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Theorem eftval 12374
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 5882 . . 3  |-  ( n  =  N  ->  ( A ^ n )  =  ( A ^ N
) )
2 fveq2 5541 . . 3  |-  ( n  =  N  ->  ( ! `  n )  =  ( ! `  N ) )
31, 2oveq12d 5892 . 2  |-  ( n  =  N  ->  (
( A ^ n
)  /  ( ! `
 n ) )  =  ( ( A ^ N )  / 
( ! `  N
) ) )
4 eftval.1 . 2  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
5 ovex 5899 . 2  |-  ( ( A ^ N )  /  ( ! `  N ) )  e. 
_V
63, 4, 5fvmpt 5618 1  |-  ( N  e.  NN0  ->  ( F `
 N )  =  ( ( A ^ N )  /  ( ! `  N )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874    / cdiv 9439   NN0cn0 9981   ^cexp 11120   !cfa 11304
This theorem is referenced by:  efcllem  12375  ef0lem  12376  eff  12379  efval2  12381  efcvg  12382  efcvgfsum  12383  reefcl  12384  efcj  12389  efaddlem  12390  eftlcvg  12402  eftlcl  12403  reeftlcl  12404  eftlub  12405  efsep  12406  effsumlt  12407  efgt1p2  12410  efgt1p  12411  eflegeo  12417  eirrlem  12498  subfaclim  23734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877
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