Theorem latjle12

Description: A join is less than or equal to a third value iff each argument is less than or equal to the third value. ( chlub analog.) (Contributed by NM, 17-Sep-2011)

	Ref	Expression
Hypotheses	latlej.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	latlej.l	⊢ ≤ = ( le ‘ 𝐾 )
	latlej.j	⊢ ∨ = ( join ‘ 𝐾 )
Assertion	latjle12	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ) ) → ( ( 𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍 ) ↔ ( 𝑋 ∨ 𝑌 ) ≤ 𝑍 ) )

Proof

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