Theorem lattr

Description: A lattice ordering is transitive. ( sstr analog.) (Contributed by NM, 17-Nov-2011)

	Ref	Expression
Hypotheses	latref.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	latref.l	⊢ ≤ = ( le ‘ 𝐾 )
Assertion	lattr	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍 ) → 𝑋 ≤ 𝑍 ) )

Proof

Step	Hyp	Ref	Expression
1		latref.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		latref.l	⊢ ≤ = ( le ‘ 𝐾 )
3		latpos	⊢ ( 𝐾 ∈ Lat → 𝐾 ∈ Poset )
4	1 2	postr	⊢ ( ( 𝐾 ∈ Poset ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍 ) → 𝑋 ≤ 𝑍 ) )
5	3 4	sylan	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍 ) → 𝑋 ≤ 𝑍 ) )