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Metamath Proof Explorer

Theorem efcvg

Description: The series that defines the exponential function converges to it. (Contributed by NM, 9-Jan-2006) (Revised by Mario Carneiro, 28-Apr-2014)

Ref Expression
Hypothesis efcvg.1 
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
Assertion efcvg 
|- ( A e. CC -> seq 0 ( + , F ) ~~> ( exp ` A ) )

Proof

Step Hyp Ref Expression
1 efcvg.1 
 |-  F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
2 nn0uz 
 |-  NN0 = ( ZZ>= ` 0 )
3 0zd 
 |-  ( A e. CC -> 0 e. ZZ )
4 1 eftval 
 |-  ( k e. NN0 -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
5 4 adantl 
 |-  ( ( A e. CC /\ k e. NN0 ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
6 eftcl 
 |-  ( ( A e. CC /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
7 1 efcllem 
 |-  ( A e. CC -> seq 0 ( + , F ) e. dom ~~> )
8 2 3 5 6 7 isumclim2 
 |-  ( A e. CC -> seq 0 ( + , F ) ~~> sum_ k e. NN0 ( ( A ^ k ) / ( ! ` k ) ) )
9 efval 
 |-  ( A e. CC -> ( exp ` A ) = sum_ k e. NN0 ( ( A ^ k ) / ( ! ` k ) ) )
10 8 9 breqtrrd 
 |-  ( A e. CC -> seq 0 ( + , F ) ~~> ( exp ` A ) )