Metamath Proof Explorer


Theorem mpaacl

Description: Minimal polynomial is a polynomial. (Contributed by Stefan O'Rear, 25-Nov-2014)

Ref Expression
Assertion mpaacl ( 𝐴 ∈ 𝔸 → ( minPolyAA ‘ 𝐴 ) ∈ ( Poly ‘ ℚ ) )

Proof

Step Hyp Ref Expression
1 mpaalem ( 𝐴 ∈ 𝔸 → ( ( minPolyAA ‘ 𝐴 ) ∈ ( Poly ‘ ℚ ) ∧ ( ( deg ‘ ( minPolyAA ‘ 𝐴 ) ) = ( degAA𝐴 ) ∧ ( ( minPolyAA ‘ 𝐴 ) ‘ 𝐴 ) = 0 ∧ ( ( coeff ‘ ( minPolyAA ‘ 𝐴 ) ) ‘ ( degAA𝐴 ) ) = 1 ) ) )
2 1 simpld ( 𝐴 ∈ 𝔸 → ( minPolyAA ‘ 𝐴 ) ∈ ( Poly ‘ ℚ ) )