Theorem dalem-cly 40047

Description: Lemma for dalem9 40048. Center of perspectivity 𝐶 is not in plane 𝑌 (when 𝑌 and 𝑍 are different planes). (Contributed by NM, 13-Aug-2012.)

Hypotheses

Ref	Expression
dalema.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)))))
dalemc.l	⊢ ≤ = (le‘𝐾)
dalemc.j	⊢ ∨ = (join‘𝐾)
dalemc.a	⊢ 𝐴 = (Atoms‘𝐾)
dalem-cly.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem-cly.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem-cly.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)

Assertion

Ref	Expression
dalem-cly	⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → ¬ 𝐶 ≤ 𝑌)

Proof of Theorem dalem-cly

Step	Hyp	Ref	Expression
1		dalema.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)))))
2	1	dalemkelat 40000	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat)
3		dalemc.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
4	1, 3	dalemceb 40014	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
5		dalem-cly.o	. . . . . . 7 ⊢ 𝑂 = (LPlanes‘𝐾)
6	1, 5	dalemyeb 40025	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
7		eqid 2737	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
8		dalemc.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
9		dalemc.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
10	7, 8, 9	latleeqj1 18386	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝐶 ≤ 𝑌 ↔ (𝐶 ∨ 𝑌) = 𝑌))
11	2, 4, 6, 10	syl3anc 1374	. . . . 5 ⊢ (𝜑 → (𝐶 ≤ 𝑌 ↔ (𝐶 ∨ 𝑌) = 𝑌))
12	1	dalemclpjs 40010	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
13	1	dalemkehl 39999	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ HL)
14		dalem-cly.y	. . . . . . . . . . . . . . 15 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
15	1, 8, 9, 3, 5, 14	dalemcea 40036	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
16	1	dalemsea 40005	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
17	1	dalempea 40002	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
18	1	dalemqea 40003	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
19	1	dalem-clpjq 40013	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
20	8, 9, 3	atnlej1 39755	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)) → 𝐶 ≠ 𝑃)
21	13, 15, 17, 18, 19, 20	syl131anc 1386	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ≠ 𝑃)
22	8, 9, 3	hlatexch1 39771	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝐶 ≠ 𝑃) → (𝐶 ≤ (𝑃 ∨ 𝑆) → 𝑆 ≤ (𝑃 ∨ 𝐶)))
23	13, 15, 16, 17, 21, 22	syl131anc 1386	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 ≤ (𝑃 ∨ 𝑆) → 𝑆 ≤ (𝑃 ∨ 𝐶)))
24	12, 23	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ≤ (𝑃 ∨ 𝐶))
25	9, 3	hlatjcom 39744	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝐶 ∨ 𝑃) = (𝑃 ∨ 𝐶))
26	13, 15, 17, 25	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∨ 𝑃) = (𝑃 ∨ 𝐶))
27	24, 26	breqtrrd 5128	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ≤ (𝐶 ∨ 𝑃))
28	1	dalemclqjt 40011	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
29	1	dalemtea 40006	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
30	1	dalemrea 40004	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
31		simp312 1323	. . . . . . . . . . . . . . . 16 ⊢ ((((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)))) → ¬ 𝐶 ≤ (𝑄 ∨ 𝑅))
32	1, 31	sylbi 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑄 ∨ 𝑅))
33	8, 9, 3	atnlej1 39755	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅)) → 𝐶 ≠ 𝑄)
34	13, 15, 18, 30, 32, 33	syl131anc 1386	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ≠ 𝑄)
35	8, 9, 3	hlatexch1 39771	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝐶 ≠ 𝑄) → (𝐶 ≤ (𝑄 ∨ 𝑇) → 𝑇 ≤ (𝑄 ∨ 𝐶)))
36	13, 15, 29, 18, 34, 35	syl131anc 1386	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 ≤ (𝑄 ∨ 𝑇) → 𝑇 ≤ (𝑄 ∨ 𝐶)))
37	28, 36	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ≤ (𝑄 ∨ 𝐶))
38	9, 3	hlatjcom 39744	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝐶 ∨ 𝑄) = (𝑄 ∨ 𝐶))
39	13, 15, 18, 38	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∨ 𝑄) = (𝑄 ∨ 𝐶))
40	37, 39	breqtrrd 5128	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ≤ (𝐶 ∨ 𝑄))
41	1, 3	dalemseb 40018	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾))
42	7, 9, 3	hlatjcl 39743	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝐶 ∨ 𝑃) ∈ (Base‘𝐾))
43	13, 15, 17, 42	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∨ 𝑃) ∈ (Base‘𝐾))
44	1, 3	dalemteb 40019	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾))
45	7, 9, 3	hlatjcl 39743	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝐶 ∨ 𝑄) ∈ (Base‘𝐾))
46	13, 15, 18, 45	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∨ 𝑄) ∈ (Base‘𝐾))
47	7, 8, 9	latjlej12 18390	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ (𝐶 ∨ 𝑃) ∈ (Base‘𝐾)) ∧ (𝑇 ∈ (Base‘𝐾) ∧ (𝐶 ∨ 𝑄) ∈ (Base‘𝐾))) → ((𝑆 ≤ (𝐶 ∨ 𝑃) ∧ 𝑇 ≤ (𝐶 ∨ 𝑄)) → (𝑆 ∨ 𝑇) ≤ ((𝐶 ∨ 𝑃) ∨ (𝐶 ∨ 𝑄))))
48	2, 41, 43, 44, 46, 47	syl122anc 1382	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑆 ≤ (𝐶 ∨ 𝑃) ∧ 𝑇 ≤ (𝐶 ∨ 𝑄)) → (𝑆 ∨ 𝑇) ≤ ((𝐶 ∨ 𝑃) ∨ (𝐶 ∨ 𝑄))))
49	27, 40, 48	mp2and 700	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ ((𝐶 ∨ 𝑃) ∨ (𝐶 ∨ 𝑄)))
50	1, 3	dalempeb 40015	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
51	1, 3	dalemqeb 40016	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
52	7, 9	latjjdi 18426	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾))) → (𝐶 ∨ (𝑃 ∨ 𝑄)) = ((𝐶 ∨ 𝑃) ∨ (𝐶 ∨ 𝑄)))
53	2, 4, 50, 51, 52	syl13anc 1375	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∨ (𝑃 ∨ 𝑄)) = ((𝐶 ∨ 𝑃) ∨ (𝐶 ∨ 𝑄)))
54	49, 53	breqtrrd 5128	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ (𝐶 ∨ (𝑃 ∨ 𝑄)))
55	1	dalemclrju 40012	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ≤ (𝑅 ∨ 𝑈))
56	1	dalemuea 40007	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
57		simp313 1324	. . . . . . . . . . . . . 14 ⊢ ((((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)))) → ¬ 𝐶 ≤ (𝑅 ∨ 𝑃))
58	1, 57	sylbi 217	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑅 ∨ 𝑃))
59	8, 9, 3	atnlej1 39755	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) → 𝐶 ≠ 𝑅)
60	13, 15, 30, 17, 58, 59	syl131anc 1386	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ≠ 𝑅)
61	8, 9, 3	hlatexch1 39771	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝐶 ≠ 𝑅) → (𝐶 ≤ (𝑅 ∨ 𝑈) → 𝑈 ≤ (𝑅 ∨ 𝐶)))
62	13, 15, 56, 30, 60, 61	syl131anc 1386	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ≤ (𝑅 ∨ 𝑈) → 𝑈 ≤ (𝑅 ∨ 𝐶)))
63	55, 62	mpd 15	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ≤ (𝑅 ∨ 𝐶))
64	9, 3	hlatjcom 39744	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝐶 ∨ 𝑅) = (𝑅 ∨ 𝐶))
65	13, 15, 30, 64	syl3anc 1374	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∨ 𝑅) = (𝑅 ∨ 𝐶))
66	63, 65	breqtrrd 5128	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ≤ (𝐶 ∨ 𝑅))
67	1, 9, 3	dalemsjteb 40022	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
68	1, 9, 3	dalempjqeb 40021	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
69	7, 9	latjcl 18374	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝐶 ∨ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾))
70	2, 4, 68, 69	syl3anc 1374	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∨ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾))
71	1, 3	dalemueb 40020	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
72	7, 9, 3	hlatjcl 39743	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝐶 ∨ 𝑅) ∈ (Base‘𝐾))
73	13, 15, 30, 72	syl3anc 1374	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∨ 𝑅) ∈ (Base‘𝐾))
74	7, 8, 9	latjlej12 18390	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ ((𝑆 ∨ 𝑇) ∈ (Base‘𝐾) ∧ (𝐶 ∨ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾)) ∧ (𝑈 ∈ (Base‘𝐾) ∧ (𝐶 ∨ 𝑅) ∈ (Base‘𝐾))) → (((𝑆 ∨ 𝑇) ≤ (𝐶 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑈 ≤ (𝐶 ∨ 𝑅)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ ((𝐶 ∨ (𝑃 ∨ 𝑄)) ∨ (𝐶 ∨ 𝑅))))
75	2, 67, 70, 71, 73, 74	syl122anc 1382	. . . . . . . . 9 ⊢ (𝜑 → (((𝑆 ∨ 𝑇) ≤ (𝐶 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑈 ≤ (𝐶 ∨ 𝑅)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ ((𝐶 ∨ (𝑃 ∨ 𝑄)) ∨ (𝐶 ∨ 𝑅))))
76	54, 66, 75	mp2and 700	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ ((𝐶 ∨ (𝑃 ∨ 𝑄)) ∨ (𝐶 ∨ 𝑅)))
77	1, 3	dalemreb 40017	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
78	7, 9	latjjdi 18426	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾))) → (𝐶 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅)) = ((𝐶 ∨ (𝑃 ∨ 𝑄)) ∨ (𝐶 ∨ 𝑅)))
79	2, 4, 68, 77, 78	syl13anc 1375	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅)) = ((𝐶 ∨ (𝑃 ∨ 𝑄)) ∨ (𝐶 ∨ 𝑅)))
80	76, 79	breqtrrd 5128	. . . . . . 7 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ (𝐶 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
81		dalem-cly.z	. . . . . . 7 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
82	14	oveq2i 7379	. . . . . . 7 ⊢ (𝐶 ∨ 𝑌) = (𝐶 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅))
83	80, 81, 82	3brtr4g 5134	. . . . . 6 ⊢ (𝜑 → 𝑍 ≤ (𝐶 ∨ 𝑌))
84		breq2 5104	. . . . . 6 ⊢ ((𝐶 ∨ 𝑌) = 𝑌 → (𝑍 ≤ (𝐶 ∨ 𝑌) ↔ 𝑍 ≤ 𝑌))
85	83, 84	syl5ibcom 245	. . . . 5 ⊢ (𝜑 → ((𝐶 ∨ 𝑌) = 𝑌 → 𝑍 ≤ 𝑌))
86	11, 85	sylbid 240	. . . 4 ⊢ (𝜑 → (𝐶 ≤ 𝑌 → 𝑍 ≤ 𝑌))
87	1	dalemzeo 40009	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑂)
88	1	dalemyeo 40008	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
89	8, 5	lplncmp 39938	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑍 ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) → (𝑍 ≤ 𝑌 ↔ 𝑍 = 𝑌))
90	13, 87, 88, 89	syl3anc 1374	. . . . 5 ⊢ (𝜑 → (𝑍 ≤ 𝑌 ↔ 𝑍 = 𝑌))
91		eqcom 2744	. . . . 5 ⊢ (𝑍 = 𝑌 ↔ 𝑌 = 𝑍)
92	90, 91	bitrdi 287	. . . 4 ⊢ (𝜑 → (𝑍 ≤ 𝑌 ↔ 𝑌 = 𝑍))
93	86, 92	sylibd 239	. . 3 ⊢ (𝜑 → (𝐶 ≤ 𝑌 → 𝑌 = 𝑍))
94	93	necon3ad 2946	. 2 ⊢ (𝜑 → (𝑌 ≠ 𝑍 → ¬ 𝐶 ≤ 𝑌))
95	94	imp 406	1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → ¬ 𝐶 ≤ 𝑌)

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  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:  dalem9  40048