Theorem dalem-cly 33975

Description: Lemma for dalem9 33976. 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 33928	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat)
3		dalemc.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
4	1, 3	dalemceb 33942	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
5		dalem-cly.o	. . . . . . 7 ⊢ 𝑂 = (LPlanes‘𝐾)
6	1, 5	dalemyeb 33953	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
7		eqid 2610	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
8		dalemc.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
9		dalemc.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
10	7, 8, 9	latleeqj1 16886	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝐶 ≤ 𝑌 ↔ (𝐶 ∨ 𝑌) = 𝑌))
11	2, 4, 6, 10	syl3anc 1318	. . . . 5 ⊢ (𝜑 → (𝐶 ≤ 𝑌 ↔ (𝐶 ∨ 𝑌) = 𝑌))
12	1	dalemclpjs 33938	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
13	1	dalemkehl 33927	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ HL)
14		dalem-cly.y	. . . . . . . . . . . . . . 15 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
15	1, 8, 9, 3, 5, 14	dalemcea 33964	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
16	1	dalemsea 33933	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
17	1	dalempea 33930	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
18	1	dalemqea 33931	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
19	1	dalem-clpjq 33941	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
20	8, 9, 3	atnlej1 33683	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)) → 𝐶 ≠ 𝑃)
21	13, 15, 17, 18, 19, 20	syl131anc 1331	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ≠ 𝑃)
22	8, 9, 3	hlatexch1 33699	. . . . . . . . . . . . . 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 1331	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 ≤ (𝑃 ∨ 𝑆) → 𝑆 ≤ (𝑃 ∨ 𝐶)))
24	12, 23	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ≤ (𝑃 ∨ 𝐶))
25	9, 3	hlatjcom 33672	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝐶 ∨ 𝑃) = (𝑃 ∨ 𝐶))
26	13, 15, 17, 25	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∨ 𝑃) = (𝑃 ∨ 𝐶))
27	24, 26	breqtrrd 4611	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ≤ (𝐶 ∨ 𝑃))
28	1	dalemclqjt 33939	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
29	1	dalemtea 33934	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
30	1	dalemrea 33932	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
31		simp312 1202	. . . . . . . . . . . . . . . 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 206	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑄 ∨ 𝑅))
33	8, 9, 3	atnlej1 33683	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅)) → 𝐶 ≠ 𝑄)
34	13, 15, 18, 30, 32, 33	syl131anc 1331	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ≠ 𝑄)
35	8, 9, 3	hlatexch1 33699	. . . . . . . . . . . . . 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 1331	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 ≤ (𝑄 ∨ 𝑇) → 𝑇 ≤ (𝑄 ∨ 𝐶)))
37	28, 36	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ≤ (𝑄 ∨ 𝐶))
38	9, 3	hlatjcom 33672	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝐶 ∨ 𝑄) = (𝑄 ∨ 𝐶))
39	13, 15, 18, 38	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∨ 𝑄) = (𝑄 ∨ 𝐶))
40	37, 39	breqtrrd 4611	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ≤ (𝐶 ∨ 𝑄))
41	1, 3	dalemseb 33946	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ (Base‘𝐾))
42	7, 9, 3	hlatjcl 33671	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝐶 ∨ 𝑃) ∈ (Base‘𝐾))
43	13, 15, 17, 42	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∨ 𝑃) ∈ (Base‘𝐾))
44	1, 3	dalemteb 33947	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾))
45	7, 9, 3	hlatjcl 33671	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝐶 ∨ 𝑄) ∈ (Base‘𝐾))
46	13, 15, 18, 45	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∨ 𝑄) ∈ (Base‘𝐾))
47	7, 8, 9	latjlej12 16890	. . . . . . . . . . . 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 1327	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑆 ≤ (𝐶 ∨ 𝑃) ∧ 𝑇 ≤ (𝐶 ∨ 𝑄)) → (𝑆 ∨ 𝑇) ≤ ((𝐶 ∨ 𝑃) ∨ (𝐶 ∨ 𝑄))))
49	27, 40, 48	mp2and 711	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ ((𝐶 ∨ 𝑃) ∨ (𝐶 ∨ 𝑄)))
50	1, 3	dalempeb 33943	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
51	1, 3	dalemqeb 33944	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
52	7, 9	latjjdi 16926	. . . . . . . . . . 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 1320	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∨ (𝑃 ∨ 𝑄)) = ((𝐶 ∨ 𝑃) ∨ (𝐶 ∨ 𝑄)))
54	49, 53	breqtrrd 4611	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ≤ (𝐶 ∨ (𝑃 ∨ 𝑄)))
55	1	dalemclrju 33940	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ≤ (𝑅 ∨ 𝑈))
56	1	dalemuea 33935	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
57		simp313 1203	. . . . . . . . . . . . . 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 206	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑅 ∨ 𝑃))
59	8, 9, 3	atnlej1 33683	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) → 𝐶 ≠ 𝑅)
60	13, 15, 30, 17, 58, 59	syl131anc 1331	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ≠ 𝑅)
61	8, 9, 3	hlatexch1 33699	. . . . . . . . . . . 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 1331	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐶 ≤ (𝑅 ∨ 𝑈) → 𝑈 ≤ (𝑅 ∨ 𝐶)))
63	55, 62	mpd 15	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ≤ (𝑅 ∨ 𝐶))
64	9, 3	hlatjcom 33672	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝐶 ∨ 𝑅) = (𝑅 ∨ 𝐶))
65	13, 15, 30, 64	syl3anc 1318	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∨ 𝑅) = (𝑅 ∨ 𝐶))
66	63, 65	breqtrrd 4611	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ≤ (𝐶 ∨ 𝑅))
67	1, 9, 3	dalemsjteb 33950	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 ∨ 𝑇) ∈ (Base‘𝐾))
68	1, 9, 3	dalempjqeb 33949	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
69	7, 9	latjcl 16874	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝐶 ∨ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾))
70	2, 4, 68, 69	syl3anc 1318	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∨ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾))
71	1, 3	dalemueb 33948	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
72	7, 9, 3	hlatjcl 33671	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝐶 ∨ 𝑅) ∈ (Base‘𝐾))
73	13, 15, 30, 72	syl3anc 1318	. . . . . . . . . 10 ⊢ (𝜑 → (𝐶 ∨ 𝑅) ∈ (Base‘𝐾))
74	7, 8, 9	latjlej12 16890	. . . . . . . . . 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 1327	. . . . . . . . 9 ⊢ (𝜑 → (((𝑆 ∨ 𝑇) ≤ (𝐶 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑈 ≤ (𝐶 ∨ 𝑅)) → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ ((𝐶 ∨ (𝑃 ∨ 𝑄)) ∨ (𝐶 ∨ 𝑅))))
76	54, 66, 75	mp2and 711	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ ((𝐶 ∨ (𝑃 ∨ 𝑄)) ∨ (𝐶 ∨ 𝑅)))
77	1, 3	dalemreb 33945	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
78	7, 9	latjjdi 16926	. . . . . . . . 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 1320	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅)) = ((𝐶 ∨ (𝑃 ∨ 𝑄)) ∨ (𝐶 ∨ 𝑅)))
80	76, 79	breqtrrd 4611	. . . . . . 7 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ≤ (𝐶 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
81		dalem-cly.z	. . . . . . 7 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
82	14	oveq2i 6560	. . . . . . 7 ⊢ (𝐶 ∨ 𝑌) = (𝐶 ∨ ((𝑃 ∨ 𝑄) ∨ 𝑅))
83	80, 81, 82	3brtr4g 4617	. . . . . 6 ⊢ (𝜑 → 𝑍 ≤ (𝐶 ∨ 𝑌))
84		breq2 4587	. . . . . 6 ⊢ ((𝐶 ∨ 𝑌) = 𝑌 → (𝑍 ≤ (𝐶 ∨ 𝑌) ↔ 𝑍 ≤ 𝑌))
85	83, 84	syl5ibcom 234	. . . . 5 ⊢ (𝜑 → ((𝐶 ∨ 𝑌) = 𝑌 → 𝑍 ≤ 𝑌))
86	11, 85	sylbid 229	. . . 4 ⊢ (𝜑 → (𝐶 ≤ 𝑌 → 𝑍 ≤ 𝑌))
87	1	dalemzeo 33937	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑂)
88	1	dalemyeo 33936	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
89	8, 5	lplncmp 33866	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑍 ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) → (𝑍 ≤ 𝑌 ↔ 𝑍 = 𝑌))
90	13, 87, 88, 89	syl3anc 1318	. . . . 5 ⊢ (𝜑 → (𝑍 ≤ 𝑌 ↔ 𝑍 = 𝑌))
91		eqcom 2617	. . . . 5 ⊢ (𝑍 = 𝑌 ↔ 𝑌 = 𝑍)
92	90, 91	syl6bb 275	. . . 4 ⊢ (𝜑 → (𝑍 ≤ 𝑌 ↔ 𝑌 = 𝑍))
93	86, 92	sylibd 228	. . 3 ⊢ (𝜑 → (𝐶 ≤ 𝑌 → 𝑌 = 𝑍))
94	93	necon3ad 2795	. 2 ⊢ (𝜑 → (𝑌 ≠ 𝑍 → ¬ 𝐶 ≤ 𝑌))
95	94	imp 444	1 ⊢ ((𝜑 ∧ 𝑌 ≠ 𝑍) → ¬ 𝐶 ≤ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  Latclat 16868  Atomscatm 33568  HLchlt 33655  LPlanesclpl 33796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803

This theorem is referenced by:  dalem9  33976