Theorem dalem55 39926

Description: Lemma for dath 39935. Lines 𝐺𝐻 and 𝑃𝑄 intersect at the auxiliary line 𝐵 (later shown to be an axis of perspectivity; see dalem60 39931). (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 39823	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ Lat)
3	2	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
4	1	dalemkehl 39822	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL)
5	4	3ad2ant1 1133	. . . . 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 39895	. . . . 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 39900	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐻 ∈ 𝐴)
18		eqid 2734	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
19	18, 7, 8	hlatjcl 39566	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
20	5, 15, 17, 19	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ∈ (Base‘𝐾))
21	1, 7, 8	dalempjqeb 39844	. . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
22	21	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
23	18, 6, 10	latmle1 18385	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ (𝐺 ∨ 𝐻))
24	3, 20, 22, 23	syl3anc 1373	. . 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 39905	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ 𝐴)
27	18, 8	atbase 39488	. . . . . . 7 ⊢ (𝐼 ∈ 𝐴 → 𝐼 ∈ (Base‘𝐾))
28	26, 27	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐼 ∈ (Base‘𝐾))
29	18, 6, 7	latlej1 18369	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ 𝐼 ∈ (Base‘𝐾)) → (𝐺 ∨ 𝐻) ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼))
30	3, 20, 28, 29	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐺 ∨ 𝐻) ≤ ((𝐺 ∨ 𝐻) ∨ 𝐼))
31	1, 8	dalemreb 39840	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ (Base‘𝐾))
32	18, 6, 7	latlej1 18369	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
33	2, 21, 31, 32	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
34	33, 12	breqtrrdi 5138	. . . . . 6 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ≤ 𝑌)
35	34	3ad2ant1 1133	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑃 ∨ 𝑄) ≤ 𝑌)
36	1, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 25	dalem42 39913	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂)
37	18, 11	lplnbase 39733	. . . . . . 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 39848	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
40	39	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾))
41	18, 6, 10	latmlem12 18392	. . . . . 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 1381	. . . . 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 5138	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ 𝐵)
46	18, 10	latmcl 18361	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝐺 ∨ 𝐻) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾))
47	3, 20, 22, 46	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ (Base‘𝐾))
48		eqid 2734	. . . . . 6 ⊢ (LLines‘𝐾) = (LLines‘𝐾)
49	1, 6, 7, 8, 9, 10, 48, 11, 12, 13, 14, 16, 25, 44	dalem53 39924	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ∈ (LLines‘𝐾))
50	18, 48	llnbase 39708	. . . . 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 18387	. . . 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 1374	. . 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 39559	. . . 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 39923	. . 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 39925	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ 𝐴)
59	6, 8	atcmp 39510	. . 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 1373	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ≤ ((𝐺 ∨ 𝐻) ∧ 𝐵) ↔ ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐵)))
61	54, 60	mpbid 232	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ≠ wne 2930   class class class wbr 5096  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  lecple 17182  joincjn 18232  meetcmee 18233  Latclat 18352  Atomscatm 39462  AtLatcal 39463  HLchlt 39549  LLinesclln 39690  LPlanesclpl 39691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  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 7313  df-ov 7359  df-oprab 7360  df-proset 18215  df-poset 18234  df-plt 18249  df-lub 18265  df-glb 18266  df-join 18267  df-meet 18268  df-p0 18344  df-lat 18353  df-clat 18420  df-oposet 39375  df-ol 39377  df-oml 39378  df-covers 39465  df-ats 39466  df-atl 39497  df-cvlat 39521  df-hlat 39550  df-llines 39697  df-lplanes 39698  df-lvols 39699

This theorem is referenced by:  dalem56  39927  dalem57  39928