efval - Metamath Proof Explorer

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Theorem efval 16111

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 ((𝐴↑𝑘) / (!‘𝑘)))

Distinct variable group: 𝐴,𝑘

Proof of Theorem efval Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7437	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥↑𝑘) = (𝐴↑𝑘))
2	1	oveq1d 7445	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥↑𝑘) / (!‘𝑘)) = ((𝐴↑𝑘) / (!‘𝑘)))
3	2	sumeq2sdv 15735	. 2 ⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ ℕ₀ ((𝑥↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ ℕ₀ ((𝐴↑𝑘) / (!‘𝑘)))
4		df-ef 16099	. 2 ⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ₀ ((𝑥↑𝑘) / (!‘𝑘)))
5		sumex 15720	. 2 ⊢ Σ𝑘 ∈ ℕ₀ ((𝐴↑𝑘) / (!‘𝑘)) ∈ V
6	3, 4, 5	fvmpt 7015	1 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ₀ ((𝐴↑𝑘) / (!‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 / cdiv 11917 ℕ₀cn0 12523 ↑cexp 14098 !cfa 14308 Σcsu 15718 expce 16093

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-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 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-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-seq 14039 df-sum 15719 df-ef 16099

This theorem is referenced by: esum 16112 efval2 16116 efcvg 16117 reefcl 16119 efaddlem 16125 eflegeo 16153 subfaclim 35172

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