Metamath Proof Explorer


Theorem dalemqrprot

Description: Lemma for dath . Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012)

Ref Expression
Hypotheses dalema.ph ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑆𝐴𝑇𝐴𝑈𝐴 ) ) ∧ ( 𝑌𝑂𝑍𝑂 ) ∧ ( ( ¬ 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 ) ) ) ) )
dalemb.j = ( join ‘ 𝐾 )
dalemb.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion dalemqrprot ( 𝜑 → ( ( 𝑄 𝑅 ) 𝑃 ) = ( ( 𝑃 𝑄 ) 𝑅 ) )

Proof

Step Hyp Ref Expression
1 dalema.ph ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ( 𝑆𝐴𝑇𝐴𝑈𝐴 ) ) ∧ ( 𝑌𝑂𝑍𝑂 ) ∧ ( ( ¬ 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 dalemb.j = ( join ‘ 𝐾 )
3 dalemb.a 𝐴 = ( Atoms ‘ 𝐾 )
4 1 dalemkehl ( 𝜑𝐾 ∈ HL )
5 1 dalemqea ( 𝜑𝑄𝐴 )
6 1 dalemrea ( 𝜑𝑅𝐴 )
7 1 dalempea ( 𝜑𝑃𝐴 )
8 2 3 hlatjrot ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴𝑅𝐴𝑃𝐴 ) ) → ( ( 𝑄 𝑅 ) 𝑃 ) = ( ( 𝑃 𝑄 ) 𝑅 ) )
9 4 5 6 7 8 syl13anc ( 𝜑 → ( ( 𝑄 𝑅 ) 𝑃 ) = ( ( 𝑃 𝑄 ) 𝑅 ) )