Theorem hlatjass

Description: Lattice join is associative. Frequently-used special case of latjass for atoms. (Contributed by NM, 27-Jul-2012)

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

Proof

Step	Hyp	Ref	Expression
1		hlatjcom.j	⊢ ∨ = ( join ‘ 𝐾 )
2		hlatjcom.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		hllat	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
4	3	adantr	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝐾 ∈ Lat )
5		simpr1	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑃 ∈ 𝐴 )
6		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
7	6 2	atbase	⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
8	5 7	syl	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
9		simpr2	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑄 ∈ 𝐴 )
10	6 2	atbase	⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
11	9 10	syl	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
12		simpr3	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑅 ∈ 𝐴 )
13	6 2	atbase	⊢ ( 𝑅 ∈ 𝐴 → 𝑅 ∈ ( Base ‘ 𝐾 ) )
14	12 13	syl	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑅 ∈ ( Base ‘ 𝐾 ) )
15	6 1	latjass	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) = ( 𝑃 ∨ ( 𝑄 ∨ 𝑅 ) ) )
16	4 8 11 14 15	syl13anc	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) = ( 𝑃 ∨ ( 𝑄 ∨ 𝑅 ) ) )