Theorem latmle1

Description: A meet is less than or equal to its first argument. (Contributed by NM, 21-Oct-2011)

	Ref	Expression
Hypotheses	latmle.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	latmle.l	⊢ ≤ = ( le ‘ 𝐾 )
	latmle.m	⊢ ∧ = ( meet ‘ 𝐾 )
Assertion	latmle1	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		latmle.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		latmle.l	⊢ ≤ = ( le ‘ 𝐾 )
3		latmle.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		simp1	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ Lat )
5		simp2	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
6		simp3	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑌 ∈ 𝐵 )
7		eqid	⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
8	1 7 3 4 5 6	latcl2	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ( join ‘ 𝐾 ) ∧ ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ ) )
9	8	simprd	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ )
10	1 2 3 4 5 6 9	lemeet1	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 )