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Theorem eftval 14851
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 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))
Assertion
Ref Expression
eftval (𝑁 ∈ ℕ0 → (𝐹𝑁) = ((𝐴𝑁) / (!‘𝑁)))
Distinct variable groups:   𝐴,𝑛   𝑛,𝑁
Allowed substitution hint:   𝐹(𝑛)

Proof of Theorem eftval
StepHypRef Expression
1 oveq2 6698 . . 3 (𝑛 = 𝑁 → (𝐴𝑛) = (𝐴𝑁))
2 fveq2 6229 . . 3 (𝑛 = 𝑁 → (!‘𝑛) = (!‘𝑁))
31, 2oveq12d 6708 . 2 (𝑛 = 𝑁 → ((𝐴𝑛) / (!‘𝑛)) = ((𝐴𝑁) / (!‘𝑁)))
4 eftval.1 . 2 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))
5 ovex 6718 . 2 ((𝐴𝑁) / (!‘𝑁)) ∈ V
63, 4, 5fvmpt 6321 1 (𝑁 ∈ ℕ0 → (𝐹𝑁) = ((𝐴𝑁) / (!‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  cmpt 4762  cfv 5926  (class class class)co 6690   / cdiv 10722  0cn0 11330  cexp 12900  !cfa 13100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693
This theorem is referenced by:  efcllem  14852  ef0lem  14853  eff  14856  efval2  14858  efcvg  14859  efcvgfsum  14860  reefcl  14861  efcj  14866  efaddlem  14867  eftlcvg  14880  eftlcl  14881  reeftlcl  14882  eftlub  14883  efsep  14884  effsumlt  14885  efgt1p2  14888  efgt1p  14889  eflegeo  14895  eirrlem  14976  subfaclim  31296
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