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Metamath Proof Explorer

Theorem eftlcvg

Description: The tail series of the exponential function are convergent. (Contributed by Mario Carneiro, 29-Apr-2014)

Ref Expression
Hypothesis eftl.1 
|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
Assertion eftlcvg 
|- ( ( A e. CC /\ M e. NN0 ) -> seq M ( + , F ) e. dom ~~> )

Proof

Step Hyp Ref Expression
1 eftl.1 
 |-  F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
2 1 efcllem 
 |-  ( A e. CC -> seq 0 ( + , F ) e. dom ~~> )
3 2 adantr 
 |-  ( ( A e. CC /\ M e. NN0 ) -> seq 0 ( + , F ) e. dom ~~> )
4 nn0uz 
 |-  NN0 = ( ZZ>= ` 0 )
5 simpr 
 |-  ( ( A e. CC /\ M e. NN0 ) -> M e. NN0 )
6 1 eftval 
 |-  ( k e. NN0 -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
7 6 adantl 
 |-  ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
8 eftcl 
 |-  ( ( A e. CC /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
9 8 adantlr 
 |-  ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( ( A ^ k ) / ( ! ` k ) ) e. CC )
10 7 9 eqeltrd 
 |-  ( ( ( A e. CC /\ M e. NN0 ) /\ k e. NN0 ) -> ( F ` k ) e. CC )
11 4 5 10 iserex 
 |-  ( ( A e. CC /\ M e. NN0 ) -> ( seq 0 ( + , F ) e. dom ~~> <-> seq M ( + , F ) e. dom ~~> ) )
12 3 11 mpbid 
 |-  ( ( A e. CC /\ M e. NN0 ) -> seq M ( + , F ) e. dom ~~> )