Metamath Proof Explorer


Theorem itgoval

Description: Value of the integral-over function. (Contributed by Stefan O'Rear, 27-Nov-2014)

Ref Expression
Assertion itgoval S IntgOver S = x | p Poly S p x = 0 coeff p deg p = 1

Proof

Step Hyp Ref Expression
1 cnex V
2 1 elpw2 S 𝒫 S
3 fveq2 s = S Poly s = Poly S
4 3 rexeqdv s = S p Poly s p x = 0 coeff p deg p = 1 p Poly S p x = 0 coeff p deg p = 1
5 4 rabbidv s = S x | p Poly s p x = 0 coeff p deg p = 1 = x | p Poly S p x = 0 coeff p deg p = 1
6 df-itgo IntgOver = s 𝒫 x | p Poly s p x = 0 coeff p deg p = 1
7 1 rabex x | p Poly S p x = 0 coeff p deg p = 1 V
8 5 6 7 fvmpt S 𝒫 IntgOver S = x | p Poly S p x = 0 coeff p deg p = 1
9 2 8 sylbir S IntgOver S = x | p Poly S p x = 0 coeff p deg p = 1