Theorem deg1vsca

Description: The degree of a scalar times a polynomial is exactly the degree of the original polynomial when the scalar is not a zero divisor. (Contributed by Stefan O'Rear, 28-Mar-2015)

	Ref	Expression
Hypotheses	deg1addle.y	|- Y = ( Poly1 ` R )
	deg1addle.d	|- D = ( deg1 ` R )
	deg1addle.r	|- ( ph -> R e. Ring )
	deg1vsca.b	|- B = ( Base ` Y )
	deg1vsca.e	|- E = ( RLReg ` R )
	deg1vsca.p	|- .x. = ( .s ` Y )
	deg1vsca.f	|- ( ph -> F e. E )
	deg1vsca.g	|- ( ph -> G e. B )
Assertion	deg1vsca	|- ( ph -> ( D ` ( F .x. G ) ) = ( D ` G ) )

Proof

Step	Hyp	Ref	Expression
1		deg1addle.y	|- Y = ( Poly1 ` R )
2		deg1addle.d	|- D = ( deg1 ` R )
3		deg1addle.r	|- ( ph -> R e. Ring )
4		deg1vsca.b	|- B = ( Base ` Y )
5		deg1vsca.e	|- E = ( RLReg ` R )
6		deg1vsca.p	|- .x. = ( .s ` Y )
7		deg1vsca.f	|- ( ph -> F e. E )
8		deg1vsca.g	|- ( ph -> G e. B )
9		eqid	|- ( 1o mPoly R ) = ( 1o mPoly R )
10	2	deg1fval	|- D = ( 1o mDeg R )
11		1on	|- 1o e. On
12	11	a1i	|- ( ph -> 1o e. On )
13	1 4	ply1bas	|- B = ( Base ` ( 1o mPoly R ) )
14	1 9 6	ply1vsca	|- .x. = ( .s ` ( 1o mPoly R ) )
15	9 10 12 3 13 5 14 7 8	mdegvsca	|- ( ph -> ( D ` ( F .x. G ) ) = ( D ` G ) )