Theorem dgraacl

Description: Closure of the degree function on algebraic numbers. (Contributed by Stefan O'Rear, 25-Nov-2014)

	Ref	Expression
Assertion	dgraacl	$$⊢A\in 𝔸\to {\mathrm{deg}}_{𝔸}\left(A\right)\in \mathbb{N}$$

Proof

Step	Hyp	Ref	Expression
1		dgraalem	$$⊢A\in 𝔸\to \left({\mathrm{deg}}_{𝔸}\left(A\right)\in \mathbb{N}\wedge \exists a\in \left(\mathrm{Poly}\left(\mathbb{Q}\right)\setminus \left\{0_{𝑝}\right\}\right)\left(\mathrm{deg}\left(a\right)={\mathrm{deg}}_{𝔸}\left(A\right)\wedge a\left(A\right)=0\right)\right)$$
2	1	simpld	$$⊢A\in 𝔸\to {\mathrm{deg}}_{𝔸}\left(A\right)\in \mathbb{N}$$