ltrnat

Description: The lattice translation of an atom is also an atom. TODO: See if this can shorten some ltrnel uses. (Contributed by NM, 25-May-2012)

	Ref	Expression
Hypotheses	ltrnel.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
	ltrnel.a	$⊢ A = Atoms (K)$
	ltrnel.h	$⊢ H = LHyp (K)$
	ltrnel.t	$⊢ T = LTrn (K) (W)$
Assertion	ltrnat	$⊢ ((K \in HL \land W \in H) \land F \in T \land P \in A) \to F (P) \in A$

Step	Hyp	Ref	Expression
1		ltrnel.l	$⊢ \underset{˙}{\leq} = \leq_{K}$
2		ltrnel.a	$⊢ A = Atoms (K)$
3		ltrnel.h	$⊢ H = LHyp (K)$
4		ltrnel.t	$⊢ T = LTrn (K) (W)$
5		simp3	$⊢ ((K \in HL \land W \in H) \land F \in T \land P \in A) \to P \in A$
6		eqid	$⊢ {Base}_{K} = {Base}_{K}$
7	6 2	atbase	$⊢ P \in A \to P \in {Base}_{K}$
8	6 2 3 4	ltrnatb	$⊢ ((K \in HL \land W \in H) \land F \in T \land P \in {Base}_{K}) \to (P \in A \leftrightarrow F (P) \in A)$
9	7 8	syl3an3	$⊢ ((K \in HL \land W \in H) \land F \in T \land P \in A) \to (P \in A \leftrightarrow F (P) \in A)$
10	5 9	mpbid	$⊢ ((K \in HL \land W \in H) \land F \in T \land P \in A) \to F (P) \in A$

Metamath Proof Explorer