Theorem deg1z

Description: Degree of the zero univariate polynomial. (Contributed by Stefan O'Rear, 23-Mar-2015)

	Ref	Expression
Hypotheses	deg1z.d	|- D = ( deg1 ` R )
	deg1z.p	|- P = ( Poly1 ` R )
	deg1z.z	|- .0. = ( 0g ` P )
Assertion	deg1z	|- ( R e. Ring -> ( D ` .0. ) = -oo )

Proof

Step	Hyp	Ref	Expression
1		deg1z.d	|- D = ( deg1 ` R )
2		deg1z.p	|- P = ( Poly1 ` R )
3		deg1z.z	|- .0. = ( 0g ` P )
4		1on	|- 1o e. On
5	1	deg1fval	|- D = ( 1o mDeg R )
6		eqid	|- ( 1o mPoly R ) = ( 1o mPoly R )
7	6 2 3	ply1mpl0	|- .0. = ( 0g ` ( 1o mPoly R ) )
8	5 6 7	mdeg0	|- ( ( 1o e. On /\ R e. Ring ) -> ( D ` .0. ) = -oo )
9	4 8	mpan	|- ( R e. Ring -> ( D ` .0. ) = -oo )