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 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( 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 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑃 ) )
4	3	3adant3r3	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑃 ) )
5	4	oveq1d	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) = ( ( 𝑄 ∨ 𝑃 ) ∨ 𝑅 ) )
6	1 2	hlatjass	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) = ( 𝑃 ∨ ( 𝑄 ∨ 𝑅 ) ) )
7		simpl	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝐾 ∈ HL )
8		simpr2	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑄 ∈ 𝐴 )
9		simpr1	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑃 ∈ 𝐴 )
10		simpr3	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑅 ∈ 𝐴 )
11	1 2	hlatjass	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑄 ∨ 𝑃 ) ∨ 𝑅 ) = ( 𝑄 ∨ ( 𝑃 ∨ 𝑅 ) ) )
12	7 8 9 10 11	syl13anc	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑄 ∨ 𝑃 ) ∨ 𝑅 ) = ( 𝑄 ∨ ( 𝑃 ∨ 𝑅 ) ) )
13	5 6 12	3eqtr3d	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( 𝑃 ∨ ( 𝑄 ∨ 𝑅 ) ) = ( 𝑄 ∨ ( 𝑃 ∨ 𝑅 ) ) )