Theorem dalem44 40153

Description: Lemma for dath 40173. 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 40152	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ≠ 𝑌)
14	13	necomd 2988	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≠ ((𝐺 ∨ 𝐻) ∨ 𝐼))
15	1	dalemkelat 40061	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Lat)
16	15	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
17	5, 4	dalemcceb 40126	. . . . . . 7 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
18	17	3ad2ant3 1136	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	dalem42 40151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂)
20		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
21	20, 7	lplnbase 39971	. . . . . . 7 ⊢ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾))
22	19, 21	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾))
23	20, 2, 3	latleeqj1 18375	. . . . . 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 1374	. . . . 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 40137	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ (𝐺 ∨ 𝑐))
26	1	dalemkehl 40060	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ HL)
27	26	3ad2ant1 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
28	5	dalemccea 40120	. . . . . . . . . . . . . 14 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
29	28	3ad2ant3 1136	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
30	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	dalem23 40133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
31	3, 4	hlatjcom 39805	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑐 ∨ 𝐺) = (𝐺 ∨ 𝑐))
32	27, 29, 30, 31	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐺) = (𝐺 ∨ 𝑐))
33	25, 32	breqtrrd 5114	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ (𝑐 ∨ 𝐺))
34	1, 2, 3, 4, 5, 6, 7, 8, 9, 11	dalem33 40142	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ≤ (𝐻 ∨ 𝑐))
35	1, 2, 3, 4, 5, 6, 7, 8, 9, 11	dalem29 40138	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)
36	3, 4	hlatjcom 39805	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝑐 ∨ 𝐻) = (𝐻 ∨ 𝑐))
37	27, 29, 35, 36	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐻) = (𝐻 ∨ 𝑐))
38	34, 37	breqtrrd 5114	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ≤ (𝑐 ∨ 𝐻))
39	1, 4	dalempeb 40076	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
40	39	3ad2ant1 1134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ (Base‘𝐾))
41	20, 3, 4	hlatjcl 39804	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴) → (𝑐 ∨ 𝐺) ∈ (Base‘𝐾))
42	27, 29, 30, 41	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐺) ∈ (Base‘𝐾))
43	1, 4	dalemqeb 40077	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
44	43	3ad2ant1 1134	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ∈ (Base‘𝐾))
45	20, 3, 4	hlatjcl 39804	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝑐 ∨ 𝐻) ∈ (Base‘𝐾))
46	27, 29, 35, 45	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐻) ∈ (Base‘𝐾))
47	20, 2, 3	latjlej12 18379	. . . . . . . . . . . 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 1382	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ≤ (𝑐 ∨ 𝐺) ∧ 𝑄 ≤ (𝑐 ∨ 𝐻)) → (𝑃 ∨ 𝑄) ≤ ((𝑐 ∨ 𝐺) ∨ (𝑐 ∨ 𝐻))))
49	33, 38, 48	mp2and 700	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ ((𝑐 ∨ 𝐺) ∨ (𝑐 ∨ 𝐻)))
50	20, 4	atbase 39726	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ 𝐴 → 𝐺 ∈ (Base‘𝐾))
51	30, 50	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ (Base‘𝐾))
52	20, 4	atbase 39726	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ 𝐴 → 𝐻 ∈ (Base‘𝐾))
53	35, 52	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ (Base‘𝐾))
54	20, 3	latjjdi 18415	. . . . . . . . . . 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 1375	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ (𝐺 ∨ 𝐻)) = ((𝑐 ∨ 𝐺) ∨ (𝑐 ∨ 𝐻)))
56	49, 55	breqtrrd 5114	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ (𝑐 ∨ (𝐺 ∨ 𝐻)))
57	1, 2, 3, 4, 5, 6, 7, 8, 9, 12	dalem37 40146	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ≤ (𝐼 ∨ 𝑐))
58	1, 2, 3, 4, 5, 6, 7, 8, 9, 12	dalem34 40143	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴)
59	3, 4	hlatjcom 39805	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) → (𝑐 ∨ 𝐼) = (𝐼 ∨ 𝑐))
60	27, 29, 58, 59	syl3anc 1374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐼) = (𝐼 ∨ 𝑐))
61	57, 60	breqtrrd 5114	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ≤ (𝑐 ∨ 𝐼))
62	1, 3, 4	dalempjqeb 40082	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
63	62	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
64	20, 3, 4	hlatjcl 39804	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
65	27, 30, 35, 64	syl3anc 1374	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
66	20, 3	latjcl 18363	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑐 ∈ (Base‘𝐾) ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾)) → (𝑐 ∨ (𝐺 ∨ 𝐻)) ∈ (Base‘𝐾))
67	16, 18, 65, 66	syl3anc 1374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ (𝐺 ∨ 𝐻)) ∈ (Base‘𝐾))
68	1, 4	dalemreb 40078	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
69	68	3ad2ant1 1134	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ∈ (Base‘𝐾))
70	20, 3, 4	hlatjcl 39804	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) → (𝑐 ∨ 𝐼) ∈ (Base‘𝐾))
71	27, 29, 58, 70	syl3anc 1374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝐼) ∈ (Base‘𝐾))
72	20, 2, 3	latjlej12 18379	. . . . . . . . . 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 1382	. . . . . . . . 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 700	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ ((𝑐 ∨ (𝐺 ∨ 𝐻)) ∨ (𝑐 ∨ 𝐼)))
75	20, 4	atbase 39726	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾))
76	58, 75	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾))
77	20, 3	latjjdi 18415	. . . . . . . . 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 1375	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) = ((𝑐 ∨ (𝐺 ∨ 𝐻)) ∨ (𝑐 ∨ 𝐼)))
79	74, 78	breqtrrd 5114	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)))
80	8, 79	eqbrtrid 5121	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≤ (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)))
81		breq2 5090	. . . . . 6 ⊢ ((𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) = ((𝐺 ∨ 𝐻) ∨ 𝐼) → (𝑌 ≤ (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) ↔ 𝑌 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼)))
82	80, 81	syl5ibcom 245	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)) = ((𝐺 ∨ 𝐻) ∨ 𝐼) → 𝑌 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼)))
83	24, 82	sylbid 240	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) → 𝑌 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼)))
84	1	dalemyeo 40069	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
85	84	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ 𝑂)
86	2, 7	lplncmp 39999	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝑂 ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂) → (𝑌 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) ↔ 𝑌 = ((𝐺 ∨ 𝐻) ∨ 𝐼)))
87	27, 85, 19, 86	syl3anc 1374	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑌 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) ↔ 𝑌 = ((𝐺 ∨ 𝐻) ∨ 𝐼)))
88	83, 87	sylibd 239	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) → 𝑌 = ((𝐺 ∨ 𝐻) ∨ 𝐼)))
89	88	necon3ad 2946	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑌 ≠ ((𝐺 ∨ 𝐻) ∨ 𝐼) → ¬ 𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼)))
90	14, 89	mpd 15	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑐 ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   class class class wbr 5086  ‘cfv 6490  (class class class)co 7358  Basecbs 17137  lecple 17185  joincjn 18235  meetcmee 18236  Latclat 18355  Atomscatm 39700  HLchlt 39787  LPlanesclpl 39929

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-proset 18218  df-poset 18237  df-plt 18252  df-lub 18268  df-glb 18269  df-join 18270  df-meet 18271  df-p0 18347  df-lat 18356  df-clat 18423  df-oposet 39613  df-ol 39615  df-oml 39616  df-covers 39703  df-ats 39704  df-atl 39735  df-cvlat 39759  df-hlat 39788  df-llines 39935  df-lplanes 39936  df-lvols 39937

This theorem is referenced by:  dalem45  40154