MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eftval Structured version   Unicode version

Theorem eftval 12680
Description: The value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
Hypothesis
Ref Expression
eftval.1  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
Assertion
Ref Expression
eftval  |-  ( N  e.  NN0  ->  ( F `
 N )  =  ( ( A ^ N )  /  ( ! `  N )
) )
Distinct variable groups:    A, n    n, N
Allowed substitution hint:    F( n)

Proof of Theorem eftval
StepHypRef Expression
1 oveq2 6090 . . 3  |-  ( n  =  N  ->  ( A ^ n )  =  ( A ^ N
) )
2 fveq2 5729 . . 3  |-  ( n  =  N  ->  ( ! `  n )  =  ( ! `  N ) )
31, 2oveq12d 6100 . 2  |-  ( n  =  N  ->  (
( A ^ n
)  /  ( ! `
 n ) )  =  ( ( A ^ N )  / 
( ! `  N
) ) )
4 eftval.1 . 2  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
5 ovex 6107 . 2  |-  ( ( A ^ N )  /  ( ! `  N ) )  e. 
_V
63, 4, 5fvmpt 5807 1  |-  ( N  e.  NN0  ->  ( F `
 N )  =  ( ( A ^ N )  /  ( ! `  N )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    e. cmpt 4267   ` cfv 5455  (class class class)co 6082    / cdiv 9678   NN0cn0 10222   ^cexp 11383   !cfa 11567
This theorem is referenced by:  efcllem  12681  ef0lem  12682  eff  12685  efval2  12687  efcvg  12688  efcvgfsum  12689  reefcl  12690  efcj  12695  efaddlem  12696  eftlcvg  12708  eftlcl  12709  reeftlcl  12710  eftlub  12711  efsep  12712  effsumlt  12713  efgt1p2  12716  efgt1p  12717  eflegeo  12723  eirrlem  12804  subfaclim  24875
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-ov 6085
  Copyright terms: Public domain W3C validator