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Metamath Proof Explorer

Theorem ltrncnvat

Description: The converse of the lattice translation of an atom is an atom. (Contributed by NM, 9-May-2013)

Ref Expression
Hypotheses ltrnel.l ⊢ ≤ = ( le ‘ 𝐾 )
ltrnel.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
ltrnel.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
ltrnel.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion ltrncnvat ( ( ( 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 ltrncnvatb ⊢ ( ( ( 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 )