Theorem dalemrot 40062

Description: Lemma for dath 40141. Rotate triangles 𝑌 = 𝑃𝑄𝑅 and 𝑍 = 𝑆𝑇𝑈 to allow reuse of analogous proofs. (Contributed by NM, 14-Aug-2012.)

Hypotheses

Ref	Expression
dalema.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)))))
dalemc.l	⊢ ≤ = (le‘𝐾)
dalemc.j	⊢ ∨ = (join‘𝐾)
dalemc.a	⊢ 𝐴 = (Atoms‘𝐾)
dalemrot.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalemrot.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)

Assertion

Ref	Expression
dalemrot	⊢ (𝜑 → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ (𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ (((𝑄 ∨ 𝑅) ∨ 𝑃) ∈ 𝑂 ∧ ((𝑇 ∨ 𝑈) ∨ 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃) ∧ ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆) ∧ ¬ 𝐶 ≤ (𝑆 ∨ 𝑇)) ∧ (𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆)))))

Proof of Theorem dalemrot

Step	Hyp	Ref	Expression
1		dalema.ph	. . . . 5 ⊢ (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 40028	. . . 4 ⊢ (𝜑 → 𝐾 ∈ HL)
3		dalemc.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
4	1, 3	dalemceb 40043	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
5	2, 4	jca 511	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)))
6	1	dalemqea 40032	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
7	1	dalemrea 40033	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
8	1	dalempea 40031	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
9	6, 7, 8	3jca 1129	. . 3 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴))
10	1	dalemtea 40035	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
11	1	dalemuea 40036	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
12	1	dalemsea 40034	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
13	10, 11, 12	3jca 1129	. . 3 ⊢ (𝜑 → (𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴))
14	5, 9, 13	3jca 1129	. 2 ⊢ (𝜑 → ((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ (𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)))
15		dalemc.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
16	1, 15, 3	dalemqrprot 40053	. . . 4 ⊢ (𝜑 → ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑃 ∨ 𝑄) ∨ 𝑅))
17		dalemrot.y	. . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
18	1	dalemyeo 40037	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
19	17, 18	eqeltrrid 2842	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ 𝑂)
20	16, 19	eqeltrd 2837	. . 3 ⊢ (𝜑 → ((𝑄 ∨ 𝑅) ∨ 𝑃) ∈ 𝑂)
21	15, 3	hlatjrot 39778	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑇 ∨ 𝑈) ∨ 𝑆) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
22	2, 10, 11, 12, 21	syl13anc 1375	. . . 4 ⊢ (𝜑 → ((𝑇 ∨ 𝑈) ∨ 𝑆) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
23		dalemrot.z	. . . . 5 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
24	1	dalemzeo 40038	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑂)
25	23, 24	eqeltrrid 2842	. . . 4 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ 𝑂)
26	22, 25	eqeltrd 2837	. . 3 ⊢ (𝜑 → ((𝑇 ∨ 𝑈) ∨ 𝑆) ∈ 𝑂)
27	20, 26	jca 511	. 2 ⊢ (𝜑 → (((𝑄 ∨ 𝑅) ∨ 𝑃) ∈ 𝑂 ∧ ((𝑇 ∨ 𝑈) ∨ 𝑆) ∈ 𝑂))
28		simp312 1323	. . . . 5 ⊢ ((((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)))) → ¬ 𝐶 ≤ (𝑄 ∨ 𝑅))
29	1, 28	sylbi 217	. . . 4 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑄 ∨ 𝑅))
30		simp313 1324	. . . . 5 ⊢ ((((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)))) → ¬ 𝐶 ≤ (𝑅 ∨ 𝑃))
31	1, 30	sylbi 217	. . . 4 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑅 ∨ 𝑃))
32	1	dalem-clpjq 40042	. . . 4 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
33	29, 31, 32	3jca 1129	. . 3 ⊢ (𝜑 → (¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃) ∧ ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)))
34		simp322 1326	. . . . 5 ⊢ ((((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)))) → ¬ 𝐶 ≤ (𝑇 ∨ 𝑈))
35	1, 34	sylbi 217	. . . 4 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑇 ∨ 𝑈))
36		simp323 1327	. . . . 5 ⊢ ((((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)))) → ¬ 𝐶 ≤ (𝑈 ∨ 𝑆))
37	1, 36	sylbi 217	. . . 4 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑈 ∨ 𝑆))
38		simp321 1325	. . . . 5 ⊢ ((((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)))) → ¬ 𝐶 ≤ (𝑆 ∨ 𝑇))
39	1, 38	sylbi 217	. . . 4 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑆 ∨ 𝑇))
40	35, 37, 39	3jca 1129	. . 3 ⊢ (𝜑 → (¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆) ∧ ¬ 𝐶 ≤ (𝑆 ∨ 𝑇)))
41	1	dalemclqjt 40040	. . . 4 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
42	1	dalemclrju 40041	. . . 4 ⊢ (𝜑 → 𝐶 ≤ (𝑅 ∨ 𝑈))
43	1	dalemclpjs 40039	. . . 4 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
44	41, 42, 43	3jca 1129	. . 3 ⊢ (𝜑 → (𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆)))
45	33, 40, 44	3jca 1129	. 2 ⊢ (𝜑 → ((¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃) ∧ ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆) ∧ ¬ 𝐶 ≤ (𝑆 ∨ 𝑇)) ∧ (𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆))))
46	14, 27, 45	3jca 1129	1 ⊢ (𝜑 → (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ (𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ (((𝑄 ∨ 𝑅) ∨ 𝑃) ∈ 𝑂 ∧ ((𝑇 ∨ 𝑈) ∨ 𝑆) ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃) ∧ ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆) ∧ ¬ 𝐶 ≤ (𝑆 ∨ 𝑇)) ∧ (𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5100  ‘cfv 6502  (class class class)co 7370  Basecbs 17150  lecple 17198  joincjn 18248  Atomscatm 39668  HLchlt 39755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-riota 7327  df-ov 7373  df-oprab 7374  df-proset 18231  df-poset 18250  df-lub 18281  df-glb 18282  df-join 18283  df-meet 18284  df-lat 18369  df-ats 39672  df-atl 39703  df-cvlat 39727  df-hlat 39756

This theorem is referenced by:  dalemeea  40068  dalem6  40073  dalem7  40074  dalem11  40079  dalem12  40080  dalem29  40106  dalem30  40107  dalem31N  40108  dalem32  40109  dalem33  40110  dalem34  40111  dalem35  40112  dalem36  40113  dalem37  40114  dalem40  40117  dalem46  40123  dalem47  40124  dalem49  40126  dalem50  40127  dalem58  40135  dalem59  40136