Theorem dalem44 40590

Description: Lemma for dath 40610. Dummy center of perspectivity 𝑐 lies outside of plane 𝐺𝐻𝐼. (Contributed by NM, 16-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))))
dalem44.m	⊢ ∧ = (meet‘𝐾)
dalem44.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem44.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem44.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem44.g	⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
dalem44.h	⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
dalem44.i	⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))

Assertion

Ref	Expression
dalem44	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼))

Proof of Theorem dalem44

Step	Hyp	Ref	Expression
1		dalem.ph	. . . 4 ⊢ (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		dalem.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		dalem.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		dalem.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5		dalem.ps	. . . 4 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
6		dalem44.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
7		dalem44.o	. . . 4 ⊢ 𝑂 = (LPlanes‘𝐾)
8		dalem44.y	. . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
9		dalem44.z	. . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
10		dalem44.g	. . . 4 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
11		dalem44.h	. . . 4 ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
12		dalem44.i	. . . 4 ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	dalem43 40589	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ≠ 𝑌)
14	13	necomd 3010	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≠ ((𝐺 ∨ 𝐻) ∨ 𝐼))
15	1	dalemkelat 40498	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Lat)
16	15	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
17	5, 4	dalemcceb 40563	. . . . . . 7 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
18	17	3ad2ant3 1153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	dalem42 40588	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂)
20		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
21	20, 7	lplnbase 40408	. . . . . . 7 ⊢ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾))
22	19, 21	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾))
23	20, 2, 3	latleeqj1 18540	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾)) → (𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) ↔ (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) = ((𝐺 ∨ 𝐻) ∨ 𝐼)))
24	16, 18, 22, 23	syl3anc 1398	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) ↔ (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) = ((𝐺 ∨ 𝐻) ∨ 𝐼)))
25	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	dalem28 40574	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ (𝐺 ∨ 𝑐))
26	1	dalemkehl 40497	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ HL)
27	26	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
28	5	dalemccea 40557	. . . . . . . . . . . . . 14 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
29	28	3ad2ant3 1153	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
30	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	dalem23 40570	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
31	3, 4	hlatjcom 40242	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑐 ∨ 𝐺) = (𝐺 ∨ 𝑐))
32	27, 29, 30, 31	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐺) = (𝐺 ∨ 𝑐))
33	25, 32	breqtrrd 5133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ (𝑐 ∨ 𝐺))
34	1, 2, 3, 4, 5, 6, 7, 8, 9, 11	dalem33 40579	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ≤ (𝐻 ∨ 𝑐))
35	1, 2, 3, 4, 5, 6, 7, 8, 9, 11	dalem29 40575	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)
36	3, 4	hlatjcom 40242	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝑐 ∨ 𝐻) = (𝐻 ∨ 𝑐))
37	27, 29, 35, 36	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐻) = (𝐻 ∨ 𝑐))
38	34, 37	breqtrrd 5133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ≤ (𝑐 ∨ 𝐻))
39	1, 4	dalempeb 40513	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
40	39	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ (Base‘𝐾))
41	20, 3, 4	hlatjcl 40241	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑐 ∨ 𝐺) ∈ (Base‘𝐾))
42	27, 29, 30, 41	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐺) ∈ (Base‘𝐾))
43	1, 4	dalemqeb 40514	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
44	43	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ∈ (Base‘𝐾))
45	20, 3, 4	hlatjcl 40241	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝑐 ∨ 𝐻) ∈ (Base‘𝐾))
46	27, 29, 35, 45	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐻) ∈ (Base‘𝐾))
47	20, 2, 3	latjlej12 18544	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ (𝑐 ∨ 𝐺) ∈ (Base‘𝐾)) ∧ (𝑄 ∈ (Base‘𝐾) ∧ (𝑐 ∨ 𝐻) ∈ (Base‘𝐾))) → ((𝑃 ≤ (𝑐 ∨ 𝐺) ∧ 𝑄 ≤ (𝑐 ∨ 𝐻)) → (𝑃 ∨ 𝑄) ≤ ((𝑐 ∨ 𝐺) ∨ (𝑐 ∨ 𝐻))))
48	16, 40, 42, 44, 46, 47	syl122anc 1406	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ≤ (𝑐 ∨ 𝐺) ∧ 𝑄 ≤ (𝑐 ∨ 𝐻)) → (𝑃 ∨ 𝑄) ≤ ((𝑐 ∨ 𝐺) ∨ (𝑐 ∨ 𝐻))))
49	33, 38, 48	mp2and 712	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ ((𝑐 ∨ 𝐺) ∨ (𝑐 ∨ 𝐻)))
50	20, 4	atbase 40163	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ 𝐴 → 𝐺 ∈ (Base‘𝐾))
51	30, 50	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ (Base‘𝐾))
52	20, 4	atbase 40163	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ 𝐴 → 𝐻 ∈ (Base‘𝐾))
53	35, 52	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ (Base‘𝐾))
54	20, 3	latjjdi 18580	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Base‘𝐾) ∧ 𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾))) → (𝑐 ∨ (𝐺 ∨ 𝐻)) = ((𝑐 ∨ 𝐺) ∨ (𝑐 ∨ 𝐻)))
55	16, 18, 51, 53, 54	syl13anc 1399	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ (𝐺 ∨ 𝐻)) = ((𝑐 ∨ 𝐺) ∨ (𝑐 ∨ 𝐻)))
56	49, 55	breqtrrd 5133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ (𝑐 ∨ (𝐺 ∨ 𝐻)))
57	1, 2, 3, 4, 5, 6, 7, 8, 9, 12	dalem37 40583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ≤ (𝐼 ∨ 𝑐))
58	1, 2, 3, 4, 5, 6, 7, 8, 9, 12	dalem34 40580	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴)
59	3, 4	hlatjcom 40242	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) → (𝑐 ∨ 𝐼) = (𝐼 ∨ 𝑐))
60	27, 29, 58, 59	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐼) = (𝐼 ∨ 𝑐))
61	57, 60	breqtrrd 5133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ≤ (𝑐 ∨ 𝐼))
62	1, 3, 4	dalempjqeb 40519	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
63	62	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
64	20, 3, 4	hlatjcl 40241	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
65	27, 30, 35, 64	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
66	20, 3	latjcl 18528	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾)) → (𝑐 ∨ (𝐺 ∨ 𝐻)) ∈ (Base‘𝐾))
67	16, 18, 65, 66	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ (𝐺 ∨ 𝐻)) ∈ (Base‘𝐾))
68	1, 4	dalemreb 40515	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
69	68	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ∈ (Base‘𝐾))
70	20, 3, 4	hlatjcl 40241	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) → (𝑐 ∨ 𝐼) ∈ (Base‘𝐾))
71	27, 29, 58, 70	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐼) ∈ (Base‘𝐾))
72	20, 2, 3	latjlej12 18544	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ ((𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝑐 ∨ (𝐺 ∨ 𝐻)) ∈ (Base‘𝐾)) ∧ (𝑅 ∈ (Base‘𝐾) ∧ (𝑐 ∨ 𝐼) ∈ (Base‘𝐾))) → (((𝑃 ∨ 𝑄) ≤ (𝑐 ∨ (𝐺 ∨ 𝐻)) ∧ 𝑅 ≤ (𝑐 ∨ 𝐼)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑐 ∨ (𝐺 ∨ 𝐻)) ∨ (𝑐 ∨ 𝐼))))
73	16, 63, 67, 69, 71, 72	syl122anc 1406	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝑃 ∨ 𝑄) ≤ (𝑐 ∨ (𝐺 ∨ 𝐻)) ∧ 𝑅 ≤ (𝑐 ∨ 𝐼)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑐 ∨ (𝐺 ∨ 𝐻)) ∨ (𝑐 ∨ 𝐼))))
74	56, 61, 73	mp2and 712	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑐 ∨ (𝐺 ∨ 𝐻)) ∨ (𝑐 ∨ 𝐼)))
75	20, 4	atbase 40163	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾))
76	58, 75	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾))
77	20, 3	latjjdi 18580	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Base‘𝐾) ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾))) → (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) = ((𝑐 ∨ (𝐺 ∨ 𝐻)) ∨ (𝑐 ∨ 𝐼)))
78	16, 18, 65, 76, 77	syl13anc 1399	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) = ((𝑐 ∨ (𝐺 ∨ 𝐻)) ∨ (𝑐 ∨ 𝐼)))
79	74, 78	breqtrrd 5133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)))
80	8, 79	eqbrtrid 5140	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≤ (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)))
81		breq2 5107	. . . . . 6 ⊢ ((𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) = ((𝐺 ∨ 𝐻) ∨ 𝐼) → (𝑌 ≤ (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) ↔ 𝑌 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼)))
82	80, 81	syl5ibcom 248	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) = ((𝐺 ∨ 𝐻) ∨ 𝐼) → 𝑌 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼)))
83	24, 82	sylbid 243	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) → 𝑌 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼)))
84	1	dalemyeo 40506	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
85	84	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ 𝑂)
86	2, 7	lplncmp 40436	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑂 ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂) → (𝑌 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) ↔ 𝑌 = ((𝐺 ∨ 𝐻) ∨ 𝐼)))
87	27, 85, 19, 86	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑌 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) ↔ 𝑌 = ((𝐺 ∨ 𝐻) ∨ 𝐼)))
88	83, 87	sylibd 242	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) → 𝑌 = ((𝐺 ∨ 𝐻) ∨ 𝐼)))
89	88	necon3ad 2968	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑌 ≠ ((𝐺 ∨ 𝐻) ∨ 𝐼) → ¬ 𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼)))
90	14, 89	mpd 16	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955   class class class wbr 5103  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  lecple 17350  joincjn 18400  meetcmee 18401  Latclat 18520  Atomscatm 40137  HLchlt 40224  LPlanesclpl 40366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-proset 18383  df-poset 18402  df-plt 18417  df-lub 18433  df-glb 18434  df-join 18435  df-meet 18436  df-p0 18512  df-lat 18521  df-clat 18588  df-oposet 40050  df-ol 40052  df-oml 40053  df-covers 40140  df-ats 40141  df-atl 40172  df-cvlat 40196  df-hlat 40225  df-llines 40372  df-lplanes 40373  df-lvols 40374

This theorem is used by:  dalem45  40591