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Theorem eftval 15522
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 7178 . . 3 (𝑛 = 𝑁 → (𝐴𝑛) = (𝐴𝑁))
2 fveq2 6674 . . 3 (𝑛 = 𝑁 → (!‘𝑛) = (!‘𝑁))
31, 2oveq12d 7188 . 2 (𝑛 = 𝑁 → ((𝐴𝑛) / (!‘𝑛)) = ((𝐴𝑁) / (!‘𝑁)))
4 eftval.1 . 2 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴𝑛) / (!‘𝑛)))
5 ovex 7203 . 2 ((𝐴𝑁) / (!‘𝑁)) ∈ V
63, 4, 5fvmpt 6775 1 (𝑁 ∈ ℕ0 → (𝐹𝑁) = ((𝐴𝑁) / (!‘𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  cmpt 5110  cfv 6339  (class class class)co 7170   / cdiv 11375  0cn0 11976  cexp 13521  !cfa 13725
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-ov 7173
This theorem is referenced by:  efcllem  15523  ef0lem  15524  eff  15527  efval2  15529  efcvg  15530  efcvgfsum  15531  reefcl  15532  efcj  15537  efaddlem  15538  eftlcvg  15551  eftlcl  15552  reeftlcl  15553  eftlub  15554  efsep  15555  effsumlt  15556  efgt1p2  15559  efgt1p  15560  eflegeo  15566  eirrlem  15649  subfaclim  32721
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