dalem54 - Mathbox for Norm Megill

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Theorem dalem54 38645

Description: Lemma for dath 38655. 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 38542	. . 3 ⊢ (𝜑 → 𝐾 ∈ HL)
3	2	3ad2ant1 1134	. 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 38615	. . 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 38620	. . 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 38632	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ≠ 𝐻)
18		eqid 2733	. . . 4 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
19	5, 6, 18	llni2 38431	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) ∧ 𝐺 ≠ 𝐻) → (𝐺 ∨ 𝐻) ∈ (LLines‘𝐾))
20	3, 13, 15, 17, 19	syl31anc 1374	. 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 38644	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ∈ (LLines‘𝐾))
23	1	dalemkelat 38543	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Lat)
24	23	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
25		eqid 2733	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
26	25, 18	llnbase 38428	. . . . . . . 8 ⊢ ((𝐺 ∨ 𝐻) ∈ (LLines‘𝐾) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
27	20, 26	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
28	1, 4, 5, 6, 7, 8, 9, 10, 11, 16	dalem34 38625	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴)
29	25, 6	atbase 38207	. . . . . . . 8 ⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾))
30	28, 29	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾))
31	25, 5	latjcl 18392	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾))
32	24, 27, 30, 31	syl3anc 1372	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾))
33	1, 9	dalemyeb 38568	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
34	33	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾))
35	25, 4, 8	latmle2 18418	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌) ≤ 𝑌)
36	24, 32, 34, 35	syl3anc 1372	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌) ≤ 𝑌)
37	21, 36	eqbrtrid 5184	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ≤ 𝑌)
38	1, 4, 5, 6, 7, 8, 9, 10, 11, 12	dalem24 38616	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝐺 ≤ 𝑌)
39	25, 6	atbase 38207	. . . . . . . 8 ⊢ (𝐺 ∈ 𝐴 → 𝐺 ∈ (Base‘𝐾))
40	13, 39	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ (Base‘𝐾))
41	25, 6	atbase 38207	. . . . . . . 8 ⊢ (𝐻 ∈ 𝐴 → 𝐻 ∈ (Base‘𝐾))
42	15, 41	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ (Base‘𝐾))
43	25, 4, 5	latjle12 18403	. . . . . . 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 1373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ≤ 𝑌 ∧ 𝐻 ≤ 𝑌) ↔ (𝐺 ∨ 𝐻) ≤ 𝑌))
45		simpl 484	. . . . . 6 ⊢ ((𝐺 ≤ 𝑌 ∧ 𝐻 ≤ 𝑌) → 𝐺 ≤ 𝑌)
46	44, 45	syl6bir 254	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ≤ 𝑌 → 𝐺 ≤ 𝑌))
47	38, 46	mtod 197	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ (𝐺 ∨ 𝐻) ≤ 𝑌)
48		nbrne2 5169	. . . 4 ⊢ ((𝐵 ≤ 𝑌 ∧ ¬ (𝐺 ∨ 𝐻) ≤ 𝑌) → 𝐵 ≠ (𝐺 ∨ 𝐻))
49	37, 47, 48	syl2anc 585	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ≠ (𝐺 ∨ 𝐻))
50	49	necomd 2997	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ≠ 𝐵)
51		hlatl 38278	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
52	3, 51	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ AtLat)
53	25, 18	llnbase 38428	. . . . 5 ⊢ (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
54	22, 53	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ∈ (Base‘𝐾))
55	25, 8	latmcl 18393	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐵 ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ (Base‘𝐾))
56	24, 27, 54, 55	syl3anc 1372	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ (Base‘𝐾))
57	1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16	dalem52 38643	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
58	1, 5, 6	dalempjqeb 38564	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
59	58	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
60	25, 4, 8	latmle1 18417	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝐺 ∨ 𝐻))
61	24, 27, 59, 60	syl3anc 1372	. . . 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 38642	. . . . . 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 496	. . . . 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 38207	. . . . . . . 8 ⊢ (𝑐 ∈ 𝐴 → 𝑐 ∈ (Base‘𝐾))
65	64	anim2i 618	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴) → (𝐾 ∈ HL ∧ 𝑐 ∈ (Base‘𝐾)))
66	65	3anim1i 1153	. . . . . 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 261	. . . . . . 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 2733	. . . . . . 7 ⊢ ((𝐺 ∨ 𝐻) ∨ 𝐼) = ((𝐺 ∨ 𝐻) ∨ 𝐼)
69		eqid 2733	. . . . . . 7 ⊢ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄))
70	67, 4, 5, 6, 8, 9, 68, 10, 21, 69	dalem10 38592	. . . . . 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 1164	. . . . 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 17	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ 𝐵)
73	25, 8	latmcl 18393	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾))
74	24, 27, 59, 73	syl3anc 1372	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾))
75	25, 4, 8	latlem12 18419	. . . . 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 1373	. . . 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 711	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ ((𝐺 ∨ 𝐻) ∧ 𝐵))
78		eqid 2733	. . . 4 ⊢ (0.‘𝐾) = (0.‘𝐾)
79	25, 4, 78, 6	atlen0 38228	. . 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 1374	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ≠ (0.‘𝐾))
81	8, 78, 6, 18	2llnmat 38443	. 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 1379	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ 𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 Basecbs 17144 lecple 17204 joincjn 18264 meetcmee 18265 0.cp0 18376 Latclat 18384 Atomscatm 38181 AtLatcal 38182 HLchlt 38268 LLinesclln 38410 LPlanesclpl 38411

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-proset 18248 df-poset 18266 df-plt 18283 df-lub 18299 df-glb 18300 df-join 18301 df-meet 18302 df-p0 18378 df-lat 18385 df-clat 18452 df-oposet 38094 df-ol 38096 df-oml 38097 df-covers 38184 df-ats 38185 df-atl 38216 df-cvlat 38240 df-hlat 38269 df-llines 38417 df-lplanes 38418 df-lvols 38419

This theorem is referenced by: dalem55 38646 dalem57 38648

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