Theorem dalem-cly 39660

Description: Lemma for dalem9 39661. 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 39613	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat)
3		dalemc.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
4	1, 3	dalemceb 39627	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
5		dalem-cly.o	. . . . . . 7 ⊢ 𝑂 = (LPlanes‘𝐾)
6	1, 5	dalemyeb 39638	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
7		eqid 2729	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
8		dalemc.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
9		dalemc.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
10	7, 8, 9	latleeqj1 18357	. . . . . 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 39623	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
13	1	dalemkehl 39612	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ HL)
14		dalem-cly.y	. . . . . . . . . . . . . . 15 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
15	1, 8, 9, 3, 5, 14	dalemcea 39649	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
16	1	dalemsea 39618	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
17	1	dalempea 39615	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
18	1	dalemqea 39616	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
19	1	dalem-clpjq 39626	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
20	8, 9, 3	atnlej1 39368	. . . . . . . . . . . . . . 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 39384	. . . . . . . . . . . . . 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 39357	. . . . . . . . . . . . 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 5120	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ≤ (𝐶 ∨ 𝑃))
28	1	dalemclqjt 39624	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
29	1	dalemtea 39619	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
30	1	dalemrea 39617	. . . . . . . . . . . . . . 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 39368	. . . . . . . . . . . . . . 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 39384	. . . . . . . . . . . . . 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 39357	. . . . . . . . . . . . 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 5120	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ≤ (𝐶 ∨ 𝑄))
41	1, 3	dalemseb 39631	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾))
42	7, 9, 3	hlatjcl 39356	. . . . . . . . . . . . 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 39632	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾))
45	7, 9, 3	hlatjcl 39356	. . . . . . . . . . . . 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 18361	. . . . . . . . . . . 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 39628	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
51	1, 3	dalemqeb 39629	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
52	7, 9	latjjdi 18397	. . . . . . . . . . 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 5120	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ (𝐶 ∨ (𝑃 ∨ 𝑄)))
55	1	dalemclrju 39625	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ≤ (𝑅 ∨ 𝑈))
56	1	dalemuea 39620	. . . . . . . . . . . 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 39368	. . . . . . . . . . . . 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 39384	. . . . . . . . . . . 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 39357	. . . . . . . . . . 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 5120	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ≤ (𝐶 ∨ 𝑅))
67	1, 9, 3	dalemsjteb 39635	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
68	1, 9, 3	dalempjqeb 39634	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
69	7, 9	latjcl 18345	. . . . . . . . . . 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 39633	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
72	7, 9, 3	hlatjcl 39356	. . . . . . . . . . 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 18361	. . . . . . . . . 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 39630	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
78	7, 9	latjjdi 18397	. . . . . . . . 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 5120	. . . . . . 7 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ (𝐶 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
81		dalem-cly.z	. . . . . . 7 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
82	14	oveq2i 7360	. . . . . . 7 ⊢ (𝐶 ∨ 𝑌) = (𝐶 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅))
83	80, 81, 82	3brtr4g 5126	. . . . . 6 ⊢ (𝜑 → 𝑍 ≤ (𝐶 ∨ 𝑌))
84		breq2 5096	. . . . . 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 39622	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑂)
88	1	dalemyeo 39621	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
89	8, 5	lplncmp 39551	. . . . . 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 2736	. . . . 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 2938	. 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 2109   ≠ wne 2925   class class class wbr 5092  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  lecple 17168  joincjn 18217  Latclat 18337  Atomscatm 39252  HLchlt 39339  LPlanesclpl 39481

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-proset 18200  df-poset 18219  df-plt 18234  df-lub 18250  df-glb 18251  df-join 18252  df-meet 18253  df-p0 18329  df-lat 18338  df-clat 18405  df-oposet 39165  df-ol 39167  df-oml 39168  df-covers 39255  df-ats 39256  df-atl 39287  df-cvlat 39311  df-hlat 39340  df-llines 39487  df-lplanes 39488

This theorem is referenced by:  dalem9  39661