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Metamath Proof Explorer

Theorem coeid2

Description: Reconstruct a polynomial as an explicit sum of the coefficient function up to the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014)

Ref Expression
Hypotheses dgrub.1 
|- A = ( coeff ` F )
dgrub.2 
|- N = ( deg ` F )
Assertion coeid2 
|- ( ( F e. ( Poly ` S ) /\ X e. CC ) -> ( F ` X ) = sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( X ^ k ) ) )

Proof

Step Hyp Ref Expression
1 dgrub.1 
 |-  A = ( coeff ` F )
2 dgrub.2 
 |-  N = ( deg ` F )
3 1 2 coeid 
 |-  ( F e. ( Poly ` S ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) )
4 3 fveq1d 
 |-  ( F e. ( Poly ` S ) -> ( F ` X ) = ( ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) ` X ) )
5 oveq1 
 |-  ( z = X -> ( z ^ k ) = ( X ^ k ) )
6 5 oveq2d 
 |-  ( z = X -> ( ( A ` k ) x. ( z ^ k ) ) = ( ( A ` k ) x. ( X ^ k ) ) )
7 6 sumeq2sdv 
 |-  ( z = X -> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) = sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( X ^ k ) ) )
8 eqid 
 |-  ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) )
9 sumex 
 |-  sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( X ^ k ) ) e. _V
10 7 8 9 fvmpt 
 |-  ( X e. CC -> ( ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( z ^ k ) ) ) ` X ) = sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( X ^ k ) ) )
11 4 10 sylan9eq 
 |-  ( ( F e. ( Poly ` S ) /\ X e. CC ) -> ( F ` X ) = sum_ k e. ( 0 ... N ) ( ( A ` k ) x. ( X ^ k ) ) )