Metamath Proof Explorer


Theorem mplsca

Description: The scalar field of a multivariate polynomial structure. (Contributed by Mario Carneiro, 9-Jan-2015)

Ref Expression
Hypotheses mplsca.p 𝑃 = ( 𝐼 mPoly 𝑅 )
mplsca.i ( 𝜑𝐼𝑉 )
mplsca.r ( 𝜑𝑅𝑊 )
Assertion mplsca ( 𝜑𝑅 = ( Scalar ‘ 𝑃 ) )

Proof

Step Hyp Ref Expression
1 mplsca.p 𝑃 = ( 𝐼 mPoly 𝑅 )
2 mplsca.i ( 𝜑𝐼𝑉 )
3 mplsca.r ( 𝜑𝑅𝑊 )
4 eqid ( 𝐼 mPwSer 𝑅 ) = ( 𝐼 mPwSer 𝑅 )
5 4 2 3 psrsca ( 𝜑𝑅 = ( Scalar ‘ ( 𝐼 mPwSer 𝑅 ) ) )
6 fvex ( Base ‘ 𝑃 ) ∈ V
7 eqid ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 )
8 1 4 7 mplval2 𝑃 = ( ( 𝐼 mPwSer 𝑅 ) ↾s ( Base ‘ 𝑃 ) )
9 eqid ( Scalar ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Scalar ‘ ( 𝐼 mPwSer 𝑅 ) )
10 8 9 resssca ( ( Base ‘ 𝑃 ) ∈ V → ( Scalar ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Scalar ‘ 𝑃 ) )
11 6 10 ax-mp ( Scalar ‘ ( 𝐼 mPwSer 𝑅 ) ) = ( Scalar ‘ 𝑃 )
12 5 11 eqtrdi ( 𝜑𝑅 = ( Scalar ‘ 𝑃 ) )