Theorem dalem39 36862

Description: Lemma for dath 36887. 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 36774	. . . 4 ⊢ (𝜑 → 𝐾 ∈ HL)
3	2	3ad2ant1 1129	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
4	1	dalemyeo 36783	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
5	4	3ad2ant1 1129	. . . 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 36834	. . . . 5 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
8	7	3ad2ant3 1131	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
9	6	dalem-ccly 36836	. . . . 5 ⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌)
10	9	3ad2ant3 1131	. . . 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 2821	. . . . 5 ⊢ (LVols‘𝐾) = (LVols‘𝐾)
16	11, 12, 13, 14, 15	lvoli3 36728	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑂 ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌) → (𝑌 ∨ 𝑐) ∈ (LVols‘𝐾))
17	3, 5, 8, 10, 16	syl31anc 1369	. . 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 36857	. . 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 36847	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
25	11, 12, 13, 15	lvolnle3at 36733	. . 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 1373	. 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 36861	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))
29	1	dalemkelat 36775	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Lat)
30	29	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
31	1, 11, 12, 13, 6, 18, 14, 19, 20, 27	dalem29 36852	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)
32		eqid 2821	. . . . . . . . . 10 ⊢ (Base‘𝐾) = (Base‘𝐾)
33	32, 12, 13	hlatjcl 36518	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
34	3, 24, 31, 33	syl3anc 1367	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
35	32, 13	atbase 36440	. . . . . . . . 9 ⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾))
36	22, 35	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾))
37	32, 12	latjcl 17661	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾))
38	30, 34, 36, 37	syl3anc 1367	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾))
39	6, 13	dalemcceb 36840	. . . . . . . 8 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
40	39	3ad2ant3 1131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
41	32, 11, 12	latlej2 17671	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → 𝑐 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))
42	30, 38, 40, 41	syl3anc 1367	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))
43	1, 14	dalemyeb 36800	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
44	43	3ad2ant1 1129	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾))
45	32, 12	latjcl 17661	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾) ∧ 𝑐 ∈ (Base‘𝐾)) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) ∈ (Base‘𝐾))
46	30, 38, 40, 45	syl3anc 1367	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) ∈ (Base‘𝐾))
47	32, 11, 12	latjle12 17672	. . . . . . 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 1368	. . . . . 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 710	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑌 ∨ 𝑐) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐))
50	12, 13	hlatjrot 36524	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴)) → ((𝐺 ∨ 𝐻) ∨ 𝐼) = ((𝐼 ∨ 𝐺) ∨ 𝐻))
51	3, 24, 31, 22, 50	syl13anc 1368	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) = ((𝐼 ∨ 𝐺) ∨ 𝐻))
52	51	oveq1d 7171	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∨ 𝑐) = (((𝐼 ∨ 𝐺) ∨ 𝐻) ∨ 𝑐))
53	49, 52	breqtrd 5092	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑌 ∨ 𝑐) ≤ (((𝐼 ∨ 𝐺) ∨ 𝐻) ∨ 𝑐))
54	53	adantr 483	. . 3 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝐻 ≤ (𝐼 ∨ 𝐺)) → (𝑌 ∨ 𝑐) ≤ (((𝐼 ∨ 𝐺) ∨ 𝐻) ∨ 𝑐))
55	32, 13	atbase 36440	. . . . . . 7 ⊢ (𝐻 ∈ 𝐴 → 𝐻 ∈ (Base‘𝐾))
56	31, 55	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ (Base‘𝐾))
57	32, 12, 13	hlatjcl 36518	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝐼 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝐼 ∨ 𝐺) ∈ (Base‘𝐾))
58	3, 22, 24, 57	syl3anc 1367	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐼 ∨ 𝐺) ∈ (Base‘𝐾))
59	32, 11, 12	latleeqj2 17674	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝐻 ∈ (Base‘𝐾) ∧ (𝐼 ∨ 𝐺) ∈ (Base‘𝐾)) → (𝐻 ≤ (𝐼 ∨ 𝐺) ↔ ((𝐼 ∨ 𝐺) ∨ 𝐻) = (𝐼 ∨ 𝐺)))
60	30, 56, 58, 59	syl3anc 1367	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐻 ≤ (𝐼 ∨ 𝐺) ↔ ((𝐼 ∨ 𝐺) ∨ 𝐻) = (𝐼 ∨ 𝐺)))
61	60	biimpa 479	. . . 4 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝐻 ≤ (𝐼 ∨ 𝐺)) → ((𝐼 ∨ 𝐺) ∨ 𝐻) = (𝐼 ∨ 𝐺))
62	61	oveq1d 7171	. . 3 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝐻 ≤ (𝐼 ∨ 𝐺)) → (((𝐼 ∨ 𝐺) ∨ 𝐻) ∨ 𝑐) = ((𝐼 ∨ 𝐺) ∨ 𝑐))
63	54, 62	breqtrd 5092	. 2 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝐻 ≤ (𝐼 ∨ 𝐺)) → (𝑌 ∨ 𝑐) ≤ ((𝐼 ∨ 𝐺) ∨ 𝑐))
64	26, 63	mtand 814	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝐻 ≤ (𝐼 ∨ 𝐺))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3016   class class class wbr 5066  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  lecple 16572  joincjn 17554  meetcmee 17555  Latclat 17655  Atomscatm 36414  HLchlt 36501  LPlanesclpl 36643  LVolsclvol 36644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-lat 17656  df-clat 17718  df-oposet 36327  df-ol 36329  df-oml 36330  df-covers 36417  df-ats 36418  df-atl 36449  df-cvlat 36473  df-hlat 36502  df-llines 36649  df-lplanes 36650  df-lvols 36651

This theorem is referenced by:  dalem40  36863  dalem41  36864