	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾))
45	32, 12	latjcl 18496	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) ∈ (Base‘𝐾))
46	30, 38, 40, 45	syl3anc 1370	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) ∈ (Base‘𝐾))
47	32, 11, 12	latjle12 18507	. . . . . . 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 1371	. . . . . 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 712	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑌 ∨ 𝑐) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))
50	12, 13	hlatjrot 39354	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴)) → ((𝐺 ∨ 𝐻) ∨ 𝐼) = ((𝐼 ∨ 𝐺) ∨ 𝐻))
51	3, 24, 31, 22, 50	syl13anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) = ((𝐼 ∨ 𝐺) ∨ 𝐻))
52	51	oveq1d 7445	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) = (((𝐼 ∨ 𝐺) ∨ 𝐻) ∨ 𝑐))
53	49, 52	breqtrd 5173	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑌 ∨ 𝑐) ≤ (((𝐼 ∨ 𝐺) ∨ 𝐻) ∨ 𝑐))
54	53	adantr 480	. . 3 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝐻 ≤ (𝐼 ∨ 𝐺)) → (𝑌 ∨ 𝑐) ≤ (((𝐼 ∨ 𝐺) ∨ 𝐻) ∨ 𝑐))
55	32, 13	atbase 39270	. . . . . . 7 ⊢ (𝐻 ∈ 𝐴 → 𝐻 ∈ (Base‘𝐾))
56	31, 55	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ (Base‘𝐾))
57	32, 12, 13	hlatjcl 39348	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝐼 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝐼 ∨ 𝐺) ∈ (Base‘𝐾))
58	3, 22, 24, 57	syl3anc 1370	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐼 ∨ 𝐺) ∈ (Base‘𝐾))
59	32, 11, 12	latleeqj2 18509	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝐻 ∈ (Base‘𝐾) ∧ (𝐼 ∨ 𝐺) ∈ (Base‘𝐾)) → (𝐻 ≤ (𝐼 ∨ 𝐺) ↔ ((𝐼 ∨ 𝐺) ∨ 𝐻) = (𝐼 ∨ 𝐺)))
60	30, 56, 58, 59	syl3anc 1370	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐻 ≤ (𝐼 ∨ 𝐺) ↔ ((𝐼 ∨ 𝐺) ∨ 𝐻) = (𝐼 ∨ 𝐺)))
61	60	biimpa 476	. . . 4 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝐻 ≤ (𝐼 ∨ 𝐺)) → ((𝐼 ∨ 𝐺) ∨ 𝐻) = (𝐼 ∨ 𝐺))
62	61	oveq1d 7445	. . 3 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝐻 ≤ (𝐼 ∨ 𝐺)) → (((𝐼 ∨ 𝐺) ∨ 𝐻) ∨ 𝑐) = ((𝐼 ∨ 𝐺) ∨ 𝑐))
63	54, 62	breqtrd 5173	. 2 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝐻 ≤ (𝐼 ∨ 𝐺)) → (𝑌 ∨ 𝑐) ≤ ((𝐼 ∨ 𝐺) ∨ 𝑐))
64	26, 63	mtand 816	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝐻 ≤ (𝐼 ∨ 𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105   ≠ wne 2937   class class class wbr 5147  ‘cfv 6562  (class class class)co 7430  Basecbs 17244  lecple 17304  joincjn 18368  meetcmee 18369  Latclat 18488  Atomscatm 39244  HLchlt 39331  LPlanesclpl 39474  LVolsclvol 39475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-proset 18351  df-poset 18370  df-plt 18387  df-lub 18403  df-glb 18404  df-join 18405  df-meet 18406  df-p0 18482  df-lat 18489  df-clat 18556  df-oposet 39157  df-ol 39159  df-oml 39160  df-covers 39247  df-ats 39248  df-atl 39279  df-cvlat 39303  df-hlat 39332  df-llines 39480  df-lplanes 39481  df-lvols 39482

This theorem is referenced by:  dalem40  39694  dalem41  39695