Theorem latleeqm2

Description: "Less than or equal to" in terms of meet. (Contributed by NM, 7-Nov-2011)

	Ref	Expression
Hypotheses	latmle.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	latmle.l	⊢ ≤ = ( le ‘ 𝐾 )
	latmle.m	⊢ ∧ = ( meet ‘ 𝐾 )
Assertion	latleeqm2	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑌 ∧ 𝑋 ) = 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		latmle.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		latmle.l	⊢ ≤ = ( le ‘ 𝐾 )
3		latmle.m	⊢ ∧ = ( meet ‘ 𝐾 )
4	1 2 3	latleeqm1	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑋 ∧ 𝑌 ) = 𝑋 ) )
5	1 3	latmcom	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) = ( 𝑌 ∧ 𝑋 ) )
6	5	eqeq1d	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑌 ) = 𝑋 ↔ ( 𝑌 ∧ 𝑋 ) = 𝑋 ) )
7	4 6	bitrd	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝑌 ∧ 𝑋 ) = 𝑋 ) )