Metamath Proof Explorer

Theorem taylplem2

Description: Lemma for taylpfval and similar theorems. (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 ) )
Assertion taylplem2 
|- ( ( ( ph /\ X e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( X - B ) ^ k ) ) e. CC )

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 0z 
 |-  0 e. ZZ
7 4 nn0zd 
 |-  ( ph -> N e. ZZ )
8 fzval2 
 |-  ( ( 0 e. ZZ /\ N e. ZZ ) -> ( 0 ... N ) = ( ( 0 [,] N ) i^i ZZ ) )
9 6 7 8 sylancr 
 |-  ( ph -> ( 0 ... N ) = ( ( 0 [,] N ) i^i ZZ ) )
10 9 eleq2d 
 |-  ( ph -> ( k e. ( 0 ... N ) <-> k e. ( ( 0 [,] N ) i^i ZZ ) ) )
11 10 adantr 
 |-  ( ( ph /\ X e. CC ) -> ( k e. ( 0 ... N ) <-> k e. ( ( 0 [,] N ) i^i ZZ ) ) )
12 11 biimpa 
 |-  ( ( ( ph /\ X e. CC ) /\ k e. ( 0 ... N ) ) -> k e. ( ( 0 [,] N ) i^i ZZ ) )
13 4 orcd 
 |-  ( ph -> ( N e. NN0 \/ N = +oo ) )
14 1 2 3 4 5 taylplem1 
 |-  ( ( ph /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> B e. dom ( ( S Dn F ) ` k ) )
15 1 2 3 13 14 taylfvallem1 
 |-  ( ( ( ph /\ X e. CC ) /\ k e. ( ( 0 [,] N ) i^i ZZ ) ) -> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( X - B ) ^ k ) ) e. CC )
16 12 15 syldan 
 |-  ( ( ( ph /\ X e. CC ) /\ k e. ( 0 ... N ) ) -> ( ( ( ( ( S Dn F ) ` k ) ` B ) / ( ! ` k ) ) x. ( ( X - B ) ^ k ) ) e. CC )