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

Theorem eftval 13663
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 6283 . . 3  |-  ( n  =  N  ->  ( A ^ n )  =  ( A ^ N
) )
2 fveq2 5857 . . 3  |-  ( n  =  N  ->  ( ! `  n )  =  ( ! `  N ) )
31, 2oveq12d 6293 . 2  |-  ( n  =  N  ->  (
( A ^ n
)  /  ( ! `
 n ) )  =  ( ( A ^ N )  / 
( ! `  N
) ) )
4 eftval.1 . 2  |-  F  =  ( n  e.  NN0  |->  ( ( A ^
n )  /  ( ! `  n )
) )
5 ovex 6300 . 2  |-  ( ( A ^ N )  /  ( ! `  N ) )  e. 
_V
63, 4, 5fvmpt 5941 1  |-  ( N  e.  NN0  ->  ( F `
 N )  =  ( ( A ^ N )  /  ( ! `  N )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762    |-> cmpt 4498   ` cfv 5579  (class class class)co 6275    / cdiv 10195   NN0cn0 10784   ^cexp 12122   !cfa 12308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-iota 5542  df-fun 5581  df-fv 5587  df-ov 6278
This theorem is referenced by:  efcllem  13664  ef0lem  13665  eff  13668  efval2  13670  efcvg  13671  efcvgfsum  13672  reefcl  13673  efcj  13678  efaddlem  13679  eftlcvg  13691  eftlcl  13692  reeftlcl  13693  eftlub  13694  efsep  13695  effsumlt  13696  efgt1p2  13699  efgt1p  13700  eflegeo  13706  eirrlem  13787  subfaclim  28258
  Copyright terms: Public domain W3C validator