Theorem dalem54 40424

Description: Lemma for dath 40434. Line 𝐺𝐻 intersects the auxiliary axis of perspectivity 𝐵. (Contributed by NM, 8-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))))
dalem54.m	⊢ ∧ = (meet‘𝐾)
dalem54.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem54.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem54.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem54.g	⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
dalem54.h	⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
dalem54.i	⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))
dalem54.b1	⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)

Assertion

Ref	Expression
dalem54	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ 𝐴)

Proof of Theorem dalem54

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	1	dalemkehl 40321	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
3	2	3ad2ant1 1149	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
4		dalem.l	. . . 4 ⊢ ≤ = (le‘𝐾)
5		dalem.j	. . . 4 ⊢ ∨ = (join‘𝐾)
6		dalem.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
7		dalem.ps	. . . 4 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
8		dalem54.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
9		dalem54.o	. . . 4 ⊢ 𝑂 = (LPlanes‘𝐾)
10		dalem54.y	. . . 4 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
11		dalem54.z	. . . 4 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
12		dalem54.g	. . . 4 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
13	1, 4, 5, 6, 7, 8, 9, 10, 11, 12	dalem23 40394	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
14		dalem54.h	. . . 4 ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
15	1, 4, 5, 6, 7, 8, 9, 10, 11, 14	dalem29 40399	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)
16		dalem54.i	. . . 4 ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))
17	1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16	dalem41 40411	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ≠ 𝐻)
18		eqid 2769	. . . 4 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
19	5, 6, 18	llni2 40210	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) ∧ 𝐺 ≠ 𝐻) → (𝐺 ∨ 𝐻) ∈ (LLines‘𝐾))
20	3, 13, 15, 17, 19	syl31anc 1398	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (LLines‘𝐾))
21		dalem54.b1	. . 3 ⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)
22	1, 4, 5, 6, 7, 8, 18, 9, 10, 11, 12, 14, 16, 21	dalem53 40423	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ∈ (LLines‘𝐾))
23	1	dalemkelat 40322	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Lat)
24	23	3ad2ant1 1149	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
25		eqid 2769	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
26	25, 18	llnbase 40207	. . . . . . . 8 ⊢ ((𝐺 ∨ 𝐻) ∈ (LLines‘𝐾) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
27	20, 26	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
28	1, 4, 5, 6, 7, 8, 9, 10, 11, 16	dalem34 40404	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴)
29	25, 6	atbase 39987	. . . . . . . 8 ⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾))
30	28, 29	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾))
31	25, 5	latjcl 18495	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾))
32	24, 27, 30, 31	syl3anc 1396	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾))
33	1, 9	dalemyeb 40347	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
34	33	3ad2ant1 1149	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾))
35	25, 4, 8	latmle2 18521	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌) ≤ 𝑌)
36	24, 32, 34, 35	syl3anc 1396	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌) ≤ 𝑌)
37	21, 36	eqbrtrid 5150	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ≤ 𝑌)
38	1, 4, 5, 6, 7, 8, 9, 10, 11, 12	dalem24 40395	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝐺 ≤ 𝑌)
39	25, 6	atbase 39987	. . . . . . . 8 ⊢ (𝐺 ∈ 𝐴 → 𝐺 ∈ (Base‘𝐾))
40	13, 39	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ (Base‘𝐾))
41	25, 6	atbase 39987	. . . . . . . 8 ⊢ (𝐻 ∈ 𝐴 → 𝐻 ∈ (Base‘𝐾))
42	15, 41	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ (Base‘𝐾))
43	25, 4, 5	latjle12 18506	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∈ (Base‘𝐾) ∧ 𝐻 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → ((𝐺 ≤ 𝑌 ∧ 𝐻 ≤ 𝑌) ↔ (𝐺 ∨ 𝐻) ≤ 𝑌))
44	24, 40, 42, 34, 43	syl13anc 1397	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ≤ 𝑌 ∧ 𝐻 ≤ 𝑌) ↔ (𝐺 ∨ 𝐻) ≤ 𝑌))
45		simpl 487	. . . . . 6 ⊢ ((𝐺 ≤ 𝑌 ∧ 𝐻 ≤ 𝑌) → 𝐺 ≤ 𝑌)
46	44, 45	biimtrrdi 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ≤ 𝑌 → 𝐺 ≤ 𝑌))
47	38, 46	mtod 201	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ (𝐺 ∨ 𝐻) ≤ 𝑌)
48		nbrne2 5135	. . . 4 ⊢ ((𝐵 ≤ 𝑌 ∧ ¬ (𝐺 ∨ 𝐻) ≤ 𝑌) → 𝐵 ≠ (𝐺 ∨ 𝐻))
49	37, 47, 48	syl2anc 595	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ≠ (𝐺 ∨ 𝐻))
50	49	necomd 3019	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ≠ 𝐵)
51		hlatl 40058	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
52	3, 51	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ AtLat)
53	25, 18	llnbase 40207	. . . . 5 ⊢ (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
54	22, 53	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ∈ (Base‘𝐾))
55	25, 8	latmcl 18496	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ (Base‘𝐾))
56	24, 27, 54, 55	syl3anc 1396	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ (Base‘𝐾))
57	1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16	dalem52 40422	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
58	1, 5, 6	dalempjqeb 40343	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
59	58	3ad2ant1 1149	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
60	25, 4, 8	latmle1 18520	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝐺 ∨ 𝐻))
61	24, 27, 59, 60	syl3anc 1396	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝐺 ∨ 𝐻))
62	1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16	dalem51 40421	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝑐 ≤ (𝐺 ∨ 𝐻) ∧ ¬ 𝑐 ≤ (𝐻 ∨ 𝐼) ∧ ¬ 𝑐 ≤ (𝐼 ∨ 𝐺)) ∧ (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)) ∧ (𝑐 ≤ (𝐺 ∨ 𝑃) ∧ 𝑐 ≤ (𝐻 ∨ 𝑄) ∧ 𝑐 ≤ (𝐼 ∨ 𝑅)))) ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ≠ 𝑌))
63	62	simpld 499	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝑐 ≤ (𝐺 ∨ 𝐻) ∧ ¬ 𝑐 ≤ (𝐻 ∨ 𝐼) ∧ ¬ 𝑐 ≤ (𝐼 ∨ 𝐺)) ∧ (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)) ∧ (𝑐 ≤ (𝐺 ∨ 𝑃) ∧ 𝑐 ≤ (𝐻 ∨ 𝑄) ∧ 𝑐 ≤ (𝐼 ∨ 𝑅)))))
64	25, 6	atbase 39987	. . . . . . . 8 ⊢ (𝑐 ∈ 𝐴 → 𝑐 ∈ (Base‘𝐾))
65	64	anim2i 628	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴) → (𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)))
66	65	3anim1i 1168	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)))
67		biid 264	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝑐 ≤ (𝐺 ∨ 𝐻) ∧ ¬ 𝑐 ≤ (𝐻 ∨ 𝐼) ∧ ¬ 𝑐 ≤ (𝐼 ∨ 𝐺)) ∧ (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)) ∧ (𝑐 ≤ (𝐺 ∨ 𝑃) ∧ 𝑐 ≤ (𝐻 ∨ 𝑄) ∧ 𝑐 ≤ (𝐼 ∨ 𝑅)))) ↔ (((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝑐 ≤ (𝐺 ∨ 𝐻) ∧ ¬ 𝑐 ≤ (𝐻 ∨ 𝐼) ∧ ¬ 𝑐 ≤ (𝐼 ∨ 𝐺)) ∧ (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)) ∧ (𝑐 ≤ (𝐺 ∨ 𝑃) ∧ 𝑐 ≤ (𝐻 ∨ 𝑄) ∧ 𝑐 ≤ (𝐼 ∨ 𝑅)))))
68		eqid 2769	. . . . . . 7 ⊢ ((𝐺 ∨ 𝐻) ∨ 𝐼) = ((𝐺 ∨ 𝐻) ∨ 𝐼)
69		eqid 2769	. . . . . . 7 ⊢ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄))
70	67, 4, 5, 6, 8, 9, 68, 10, 21, 69	dalem10 40371	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝑐 ≤ (𝐺 ∨ 𝐻) ∧ ¬ 𝑐 ≤ (𝐻 ∨ 𝐼) ∧ ¬ 𝑐 ≤ (𝐼 ∨ 𝐺)) ∧ (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)) ∧ (𝑐 ≤ (𝐺 ∨ 𝑃) ∧ 𝑐 ≤ (𝐻 ∨ 𝑄) ∧ 𝑐 ≤ (𝐼 ∨ 𝑅)))) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ 𝐵)
71	66, 70	syl3an1 1179	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝑐 ≤ (𝐺 ∨ 𝐻) ∧ ¬ 𝑐 ≤ (𝐻 ∨ 𝐼) ∧ ¬ 𝑐 ≤ (𝐼 ∨ 𝐺)) ∧ (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)) ∧ (𝑐 ≤ (𝐺 ∨ 𝑃) ∧ 𝑐 ≤ (𝐻 ∨ 𝑄) ∧ 𝑐 ≤ (𝐼 ∨ 𝑅)))) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ 𝐵)
72	63, 71	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ 𝐵)
73	25, 8	latmcl 18496	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾))
74	24, 27, 59, 73	syl3anc 1396	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾))
75	25, 4, 8	latlem12 18522	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾) ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾))) → ((((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝐺 ∨ 𝐻) ∧ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ 𝐵) ↔ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ ((𝐺 ∨ 𝐻) ∧ 𝐵)))
76	24, 74, 27, 54, 75	syl13anc 1397	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝐺 ∨ 𝐻) ∧ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ 𝐵) ↔ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ ((𝐺 ∨ 𝐻) ∧ 𝐵)))
77	61, 72, 76	mpbi2and 724	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ ((𝐺 ∨ 𝐻) ∧ 𝐵))
78		eqid 2769	. . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾)
79	25, 4, 78, 6	atlen0 40008	. . 3 ⊢ (((𝐾 ∈ AtLat ∧ ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ (Base‘𝐾) ∧ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴) ∧ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ ((𝐺 ∨ 𝐻) ∧ 𝐵)) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ≠ (0.‘𝐾))
80	52, 56, 57, 77, 79	syl31anc 1398	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ≠ (0.‘𝐾))
81	8, 78, 6, 18	2llnmat 40222	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝐺 ∨ 𝐻) ∈ (LLines‘𝐾) ∧ 𝐵 ∈ (LLines‘𝐾)) ∧ ((𝐺 ∨ 𝐻) ≠ 𝐵 ∧ ((𝐺 ∨ 𝐻) ∧ 𝐵) ≠ (0.‘𝐾))) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ 𝐴)
82	3, 20, 22, 50, 80, 81	syl32anc 1403	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964   class class class wbr 5113  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  lecple 17317  joincjn 18367  meetcmee 18368  0.cp0 18477  Latclat 18487  Atomscatm 39961  AtLatcal 39962  HLchlt 40048  LLinesclln 40189  LPlanesclpl 40190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-proset 18350  df-poset 18369  df-plt 18384  df-lub 18400  df-glb 18401  df-join 18402  df-meet 18403  df-p0 18479  df-lat 18488  df-clat 18555  df-oposet 39874  df-ol 39876  df-oml 39877  df-covers 39964  df-ats 39965  df-atl 39996  df-cvlat 40020  df-hlat 40049  df-llines 40196  df-lplanes 40197  df-lvols 40198

This theorem is referenced by:  dalem55  40425  dalem57  40427