Theorem hlatjcom

Description: Commutatitivity of join operation. Frequently-used special case of latjcom for atoms. (Contributed by NM, 15-Jun-2012)

	Ref	Expression
Hypotheses	hlatjcom.j	⊢ ∨ = ( join ‘ 𝐾 )
	hlatjcom.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	hlatjcom	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 ∨ 𝑌 ) = ( 𝑌 ∨ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		hlatjcom.j	⊢ ∨ = ( join ‘ 𝐾 )
2		hlatjcom.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		hllat	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
4		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
5	4 2	atbase	⊢ ( 𝑋 ∈ 𝐴 → 𝑋 ∈ ( Base ‘ 𝐾 ) )
6	4 2	atbase	⊢ ( 𝑌 ∈ 𝐴 → 𝑌 ∈ ( Base ‘ 𝐾 ) )
7	4 1	latjcom	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ ( Base ‘ 𝐾 ) ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑋 ∨ 𝑌 ) = ( 𝑌 ∨ 𝑋 ) )
8	3 5 6 7	syl3an	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 ∨ 𝑌 ) = ( 𝑌 ∨ 𝑋 ) )