Metamath Proof Explorer

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	$⊢ \underset{˙}{\leq} = \leq_{K}$
		latlej.j	$⊢ \underset{˙}{\lor} = join (K)$
	Assertion	latjle12	$⊢ (K \in Lat \land (X \in B \land Y \in B \land Z \in B)) \to ((X \underset{˙}{\leq} Z \land Y \underset{˙}{\leq} Z) \leftrightarrow (X \underset{˙}{\lor} Y) \underset{˙}{\leq} Z)$

Proof

Step	Hyp	Ref	Expression
1		latlej.b	$⊢ B = {Base}_{K}$
2		latlej.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
3		latlej.j	$⊢ \underset{˙}{\lor} = join (K)$
4		latpos	$⊢ K \in Lat \to K \in Poset$
5	4	adantr	$⊢ (K \in Lat \land (X \in B \land Y \in B \land Z \in B)) \to K \in Poset$
6		simpr1	$⊢ (K \in Lat \land (X \in B \land Y \in B \land Z \in B)) \to X \in B$
7		simpr2	$⊢ (K \in Lat \land (X \in B \land Y \in B \land Z \in B)) \to Y \in B$
8		simpr3	$⊢ (K \in Lat \land (X \in B \land Y \in B \land Z \in B)) \to Z \in B$
9		eqid	$⊢ meet (K) = meet (K)$
10		simpl	$⊢ (K \in Lat \land (X \in B \land Y \in B \land Z \in B)) \to K \in Lat$
11	1 3 9 10 6 7	latcl2	$⊢ (K \in Lat \land (X \in B \land Y \in B \land Z \in B)) \to (⟨X, Y⟩ \in dom \underset{˙}{\lor} \land ⟨X, Y⟩ \in dom meet (K))$
12	11	simpld	$⊢ (K \in Lat \land (X \in B \land Y \in B \land Z \in B)) \to ⟨X, Y⟩ \in dom \underset{˙}{\lor}$
13	1 2 3 5 6 7 8 12	joinle	$⊢ (K \in Lat \land (X \in B \land Y \in B \land Z \in B)) \to ((X \underset{˙}{\leq} Z \land Y \underset{˙}{\leq} Z) \leftrightarrow (X \underset{˙}{\lor} Y) \underset{˙}{\leq} Z)$