Theorem joincom

Description: The join of a poset commutes. (The antecedent <. X , Y >. e. dom .\/ /\ <. Y , X >. e. dom .\/ i.e. "the joins exist" could be omitted as an artifact of our particular join definition, but other definitions may require it.) (Contributed by NM, 16-Sep-2011) (Revised by NM, 12-Sep-2018)

	Ref	Expression
Hypotheses	joincom.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	joincom.j	⊢ ∨ = ( join ‘ 𝐾 )
Assertion	joincom	⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ ∧ ⟨ 𝑌 , 𝑋 ⟩ ∈ dom ∨ ) ) → ( 𝑋 ∨ 𝑌 ) = ( 𝑌 ∨ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		joincom.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		joincom.j	⊢ ∨ = ( join ‘ 𝐾 )
3	1 2	joincomALT	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑌 ) = ( 𝑌 ∨ 𝑋 ) )
4	3	adantr	⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ ∧ ⟨ 𝑌 , 𝑋 ⟩ ∈ dom ∨ ) ) → ( 𝑋 ∨ 𝑌 ) = ( 𝑌 ∨ 𝑋 ) )