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Theorem eftval 16019
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 7416 . . 3 (𝑛 = 𝑁 → (𝐴𝑛) = (𝐴𝑁))
2 fveq2 6891 . . 3 (𝑛 = 𝑁 → (!‘𝑛) = (!‘𝑁))
31, 2oveq12d 7426 . 2 (𝑛 = 𝑁 → ((𝐴𝑛) / (!‘𝑛)) = ((𝐴𝑁) / (!‘𝑁)))
4 eftval.1 . 2 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))
5 ovex 7441 . 2 ((𝐴𝑁) / (!‘𝑁)) ∈ V
63, 4, 5fvmpt 6998 1 (𝑁 ∈ ℕ0 → (𝐹𝑁) = ((𝐴𝑁) / (!‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cmpt 5231  cfv 6543  (class class class)co 7408   / cdiv 11870  0cn0 12471  cexp 14026  !cfa 14232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411
This theorem is referenced by:  efcllem  16020  ef0lem  16021  eff  16024  efval2  16026  efcvg  16027  efcvgfsum  16028  reefcl  16029  efcj  16034  efaddlem  16035  eftlcvg  16048  eftlcl  16049  reeftlcl  16050  eftlub  16051  efsep  16052  effsumlt  16053  efgt1p2  16056  efgt1p  16057  eflegeo  16063  eirrlem  16146  subfaclim  34174
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