Theorem latlem12

Description: An element is less than or equal to a meet iff the element is less than or equal to each argument of the meet. (Contributed by NM, 21-Oct-2011)

	Ref	Expression
Hypotheses	latmle.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	latmle.l	⊢ ≤ = ( le ‘ 𝐾 )
	latmle.m	⊢ ∧ = ( meet ‘ 𝐾 )
Assertion	latlem12	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑍 ) ↔ 𝑋 ≤ ( 𝑌 ∧ 𝑍 ) ) )

Proof

Step	Hyp	Ref	Expression
1		latmle.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		latmle.l	⊢ ≤ = ( le ‘ 𝐾 )
3		latmle.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		latpos	⊢ ( 𝐾 ∈ Lat → 𝐾 ∈ Poset )
5	4	adantr	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐾 ∈ Poset )
6		simpr2	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
7		simpr3	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑍 ∈ 𝐵 )
8		simpr1	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
9		eqid	⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
10		simpl	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → 𝐾 ∈ Lat )
11	1 9 3 10 6 7	latcl2	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ⟨ 𝑌 , 𝑍 ⟩ ∈ dom ( join ‘ 𝐾 ) ∧ ⟨ 𝑌 , 𝑍 ⟩ ∈ dom ∧ ) )
12	11	simprd	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ⟨ 𝑌 , 𝑍 ⟩ ∈ dom ∧ )
13	1 2 3 5 6 7 8 12	meetle	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑍 ) ↔ 𝑋 ≤ ( 𝑌 ∧ 𝑍 ) ) )