efval - Metamath Proof Explorer

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Theorem efval 16052

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 7397	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥↑𝑘) = (𝐴↑𝑘))
2	1	oveq1d 7405	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥↑𝑘) / (!‘𝑘)) = ((𝐴↑𝑘) / (!‘𝑘)))
3	2	sumeq2sdv 15676	. 2 ⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ ℕ₀ ((𝑥↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ ℕ₀ ((𝐴↑𝑘) / (!‘𝑘)))
4		df-ef 16040	. 2 ⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ₀ ((𝑥↑𝑘) / (!‘𝑘)))
5		sumex 15661	. 2 ⊢ Σ𝑘 ∈ ℕ₀ ((𝐴↑𝑘) / (!‘𝑘)) ∈ V
6	3, 4, 5	fvmpt 6971	1 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ₀ ((𝐴↑𝑘) / (!‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 / cdiv 11842 ℕ₀cn0 12449 ↑cexp 14033 !cfa 14245 Σcsu 15659 expce 16034

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-iota 6467 df-fun 6516 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-seq 13974 df-sum 15660 df-ef 16040

This theorem is referenced by: esum 16053 efval2 16057 efcvg 16058 reefcl 16060 efaddlem 16066 eflegeo 16096 subfaclim 35182

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