ltrnm

Description: Lattice translation of a meet. (Contributed by NM, 20-May-2012)

	Ref	Expression
Hypotheses	ltrnm.b	$⊢ B = {Base}_{K}$
	ltrnm.m	$⊢ \underset{˙}{\land} = meet (K)$
	ltrnm.h	$⊢ H = LHyp (K)$
	ltrnm.t	$⊢ T = LTrn (K) (W)$
Assertion	ltrnm	$⊢ ((K \in HL \land W \in H) \land F \in T \land (X \in B \land Y \in B)) \to F (X \underset{˙}{\land} Y) = F (X) \underset{˙}{\land} F (Y)$

Step	Hyp	Ref	Expression
1		ltrnm.b	$⊢ B = {Base}_{K}$
2		ltrnm.m	$⊢ \underset{˙}{\land} = meet (K)$
3		ltrnm.h	$⊢ H = LHyp (K)$
4		ltrnm.t	$⊢ T = LTrn (K) (W)$
5		simp1l	$⊢ ((K \in HL \land W \in H) \land F \in T \land (X \in B \land Y \in B)) \to K \in HL$
6	5	hllatd	$⊢ ((K \in HL \land W \in H) \land F \in T \land (X \in B \land Y \in B)) \to K \in Lat$
7		eqid	$⊢ LAut (K) = LAut (K)$
8	3 7 4	ltrnlaut	$⊢ ((K \in HL \land W \in H) \land F \in T) \to F \in LAut (K)$
9	8	3adant3	$⊢ ((K \in HL \land W \in H) \land F \in T \land (X \in B \land Y \in B)) \to F \in LAut (K)$
10		simp3l	$⊢ ((K \in HL \land W \in H) \land F \in T \land (X \in B \land Y \in B)) \to X \in B$
11		simp3r	$⊢ ((K \in HL \land W \in H) \land F \in T \land (X \in B \land Y \in B)) \to Y \in B$
12	1 2 7	lautm	$⊢ (K \in Lat \land (F \in LAut (K) \land X \in B \land Y \in B)) \to F (X \underset{˙}{\land} Y) = F (X) \underset{˙}{\land} F (Y)$
13	6 9 10 11 12	syl13anc	$⊢ ((K \in HL \land W \in H) \land F \in T \land (X \in B \land Y \in B)) \to F (X \underset{˙}{\land} Y) = F (X) \underset{˙}{\land} F (Y)$

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