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Metamath Proof Explorer

Theorem latmmdir

Description: Lattice meet distributes over itself. ( inindir analog.) (Contributed by NM, 6-Jun-2012)

		Ref	Expression
	Hypotheses	olmass.b	$⊢ B = {Base}_{K}$
		olmass.m	$⊢ \underset{˙}{\land} = meet (K)$
	Assertion	latmmdir	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to (X \underset{˙}{\land} Y) \underset{˙}{\land} Z = (X \underset{˙}{\land} Z) \underset{˙}{\land} (Y \underset{˙}{\land} Z)$

Proof

Step	Hyp	Ref	Expression
1		olmass.b	$⊢ B = {Base}_{K}$
2		olmass.m	$⊢ \underset{˙}{\land} = meet (K)$
3		ollat	$⊢ K \in OL \to K \in Lat$
4	3	adantr	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to K \in Lat$
5		simpr3	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to Z \in B$
6	1 2	latmidm	$⊢ (K \in Lat \land Z \in B) \to Z \underset{˙}{\land} Z = Z$
7	4 5 6	syl2anc	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to Z \underset{˙}{\land} Z = Z$
8	7	oveq2d	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to (X \underset{˙}{\land} Y) \underset{˙}{\land} (Z \underset{˙}{\land} Z) = (X \underset{˙}{\land} Y) \underset{˙}{\land} Z$
9		simpl	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to K \in OL$
10		simpr1	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to X \in B$
11		simpr2	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to Y \in B$
12	1 2	latm4	$⊢ (K \in OL \land (X \in B \land Y \in B) \land (Z \in B \land Z \in B)) \to (X \underset{˙}{\land} Y) \underset{˙}{\land} (Z \underset{˙}{\land} Z) = (X \underset{˙}{\land} Z) \underset{˙}{\land} (Y \underset{˙}{\land} Z)$
13	9 10 11 5 5 12	syl122anc	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to (X \underset{˙}{\land} Y) \underset{˙}{\land} (Z \underset{˙}{\land} Z) = (X \underset{˙}{\land} Z) \underset{˙}{\land} (Y \underset{˙}{\land} Z)$
14	8 13	eqtr3d	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to (X \underset{˙}{\land} Y) \underset{˙}{\land} Z = (X \underset{˙}{\land} Z) \underset{˙}{\land} (Y \underset{˙}{\land} Z)$