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Theorem eftval 12358
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 5866 . . 3  |-  ( n  =  N  ->  ( A ^ n )  =  ( A ^ N
) )
2 fveq2 5525 . . 3  |-  ( n  =  N  ->  ( ! `  n )  =  ( ! `  N ) )
31, 2oveq12d 5876 . 2  |-  ( n  =  N  ->  (
( A ^ n
)  /  ( ! `
 n ) )  =  ( ( A ^ N )  / 
( ! `  N
) ) )
4 eftval.1 . 2  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
5 ovex 5883 . 2  |-  ( ( A ^ N )  /  ( ! `  N ) )  e. 
_V
63, 4, 5fvmpt 5602 1  |-  ( N  e.  NN0  ->  ( F `
 N )  =  ( ( A ^ N )  /  ( ! `  N )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    / cdiv 9423   NN0cn0 9965   ^cexp 11104   !cfa 11288
This theorem is referenced by:  efcllem  12359  ef0lem  12360  eff  12363  efval2  12365  efcvg  12366  efcvgfsum  12367  reefcl  12368  efcj  12373  efaddlem  12374  eftlcvg  12386  eftlcl  12387  reeftlcl  12388  eftlub  12389  efsep  12390  effsumlt  12391  efgt1p2  12394  efgt1p  12395  eflegeo  12401  eirrlem  12482  subfaclim  23719
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861
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