Theorem dalemrot 39917

Description: Lemma for dath 39996. 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 39883	. . . 4 ⊢ (𝜑 → 𝐾 ∈ HL)
3		dalemc.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
4	1, 3	dalemceb 39898	. . . 4 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
5	2, 4	jca 511	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)))
6	1	dalemqea 39887	. . . 4 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
7	1	dalemrea 39888	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
8	1	dalempea 39886	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
9	6, 7, 8	3jca 1128	. . 3 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴))
10	1	dalemtea 39890	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
11	1	dalemuea 39891	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝐴)
12	1	dalemsea 39889	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
13	10, 11, 12	3jca 1128	. . 3 ⊢ (𝜑 → (𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴))
14	5, 9, 13	3jca 1128	. 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 39908	. . . 4 ⊢ (𝜑 → ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑃 ∨ 𝑄) ∨ 𝑅))
17		dalemrot.y	. . . . 5 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
18	1	dalemyeo 39892	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
19	17, 18	eqeltrrid 2841	. . . 4 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ 𝑂)
20	16, 19	eqeltrd 2836	. . 3 ⊢ (𝜑 → ((𝑄 ∨ 𝑅) ∨ 𝑃) ∈ 𝑂)
21	15, 3	hlatjrot 39633	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑇 ∨ 𝑈) ∨ 𝑆) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
22	2, 10, 11, 12, 21	syl13anc 1374	. . . 4 ⊢ (𝜑 → ((𝑇 ∨ 𝑈) ∨ 𝑆) = ((𝑆 ∨ 𝑇) ∨ 𝑈))
23		dalemrot.z	. . . . 5 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
24	1	dalemzeo 39893	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑂)
25	23, 24	eqeltrrid 2841	. . . 4 ⊢ (𝜑 → ((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ 𝑂)
26	22, 25	eqeltrd 2836	. . 3 ⊢ (𝜑 → ((𝑇 ∨ 𝑈) ∨ 𝑆) ∈ 𝑂)
27	20, 26	jca 511	. 2 ⊢ (𝜑 → (((𝑄 ∨ 𝑅) ∨ 𝑃) ∈ 𝑂 ∧ ((𝑇 ∨ 𝑈) ∨ 𝑆) ∈ 𝑂))
28		simp312 1322	. . . . 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 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 ≤ (𝑅 ∨ 𝑃))
31	1, 30	sylbi 217	. . . 4 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑅 ∨ 𝑃))
32	1	dalem-clpjq 39897	. . . 4 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
33	29, 31, 32	3jca 1128	. . 3 ⊢ (𝜑 → (¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃) ∧ ¬ 𝐶 ≤ (𝑃 ∨ 𝑄)))
34		simp322 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 ≤ (𝑇 ∨ 𝑈))
35	1, 34	sylbi 217	. . . 4 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑇 ∨ 𝑈))
36		simp323 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 ≤ (𝑈 ∨ 𝑆))
37	1, 36	sylbi 217	. . . 4 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑈 ∨ 𝑆))
38		simp321 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 ≤ (𝑆 ∨ 𝑇))
39	1, 38	sylbi 217	. . . 4 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑆 ∨ 𝑇))
40	35, 37, 39	3jca 1128	. . 3 ⊢ (𝜑 → (¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆) ∧ ¬ 𝐶 ≤ (𝑆 ∨ 𝑇)))
41	1	dalemclqjt 39895	. . . 4 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
42	1	dalemclrju 39896	. . . 4 ⊢ (𝜑 → 𝐶 ≤ (𝑅 ∨ 𝑈))
43	1	dalemclpjs 39894	. . . 4 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
44	41, 42, 43	3jca 1128	. . 3 ⊢ (𝜑 → (𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈) ∧ 𝐶 ≤ (𝑃 ∨ 𝑆)))
45	33, 40, 44	3jca 1128	. 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 1128	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 1086   = wceq 1541   ∈ wcel 2113   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  lecple 17184  joincjn 18234  Atomscatm 39523  HLchlt 39610

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-proset 18217  df-poset 18236  df-lub 18267  df-glb 18268  df-join 18269  df-meet 18270  df-lat 18355  df-ats 39527  df-atl 39558  df-cvlat 39582  df-hlat 39611

This theorem is referenced by:  dalemeea  39923  dalem6  39928  dalem7  39929  dalem11  39934  dalem12  39935  dalem29  39961  dalem30  39962  dalem31N  39963  dalem32  39964  dalem33  39965  dalem34  39966  dalem35  39967  dalem36  39968  dalem37  39969  dalem40  39972  dalem46  39978  dalem47  39979  dalem49  39981  dalem50  39982  dalem58  39990  dalem59  39991