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Theorem eftval 15422
 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 7156 . . 3 (𝑛 = 𝑁 → (𝐴𝑛) = (𝐴𝑁))
2 fveq2 6663 . . 3 (𝑛 = 𝑁 → (!‘𝑛) = (!‘𝑁))
31, 2oveq12d 7166 . 2 (𝑛 = 𝑁 → ((𝐴𝑛) / (!‘𝑛)) = ((𝐴𝑁) / (!‘𝑁)))
4 eftval.1 . 2 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))
5 ovex 7181 . 2 ((𝐴𝑁) / (!‘𝑁)) ∈ V
63, 4, 5fvmpt 6761 1 (𝑁 ∈ ℕ0 → (𝐹𝑁) = ((𝐴𝑁) / (!‘𝑁)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1530   ∈ wcel 2107   ↦ cmpt 5137  ‘cfv 6348  (class class class)co 7148   / cdiv 11289  ℕ0cn0 11889  ↑cexp 13421  !cfa 13625 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151 This theorem is referenced by:  efcllem  15423  ef0lem  15424  eff  15427  efval2  15429  efcvg  15430  efcvgfsum  15431  reefcl  15432  efcj  15437  efaddlem  15438  eftlcvg  15451  eftlcl  15452  reeftlcl  15453  eftlub  15454  efsep  15455  effsumlt  15456  efgt1p2  15459  efgt1p  15460  eflegeo  15466  eirrlem  15549  subfaclim  32423
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