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Metamath Proof Explorer

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 ) )