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Metamath Proof Explorer

Theorem latm12

Description: A rearrangement of lattice meet. ( in12 analog.) (Contributed by NM, 8-Nov-2011)

		Ref	Expression
	Hypotheses	olmass.b	$⊢ B = {Base}_{K}$
		olmass.m	$⊢ \underset{˙}{\land} = meet (K)$
	Assertion	latm12	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to X \underset{˙}{\land} (Y \underset{˙}{\land} Z) = Y \underset{˙}{\land} (X \underset{˙}{\land} Z)$

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		simpr1	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to X \in B$
6		simpr2	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to Y \in B$
7	1 2	latmcom	$⊢ (K \in Lat \land X \in B \land Y \in B) \to X \underset{˙}{\land} Y = Y \underset{˙}{\land} X$
8	4 5 6 7	syl3anc	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to X \underset{˙}{\land} Y = Y \underset{˙}{\land} X$
9	8	oveq1d	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to (X \underset{˙}{\land} Y) \underset{˙}{\land} Z = (Y \underset{˙}{\land} X) \underset{˙}{\land} Z$
10	1 2	latmassOLD	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to (X \underset{˙}{\land} Y) \underset{˙}{\land} Z = X \underset{˙}{\land} (Y \underset{˙}{\land} Z)$
11		simpr3	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to Z \in B$
12	6 5 11	3jca	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to (Y \in B \land X \in B \land Z \in B)$
13	1 2	latmassOLD	$⊢ (K \in OL \land (Y \in B \land X \in B \land Z \in B)) \to (Y \underset{˙}{\land} X) \underset{˙}{\land} Z = Y \underset{˙}{\land} (X \underset{˙}{\land} Z)$
14	12 13	syldan	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to (Y \underset{˙}{\land} X) \underset{˙}{\land} Z = Y \underset{˙}{\land} (X \underset{˙}{\land} Z)$
15	9 10 14	3eqtr3d	$⊢ (K \in OL \land (X \in B \land Y \in B \land Z \in B)) \to X \underset{˙}{\land} (Y \underset{˙}{\land} Z) = Y \underset{˙}{\land} (X \underset{˙}{\land} Z)$