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Theorem eftval 16108
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 7438 . . 3 (𝑛 = 𝑁 → (𝐴𝑛) = (𝐴𝑁))
2 fveq2 6906 . . 3 (𝑛 = 𝑁 → (!‘𝑛) = (!‘𝑁))
31, 2oveq12d 7448 . 2 (𝑛 = 𝑁 → ((𝐴𝑛) / (!‘𝑛)) = ((𝐴𝑁) / (!‘𝑁)))
4 eftval.1 . 2 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))
5 ovex 7463 . 2 ((𝐴𝑁) / (!‘𝑁)) ∈ V
63, 4, 5fvmpt 7015 1 (𝑁 ∈ ℕ0 → (𝐹𝑁) = ((𝐴𝑁) / (!‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  cmpt 5230  cfv 6562  (class class class)co 7430   / cdiv 11917  0cn0 12523  cexp 14098  !cfa 14308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433
This theorem is referenced by:  efcllem  16109  ef0lem  16110  eff  16113  efval2  16116  efcvg  16117  efcvgfsum  16118  reefcl  16119  efcj  16124  efaddlem  16125  eftlcvg  16138  eftlcl  16139  reeftlcl  16140  eftlub  16141  efsep  16142  effsumlt  16143  efgt1p2  16146  efgt1p  16147  eflegeo  16153  eirrlem  16236  subfaclim  35172
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