Metamath Proof Explorer

Theorem taylpval

Description: Value of the Taylor polynomial. (Contributed by Mario Carneiro, 31-Dec-2016)

Ref Expression
Hypotheses taylpfval.s 
|- ( ph -> S e. { RR , CC } )
taylpfval.f 
|- ( ph -> F : A --> CC )
taylpfval.a 
|- ( ph -> A C_ S )
taylpfval.n 
|- ( ph -> N e. NN0 )
taylpfval.b 
|- ( ph -> B e. dom ( ( S Dn F ) ` N ) )
taylpfval.t 
|- T = ( N ( S Tayl F ) B )
taylpval.x 
|- ( ph -> X e. CC )
Assertion taylpval 
|- ( ph -> ( T ` X ) = sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( X - B ) ^ k ) ) )

Proof

Step Hyp Ref Expression
1 taylpfval.s 
 |-  ( ph -> S e. { RR , CC } )
2 taylpfval.f 
 |-  ( ph -> F : A --> CC )
3 taylpfval.a 
 |-  ( ph -> A C_ S )
4 taylpfval.n 
 |-  ( ph -> N e. NN0 )
5 taylpfval.b 
 |-  ( ph -> B e. dom ( ( S Dn F ) ` N ) )
6 taylpfval.t 
 |-  T = ( N ( S Tayl F ) B )
7 taylpval.x 
 |-  ( ph -> X e. CC )
8 1 2 3 4 5 6 taylpfval 
 |-  ( ph -> T = ( x e. CC |-> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) ) )
9 simplr 
 |-  ( ( ( ph /\ x = X ) /\ k e. ( 0 ... N ) ) -> x = X )
10 9 oveq1d 
 |-  ( ( ( ph /\ x = X ) /\ k e. ( 0 ... N ) ) -> ( x - B ) = ( X - B ) )
11 10 oveq1d 
 |-  ( ( ( ph /\ x = X ) /\ k e. ( 0 ... N ) ) -> ( ( x - B ) ^ k ) = ( ( X - B ) ^ k ) )
12 11 oveq2d 
 |-  ( ( ( ph /\ x = X ) /\ k e. ( 0 ... N ) ) -> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) = ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( X - B ) ^ k ) ) )
13 12 sumeq2dv 
 |-  ( ( ph /\ x = X ) -> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( x - B ) ^ k ) ) = sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( X - B ) ^ k ) ) )
14 sumex 
 |-  sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( X - B ) ^ k ) ) e. _V
15 14 a1i 
 |-  ( ph -> sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( X - B ) ^ k ) ) e. _V )
16 8 13 7 15 fvmptd 
 |-  ( ph -> ( T ` X ) = sum_ k e. ( 0 ... N ) ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( X - B ) ^ k ) ) )