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Theorem efval 16115
Description: Value of the exponential function. (Contributed by NM, 8-Jan-2006.) (Revised by Mario Carneiro, 10-Nov-2013.)
Assertion
Ref Expression
efval (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 ((𝐴𝑘) / (!‘𝑘)))
Distinct variable group:   𝐴,𝑘

Proof of Theorem efval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oveq1 7438 . . . 4 (𝑥 = 𝐴 → (𝑥𝑘) = (𝐴𝑘))
21oveq1d 7446 . . 3 (𝑥 = 𝐴 → ((𝑥𝑘) / (!‘𝑘)) = ((𝐴𝑘) / (!‘𝑘)))
32sumeq2sdv 15739 . 2 (𝑥 = 𝐴 → Σ𝑘 ∈ ℕ0 ((𝑥𝑘) / (!‘𝑘)) = Σ𝑘 ∈ ℕ0 ((𝐴𝑘) / (!‘𝑘)))
4 df-ef 16103 . 2 exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ0 ((𝑥𝑘) / (!‘𝑘)))
5 sumex 15724 . 2 Σ𝑘 ∈ ℕ0 ((𝐴𝑘) / (!‘𝑘)) ∈ V
63, 4, 5fvmpt 7016 1 (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 ((𝐴𝑘) / (!‘𝑘)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431  cc 11153   / cdiv 11920  0cn0 12526  cexp 14102  !cfa 14312  Σcsu 15722  expce 16097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seq 14043  df-sum 15723  df-ef 16103
This theorem is referenced by:  esum  16116  efval2  16120  efcvg  16121  reefcl  16123  efaddlem  16129  eflegeo  16157  subfaclim  35193
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