Theorem 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	⊢ ≤ = ( le ‘ 𝐾 )
	ltrnel.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	ltrnel.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	ltrnel.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion	ltrnat	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		ltrnel.l	⊢ ≤ = ( le ‘ 𝐾 )
2		ltrnel.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		ltrnel.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		ltrnel.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		simp3	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → 𝑃 ∈ 𝐴 )
6		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
7	6 2	atbase	⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
8	6 2 3 4	ltrnatb	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∈ 𝐴 ↔ ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ) )
9	7 8	syl3an3	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑃 ∈ 𝐴 ↔ ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ) )
10	5 9	mpbid	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )