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Theorem eftval 14023
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 6288 . . 3  |-  ( n  =  N  ->  ( A ^ n )  =  ( A ^ N
) )
2 fveq2 5851 . . 3  |-  ( n  =  N  ->  ( ! `  n )  =  ( ! `  N ) )
31, 2oveq12d 6298 . 2  |-  ( n  =  N  ->  (
( A ^ n
)  /  ( ! `
 n ) )  =  ( ( A ^ N )  / 
( ! `  N
) ) )
4 eftval.1 . 2  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
5 ovex 6308 . 2  |-  ( ( A ^ N )  /  ( ! `  N ) )  e. 
_V
63, 4, 5fvmpt 5934 1  |-  ( N  e.  NN0  ->  ( F `
 N )  =  ( ( A ^ N )  /  ( ! `  N )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1407    e. wcel 1844    |-> cmpt 4455   ` cfv 5571  (class class class)co 6280    / cdiv 10249   NN0cn0 10838   ^cexp 12212   !cfa 12399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-iota 5535  df-fun 5573  df-fv 5579  df-ov 6283
This theorem is referenced by:  efcllem  14024  ef0lem  14025  eff  14028  efval2  14030  efcvg  14031  efcvgfsum  14032  reefcl  14033  efcj  14038  efaddlem  14039  eftlcvg  14052  eftlcl  14053  reeftlcl  14054  eftlub  14055  efsep  14056  effsumlt  14057  efgt1p2  14060  efgt1p  14061  eflegeo  14067  eirrlem  14148  subfaclim  29498
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