Metamath Proof Explorer


Theorem mplmul

Description: The multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015)

Ref Expression
Hypotheses mplmul.p P = I mPoly R
mplmul.b B = Base P
mplmul.m · ˙ = R
mplmul.t ˙ = P
mplmul.d D = h 0 I | h -1 Fin
mplmul.f φ F B
mplmul.g φ G B
Assertion mplmul φ F ˙ G = k D R x y D | y f k F x · ˙ G k f x

Proof

Step Hyp Ref Expression
1 mplmul.p P = I mPoly R
2 mplmul.b B = Base P
3 mplmul.m · ˙ = R
4 mplmul.t ˙ = P
5 mplmul.d D = h 0 I | h -1 Fin
6 mplmul.f φ F B
7 mplmul.g φ G B
8 eqid I mPwSer R = I mPwSer R
9 eqid Base I mPwSer R = Base I mPwSer R
10 1 8 4 mplmulr ˙ = I mPwSer R
11 1 8 2 9 mplbasss B Base I mPwSer R
12 11 6 sselid φ F Base I mPwSer R
13 11 7 sselid φ G Base I mPwSer R
14 8 9 3 10 5 12 13 psrmulfval φ F ˙ G = k D R x y D | y f k F x · ˙ G k f x