Theorem dalem-cly 39690

Description: Lemma for dalem9 39691. 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 39643	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat)
3		dalemc.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
4	1, 3	dalemceb 39657	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
5		dalem-cly.o	. . . . . . 7 ⊢ 𝑂 = (LPlanes‘𝐾)
6	1, 5	dalemyeb 39668	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
7		eqid 2735	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
8		dalemc.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
9		dalemc.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
10	7, 8, 9	latleeqj1 18461	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝐶 ≤ 𝑌 ↔ (𝐶 ∨ 𝑌) = 𝑌))
11	2, 4, 6, 10	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝐶 ≤ 𝑌 ↔ (𝐶 ∨ 𝑌) = 𝑌))
12	1	dalemclpjs 39653	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
13	1	dalemkehl 39642	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ HL)
14		dalem-cly.y	. . . . . . . . . . . . . . 15 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
15	1, 8, 9, 3, 5, 14	dalemcea 39679	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
16	1	dalemsea 39648	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
17	1	dalempea 39645	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
18	1	dalemqea 39646	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
19	1	dalem-clpjq 39656	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
20	8, 9, 3	atnlej1 39398	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)) → 𝐶 ≠ 𝑃)
21	13, 15, 17, 18, 19, 20	syl131anc 1385	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ≠ 𝑃)
22	8, 9, 3	hlatexch1 39414	. . . . . . . . . . . . . 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 1385	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 ≤ (𝑃 ∨ 𝑆) → 𝑆 ≤ (𝑃 ∨ 𝐶)))
24	12, 23	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ≤ (𝑃 ∨ 𝐶))
25	9, 3	hlatjcom 39386	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝐶 ∨ 𝑃) = (𝑃 ∨ 𝐶))
26	13, 15, 17, 25	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∨ 𝑃) = (𝑃 ∨ 𝐶))
27	24, 26	breqtrrd 5147	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ≤ (𝐶 ∨ 𝑃))
28	1	dalemclqjt 39654	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
29	1	dalemtea 39649	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
30	1	dalemrea 39647	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
31		simp312 1322	. . . . . . . . . . . . . . . 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 39398	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅)) → 𝐶 ≠ 𝑄)
34	13, 15, 18, 30, 32, 33	syl131anc 1385	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ≠ 𝑄)
35	8, 9, 3	hlatexch1 39414	. . . . . . . . . . . . . 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 1385	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 ≤ (𝑄 ∨ 𝑇) → 𝑇 ≤ (𝑄 ∨ 𝐶)))
37	28, 36	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ≤ (𝑄 ∨ 𝐶))
38	9, 3	hlatjcom 39386	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝐶 ∨ 𝑄) = (𝑄 ∨ 𝐶))
39	13, 15, 18, 38	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∨ 𝑄) = (𝑄 ∨ 𝐶))
40	37, 39	breqtrrd 5147	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ≤ (𝐶 ∨ 𝑄))
41	1, 3	dalemseb 39661	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾))
42	7, 9, 3	hlatjcl 39385	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝐶 ∨ 𝑃) ∈ (Base‘𝐾))
43	13, 15, 17, 42	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∨ 𝑃) ∈ (Base‘𝐾))
44	1, 3	dalemteb 39662	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾))
45	7, 9, 3	hlatjcl 39385	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝐶 ∨ 𝑄) ∈ (Base‘𝐾))
46	13, 15, 18, 45	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∨ 𝑄) ∈ (Base‘𝐾))
47	7, 8, 9	latjlej12 18465	. . . . . . . . . . . 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 1381	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑆 ≤ (𝐶 ∨ 𝑃) ∧ 𝑇 ≤ (𝐶 ∨ 𝑄)) → (𝑆 ∨ 𝑇) ≤ ((𝐶 ∨ 𝑃) ∨ (𝐶 ∨ 𝑄))))
49	27, 40, 48	mp2and 699	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ ((𝐶 ∨ 𝑃) ∨ (𝐶 ∨ 𝑄)))
50	1, 3	dalempeb 39658	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
51	1, 3	dalemqeb 39659	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
52	7, 9	latjjdi 18501	. . . . . . . . . . 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 1374	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∨ (𝑃 ∨ 𝑄)) = ((𝐶 ∨ 𝑃) ∨ (𝐶 ∨ 𝑄)))
54	49, 53	breqtrrd 5147	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ (𝐶 ∨ (𝑃 ∨ 𝑄)))
55	1	dalemclrju 39655	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ≤ (𝑅 ∨ 𝑈))
56	1	dalemuea 39650	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
57		simp313 1323	. . . . . . . . . . . . . 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 39398	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) → 𝐶 ≠ 𝑅)
60	13, 15, 30, 17, 58, 59	syl131anc 1385	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ≠ 𝑅)
61	8, 9, 3	hlatexch1 39414	. . . . . . . . . . . 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 1385	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ≤ (𝑅 ∨ 𝑈) → 𝑈 ≤ (𝑅 ∨ 𝐶)))
63	55, 62	mpd 15	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ≤ (𝑅 ∨ 𝐶))
64	9, 3	hlatjcom 39386	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝐶 ∨ 𝑅) = (𝑅 ∨ 𝐶))
65	13, 15, 30, 64	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∨ 𝑅) = (𝑅 ∨ 𝐶))
66	63, 65	breqtrrd 5147	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ≤ (𝐶 ∨ 𝑅))
67	1, 9, 3	dalemsjteb 39665	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
68	1, 9, 3	dalempjqeb 39664	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
69	7, 9	latjcl 18449	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝐶 ∨ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾))
70	2, 4, 68, 69	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∨ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾))
71	1, 3	dalemueb 39663	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
72	7, 9, 3	hlatjcl 39385	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝐶 ∨ 𝑅) ∈ (Base‘𝐾))
73	13, 15, 30, 72	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∨ 𝑅) ∈ (Base‘𝐾))
74	7, 8, 9	latjlej12 18465	. . . . . . . . . 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 1381	. . . . . . . . 9 ⊢ (𝜑 → (((𝑆 ∨ 𝑇) ≤ (𝐶 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑈 ≤ (𝐶 ∨ 𝑅)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ ((𝐶 ∨ (𝑃 ∨ 𝑄)) ∨ (𝐶 ∨ 𝑅))))
76	54, 66, 75	mp2and 699	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ ((𝐶 ∨ (𝑃 ∨ 𝑄)) ∨ (𝐶 ∨ 𝑅)))
77	1, 3	dalemreb 39660	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
78	7, 9	latjjdi 18501	. . . . . . . . 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 1374	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅)) = ((𝐶 ∨ (𝑃 ∨ 𝑄)) ∨ (𝐶 ∨ 𝑅)))
80	76, 79	breqtrrd 5147	. . . . . . 7 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ (𝐶 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
81		dalem-cly.z	. . . . . . 7 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
82	14	oveq2i 7416	. . . . . . 7 ⊢ (𝐶 ∨ 𝑌) = (𝐶 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅))
83	80, 81, 82	3brtr4g 5153	. . . . . 6 ⊢ (𝜑 → 𝑍 ≤ (𝐶 ∨ 𝑌))
84		breq2 5123	. . . . . 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 39652	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑂)
88	1	dalemyeo 39651	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
89	8, 5	lplncmp 39581	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑍 ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) → (𝑍 ≤ 𝑌 ↔ 𝑍 = 𝑌))
90	13, 87, 88, 89	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑍 ≤ 𝑌 ↔ 𝑍 = 𝑌))
91		eqcom 2742	. . . . 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 2945	. 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 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  lecple 17278  joincjn 18323  Latclat 18441  Atomscatm 39281  HLchlt 39368  LPlanesclpl 39511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-proset 18306  df-poset 18325  df-plt 18340  df-lub 18356  df-glb 18357  df-join 18358  df-meet 18359  df-p0 18435  df-lat 18442  df-clat 18509  df-oposet 39194  df-ol 39196  df-oml 39197  df-covers 39284  df-ats 39285  df-atl 39316  df-cvlat 39340  df-hlat 39369  df-llines 39517  df-lplanes 39518

This theorem is referenced by:  dalem9  39691