Theorem mpladd

Description: The addition operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015) (Revised by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	mpladd.p	|- P = ( I mPoly R )
	mpladd.b	|- B = ( Base ` P )
	mpladd.a	|- .+ = ( +g ` R )
	mpladd.g	|- .+b = ( +g ` P )
	mpladd.x	|- ( ph -> X e. B )
	mpladd.y	|- ( ph -> Y e. B )
Assertion	mpladd	|- ( ph -> ( X .+b Y ) = ( X oF .+ Y ) )

Proof

Step	Hyp	Ref	Expression
1		mpladd.p	|- P = ( I mPoly R )
2		mpladd.b	|- B = ( Base ` P )
3		mpladd.a	|- .+ = ( +g ` R )
4		mpladd.g	|- .+b = ( +g ` P )
5		mpladd.x	|- ( ph -> X e. B )
6		mpladd.y	|- ( ph -> Y e. B )
7		eqid	|- ( I mPwSer R ) = ( I mPwSer R )
8		eqid	|- ( Base ` ( I mPwSer R ) ) = ( Base ` ( I mPwSer R ) )
9	1 7 4	mplplusg	|- .+b = ( +g ` ( I mPwSer R ) )
10	1 7 2 8	mplbasss	|- B C_ ( Base ` ( I mPwSer R ) )
11	10 5	sselid	|- ( ph -> X e. ( Base ` ( I mPwSer R ) ) )
12	10 6	sselid	|- ( ph -> Y e. ( Base ` ( I mPwSer R ) ) )
13	7 8 3 9 11 12	psradd	|- ( ph -> ( X .+b Y ) = ( X oF .+ Y ) )