Theorem dalemrotyz

Description: Lemma for dath . Rotate triangles Y = P Q R and Z = S T U to allow reuse of analogous proofs. (Contributed by NM, 19-Aug-2012)

	Ref	Expression
Hypotheses	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	dalemrotyz	⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) = ( ( 𝑇 ∨ 𝑈 ) ∨ 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		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 ) ) ) ) )
2		dalemc.l	⊢ ≤ = ( le ‘ 𝐾 )
3		dalemc.j	⊢ ∨ = ( join ‘ 𝐾 )
4		dalemc.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		dalemrot.y	⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
6		dalemrot.z	⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
7		simpr	⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝑌 = 𝑍 )
8	1 3 4	dalemqrprot	⊢ ( 𝜑 → ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) )
9	5 8	eqtr4id	⊢ ( 𝜑 → 𝑌 = ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) )
10	9	adantr	⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝑌 = ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) )
11	1	dalemkehl	⊢ ( 𝜑 → 𝐾 ∈ HL )
12	1	dalemtea	⊢ ( 𝜑 → 𝑇 ∈ 𝐴 )
13	1	dalemuea	⊢ ( 𝜑 → 𝑈 ∈ 𝐴 )
14	1	dalemsea	⊢ ( 𝜑 → 𝑆 ∈ 𝐴 )
15	3 4	hlatjrot	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( ( 𝑇 ∨ 𝑈 ) ∨ 𝑆 ) = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) )
16	11 12 13 14 15	syl13anc	⊢ ( 𝜑 → ( ( 𝑇 ∨ 𝑈 ) ∨ 𝑆 ) = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) )
17	6 16	eqtr4id	⊢ ( 𝜑 → 𝑍 = ( ( 𝑇 ∨ 𝑈 ) ∨ 𝑆 ) )
18	17	adantr	⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝑍 = ( ( 𝑇 ∨ 𝑈 ) ∨ 𝑆 ) )
19	7 10 18	3eqtr3d	⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑃 ) = ( ( 𝑇 ∨ 𝑈 ) ∨ 𝑆 ) )