Metamath Proof Explorer


Theorem hlatj12

Description: Swap 1st and 2nd members of lattice join. Frequently-used special case of latj32 for atoms. (Contributed by NM, 4-Jun-2012)

Ref Expression
Hypotheses hlatjcom.j = ( join ‘ 𝐾 )
hlatjcom.a 𝐴 = ( Atoms ‘ 𝐾 )
Assertion hlatj12 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ) → ( 𝑃 ( 𝑄 𝑅 ) ) = ( 𝑄 ( 𝑃 𝑅 ) ) )

Proof

Step Hyp Ref Expression
1 hlatjcom.j = ( join ‘ 𝐾 )
2 hlatjcom.a 𝐴 = ( Atoms ‘ 𝐾 )
3 1 2 hlatjcom ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
4 3 3adant3r3 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
5 4 oveq1d ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ) → ( ( 𝑃 𝑄 ) 𝑅 ) = ( ( 𝑄 𝑃 ) 𝑅 ) )
6 1 2 hlatjass ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ) → ( ( 𝑃 𝑄 ) 𝑅 ) = ( 𝑃 ( 𝑄 𝑅 ) ) )
7 simpl ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ) → 𝐾 ∈ HL )
8 simpr2 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ) → 𝑄𝐴 )
9 simpr1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ) → 𝑃𝐴 )
10 simpr3 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ) → 𝑅𝐴 )
11 1 2 hlatjass ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴𝑃𝐴𝑅𝐴 ) ) → ( ( 𝑄 𝑃 ) 𝑅 ) = ( 𝑄 ( 𝑃 𝑅 ) ) )
12 7 8 9 10 11 syl13anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ) → ( ( 𝑄 𝑃 ) 𝑅 ) = ( 𝑄 ( 𝑃 𝑅 ) ) )
13 5 6 12 3eqtr3d ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ) → ( 𝑃 ( 𝑄 𝑅 ) ) = ( 𝑄 ( 𝑃 𝑅 ) ) )