Theorem mpaadgr

Description: Minimal polynomial has degree the degree of the number. (Contributed by Stefan O'Rear, 25-Nov-2014)

	Ref	Expression
Assertion	mpaadgr	$$⊢A\in 𝔸\to \mathrm{deg}\left({\mathrm{minPoly}}_{𝔸}\left(A\right)\right)={\mathrm{deg}}_{𝔸}\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		mpaalem	$$⊢A\in 𝔸\to \left({\mathrm{minPoly}}_{𝔸}\left(A\right)\in \mathrm{Poly}\left(\mathbb{Q}\right)\wedge \left(\mathrm{deg}\left({\mathrm{minPoly}}_{𝔸}\left(A\right)\right)={\mathrm{deg}}_{𝔸}\left(A\right)\wedge {\mathrm{minPoly}}_{𝔸}\left(A\right)\left(A\right)=0\wedge \mathrm{coeff}\left({\mathrm{minPoly}}_{𝔸}\left(A\right)\right)\left({\mathrm{deg}}_{𝔸}\left(A\right)\right)=1\right)\right)$$
2		simpr1	$$⊢\left({\mathrm{minPoly}}_{𝔸}\left(A\right)\in \mathrm{Poly}\left(\mathbb{Q}\right)\wedge \left(\mathrm{deg}\left({\mathrm{minPoly}}_{𝔸}\left(A\right)\right)={\mathrm{deg}}_{𝔸}\left(A\right)\wedge {\mathrm{minPoly}}_{𝔸}\left(A\right)\left(A\right)=0\wedge \mathrm{coeff}\left({\mathrm{minPoly}}_{𝔸}\left(A\right)\right)\left({\mathrm{deg}}_{𝔸}\left(A\right)\right)=1\right)\right)\to \mathrm{deg}\left({\mathrm{minPoly}}_{𝔸}\left(A\right)\right)={\mathrm{deg}}_{𝔸}\left(A\right)$$
3	1 2	syl	$$⊢A\in 𝔸\to \mathrm{deg}\left({\mathrm{minPoly}}_{𝔸}\left(A\right)\right)={\mathrm{deg}}_{𝔸}\left(A\right)$$