Theorem dalem44 40550

Description: Lemma for dath 40570. 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 40549	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ≠ 𝑌)
14	13	necomd 3015	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≠ ((𝐺 ∨ 𝐻) ∨ 𝐼))
15	1	dalemkelat 40458	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Lat)
16	15	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
17	5, 4	dalemcceb 40523	. . . . . . 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 40548	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂)
20		eqid 2765	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
21	20, 7	lplnbase 40368	. . . . . . 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 18531	. . . . . 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 40534	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ (𝐺 ∨ 𝑐))
26	1	dalemkehl 40457	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ HL)
27	26	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
28	5	dalemccea 40517	. . . . . . . . . . . . . 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 40530	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
31	3, 4	hlatjcom 40202	. . . . . . . . . . . . 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 5141	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ≤ (𝑐 ∨ 𝐺))
34	1, 2, 3, 4, 5, 6, 7, 8, 9, 11	dalem33 40539	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ≤ (𝐻 ∨ 𝑐))
35	1, 2, 3, 4, 5, 6, 7, 8, 9, 11	dalem29 40535	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)
36	3, 4	hlatjcom 40202	. . . . . . . . . . . . 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 5141	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ≤ (𝑐 ∨ 𝐻))
39	1, 4	dalempeb 40473	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
40	39	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ (Base‘𝐾))
41	20, 3, 4	hlatjcl 40201	. . . . . . . . . . . . 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 40474	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
44	43	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑄 ∈ (Base‘𝐾))
45	20, 3, 4	hlatjcl 40201	. . . . . . . . . . . . 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 18535	. . . . . . . . . . . 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 40123	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ 𝐴 → 𝐺 ∈ (Base‘𝐾))
51	30, 50	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ (Base‘𝐾))
52	20, 4	atbase 40123	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ 𝐴 → 𝐻 ∈ (Base‘𝐾))
53	35, 52	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ (Base‘𝐾))
54	20, 3	latjjdi 18571	. . . . . . . . . . 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 5141	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ (𝑐 ∨ (𝐺 ∨ 𝐻)))
57	1, 2, 3, 4, 5, 6, 7, 8, 9, 12	dalem37 40543	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ≤ (𝐼 ∨ 𝑐))
58	1, 2, 3, 4, 5, 6, 7, 8, 9, 12	dalem34 40540	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴)
59	3, 4	hlatjcom 40202	. . . . . . . . . . 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 5141	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ≤ (𝑐 ∨ 𝐼))
62	1, 3, 4	dalempjqeb 40479	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
63	62	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
64	20, 3, 4	hlatjcl 40201	. . . . . . . . . . . 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 18519	. . . . . . . . . . 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 40475	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
69	68	3ad2ant1 1151	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑅 ∈ (Base‘𝐾))
70	20, 3, 4	hlatjcl 40201	. . . . . . . . . . 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 18535	. . . . . . . . . 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 40123	. . . . . . . . . 10 ⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾))
76	58, 75	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾))
77	20, 3	latjjdi 18571	. . . . . . . . 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 5141	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑃 ∨ 𝑄) ∨ 𝑅) ≤ (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)))
80	8, 79	eqbrtrid 5148	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ≤ (𝑐 ∨ ((𝐺 ∨ 𝐻) ∨ 𝐼)))
81		breq2 5115	. . . . . 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 40466	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
85	84	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ 𝑂)
86	2, 7	lplncmp 40396	. . . . 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 2973	. 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 2146   ≠ wne 2960   class class class wbr 5111  ‘cfv 6540  (class class class)co 7419  Basecbs 17293  lecple 17341  joincjn 18391  meetcmee 18392  Latclat 18511  Atomscatm 40097  HLchlt 40184  LPlanesclpl 40326

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-proset 18374  df-poset 18393  df-plt 18408  df-lub 18424  df-glb 18425  df-join 18426  df-meet 18427  df-p0 18503  df-lat 18512  df-clat 18579  df-oposet 40010  df-ol 40012  df-oml 40013  df-covers 40100  df-ats 40101  df-atl 40132  df-cvlat 40156  df-hlat 40185  df-llines 40332  df-lplanes 40333  df-lvols 40334

This theorem is used by:  dalem45  40551