Theorem mplplusg

Description: Value of addition in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015) (Revised by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	mplplusg.y	$$⊢Y=I\mathrm{mPoly}R$$
	mplplusg.s	$$⊢S=I\mathrm{mPwSer}R$$
	mplplusg.p	$$⊢\underset{\dot{}}{+}={+}_Y$$
Assertion	mplplusg	$$⊢\underset{\dot{}}{+}={+}_S$$

Proof

Step	Hyp	Ref	Expression
1		mplplusg.y	$$⊢Y=I\mathrm{mPoly}R$$
2		mplplusg.s	$$⊢S=I\mathrm{mPwSer}R$$
3		mplplusg.p	$$⊢\underset{\dot{}}{+}={+}_Y$$
4		fvex	$$⊢{\mathrm{Base}}_Y\in \mathrm{V}$$
5		eqid	$$⊢{\mathrm{Base}}_Y={\mathrm{Base}}_Y$$
6	1 2 5	mplval2	$$⊢Y=S{\upharpoonright }_{𝑠}{\mathrm{Base}}_Y$$
7		eqid	$$⊢{+}_S={+}_S$$
8	6 7	ressplusg	$$⊢{\mathrm{Base}}_Y\in \mathrm{V}\to {+}_S={+}_Y$$
9	4 8	ax-mp	$$⊢{+}_S={+}_Y$$
10	3 9	eqtr4i	$$⊢\underset{\dot{}}{+}={+}_S$$