Theorem latleeqj2

Description: "Less than or equal to" in terms of join. ( chlejb2 analog.) (Contributed by NM, 14-Nov-2011)

	Ref	Expression
Hypotheses	latlej.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	latlej.l	⊢ ≤ = ( le ‘ 𝐾 )
	latlej.j	⊢ ∨ = ( join ‘ 𝐾 )
Assertion	latleeqj2	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑌 ∨ 𝑋 ) = 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		latlej.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		latlej.l	⊢ ≤ = ( le ‘ 𝐾 )
3		latlej.j	⊢ ∨ = ( join ‘ 𝐾 )
4	1 2 3	latleeqj1	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 ∨ 𝑌 ) = 𝑌 ) )
5	1 3	latjcom	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑌 ) = ( 𝑌 ∨ 𝑋 ) )
6	5	eqeq1d	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ∨ 𝑌 ) = 𝑌 ↔ ( 𝑌 ∨ 𝑋 ) = 𝑌 ) )
7	4 6	bitrd	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑌 ∨ 𝑋 ) = 𝑌 ) )