Theorem dalem55 37364

Description: Lemma for dath 37373. Lines 𝐺𝐻 and 𝑃𝑄 intersect at the auxiliary line 𝐵 (later shown to be an axis of perspectivity; see dalem60 37369). (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
dalem55	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐵))

Proof of Theorem dalem55

Step	Hyp	Ref	Expression
1		dalem.ph	. . . . . 6 ⊢ (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	dalemkelat 37261	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat)
3	2	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
4	1	dalemkehl 37260	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL)
5	4	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
6		dalem.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
7		dalem.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
8		dalem.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
9		dalem.ps	. . . . . 6 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
10		dalem54.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
11		dalem54.o	. . . . . 6 ⊢ 𝑂 = (LPlanes‘𝐾)
12		dalem54.y	. . . . . 6 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
13		dalem54.z	. . . . . 6 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
14		dalem54.g	. . . . . 6 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
15	1, 6, 7, 8, 9, 10, 11, 12, 13, 14	dalem23 37333	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ∈ 𝐴)
16		dalem54.h	. . . . . 6 ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))
17	1, 6, 7, 8, 9, 10, 11, 12, 13, 16	dalem29 37338	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)
18		eqid 2738	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
19	18, 7, 8	hlatjcl 37004	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
20	5, 15, 17, 19	syl3anc 1372	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
21	1, 7, 8	dalempjqeb 37282	. . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
22	21	3ad2ant1 1134	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
23	18, 6, 10	latmle1 17802	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝐺 ∨ 𝐻))
24	3, 20, 22, 23	syl3anc 1372	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝐺 ∨ 𝐻))
25		dalem54.i	. . . . . . . 8 ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))
26	1, 6, 7, 8, 9, 10, 11, 12, 13, 25	dalem34 37343	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴)
27	18, 8	atbase 36926	. . . . . . 7 ⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾))
28	26, 27	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾))
29	18, 6, 7	latlej1 17786	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → (𝐺 ∨ 𝐻) ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼))
30	3, 20, 28, 29	syl3anc 1372	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼))
31	1, 8	dalemreb 37278	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
32	18, 6, 7	latlej1 17786	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
33	2, 21, 31, 32	syl3anc 1372	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
34	33, 12	breqtrrdi 5072	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ≤ 𝑌)
35	34	3ad2ant1 1134	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ 𝑌)
36	1, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25	dalem42 37351	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂)
37	18, 11	lplnbase 37171	. . . . . . 7 ⊢ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾))
38	36, 37	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾))
39	1, 11	dalemyeb 37286	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
40	39	3ad2ant1 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾))
41	18, 6, 10	latmlem12 17809	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ ((𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ (Base‘𝐾)) ∧ ((𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → (((𝐺 ∨ 𝐻) ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ (𝑃 ∨ 𝑄) ≤ 𝑌) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)))
42	3, 20, 38, 22, 40, 41	syl122anc 1380	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ (𝑃 ∨ 𝑄) ≤ 𝑌) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)))
43	30, 35, 42	mp2and 699	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌))
44		dalem54.b1	. . . 4 ⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)
45	43, 44	breqtrrdi 5072	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ 𝐵)
46	18, 10	latmcl 17778	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾))
47	3, 20, 22, 46	syl3anc 1372	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾))
48		eqid 2738	. . . . . 6 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
49	1, 6, 7, 8, 9, 10, 48, 11, 12, 13, 14, 16, 25, 44	dalem53 37362	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ∈ (LLines‘𝐾))
50	18, 48	llnbase 37146	. . . . 5 ⊢ (𝐵 ∈ (LLines‘𝐾) → 𝐵 ∈ (Base‘𝐾))
51	49, 50	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ∈ (Base‘𝐾))
52	18, 6, 10	latlem12 17804	. . . 4 ⊢ ((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)))
53	3, 47, 20, 51, 52	syl13anc 1373	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝐺 ∨ 𝐻) ∧ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ 𝐵) ↔ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ ((𝐺 ∨ 𝐻) ∧ 𝐵)))
54	24, 45, 53	mpbi2and 712	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ ((𝐺 ∨ 𝐻) ∧ 𝐵))
55		hlatl 36997	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
56	5, 55	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ AtLat)
57	1, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25	dalem52 37361	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
58	1, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25, 44	dalem54 37363	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ 𝐴)
59	6, 8	atcmp 36948	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴 ∧ ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ 𝐴) → (((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ ((𝐺 ∨ 𝐻) ∧ 𝐵) ↔ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐵)))
60	56, 57, 58, 59	syl3anc 1372	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ ((𝐺 ∨ 𝐻) ∧ 𝐵) ↔ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐵)))
61	54, 60	mpbid 235	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114   ≠ wne 2934   class class class wbr 5030  ‘cfv 6339  (class class class)co 7170  Basecbs 16586  lecple 16675  joincjn 17670  meetcmee 17671  Latclat 17771  Atomscatm 36900  AtLatcal 36901  HLchlt 36987  LLinesclln 37128  LPlanesclpl 37129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-proset 17654  df-poset 17672  df-plt 17684  df-lub 17700  df-glb 17701  df-join 17702  df-meet 17703  df-p0 17765  df-lat 17772  df-clat 17834  df-oposet 36813  df-ol 36815  df-oml 36816  df-covers 36903  df-ats 36904  df-atl 36935  df-cvlat 36959  df-hlat 36988  df-llines 37135  df-lplanes 37136  df-lvols 37137

This theorem is referenced by:  dalem56  37365  dalem57  37366