Theorem dalemrotps

Description: Lemma for dath . Rotate triangles Y = P Q R and Z = S T U to allow reuse of analogous proofs. (Contributed by NM, 15-Aug-2012)

	Ref	Expression
Hypotheses	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 ) ) ) )
	dalemrotps.y	⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
Assertion	dalemrotps	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		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 ) ) ) ) )
2		dalem.l	⊢ ≤ = ( le ‘ 𝐾 )
3		dalem.j	⊢ ∨ = ( join ‘ 𝐾 )
4		dalem.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		dalem.ps	⊢ ( 𝜓 ↔ ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) )
6		dalemrotps.y	⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
7	5	dalemccea	⊢ ( 𝜓 → 𝑐 ∈ 𝐴 )
8	5	dalemddea	⊢ ( 𝜓 → 𝑑 ∈ 𝐴 )
9	7 8	jca	⊢ ( 𝜓 → ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) )
10	9	adantl	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) )
11	5	dalem-ccly	⊢ ( 𝜓 → ¬ 𝑐 ≤ 𝑌 )
12	11	adantl	⊢ ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝑐 ≤ 𝑌 )
13	1 3 4	dalemqrprot	⊢ ( 𝜑 → ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) )
14	6 13	eqtr4id	⊢ ( 𝜑 → 𝑌 = ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) )
15	14	breq2d	⊢ ( 𝜑 → ( 𝑐 ≤ 𝑌 ↔ 𝑐 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) ) )
16	15	adantr	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑐 ≤ 𝑌 ↔ 𝑐 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) ) )
17	12 16	mtbid	⊢ ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) )
18	5	dalemccnedd	⊢ ( 𝜓 → 𝑐 ≠ 𝑑 )
19	18	necomd	⊢ ( 𝜓 → 𝑑 ≠ 𝑐 )
20	19	adantl	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑑 ≠ 𝑐 )
21	5	dalem-ddly	⊢ ( 𝜓 → ¬ 𝑑 ≤ 𝑌 )
22	21	adantl	⊢ ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝑑 ≤ 𝑌 )
23	14	breq2d	⊢ ( 𝜑 → ( 𝑑 ≤ 𝑌 ↔ 𝑑 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) ) )
24	23	adantr	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑑 ≤ 𝑌 ↔ 𝑑 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) ) )
25	22 24	mtbid	⊢ ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝑑 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) )
26	5	dalemclccjdd	⊢ ( 𝜓 → 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) )
27	26	adantl	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) )
28	20 25 27	3jca	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) )
29	10 17 28	3jca	⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) )