efval - Metamath Proof Explorer

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Theorem efval 16171

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 7424	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥↑𝑘) = (𝐴↑𝑘))
2	1	oveq1d 7432	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥↑𝑘) / (!‘𝑘)) = ((𝐴↑𝑘) / (!‘𝑘)))
3	2	sumeq2sdv 15794	. 2 ⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ ℕ₀ ((𝑥↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ ℕ₀ ((𝐴↑𝑘) / (!‘𝑘)))
4		df-ef 16159	. 2 ⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ₀ ((𝑥↑𝑘) / (!‘𝑘)))
5		sumex 15779	. 2 ⊢ Σ𝑘 ∈ ℕ₀ ((𝐴↑𝑘) / (!‘𝑘)) ∈ V
6	3, 4, 5	fvmpt 6990	1 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ₀ ((𝐴↑𝑘) / (!‘𝑘)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 ‘cfv 6537 (class class class)co 7417 ℂcc 11126 / cdiv 11899 ℕ₀cn0 12532 ↑cexp 14129 !cfa 14341 Σcsu 15777 expce 16153

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pr 5402

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-iota 6493 df-fun 6539 df-fv 6545 df-ov 7420 df-oprab 7421 df-mpo 7422 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-seq 14070 df-sum 15778 df-ef 16159

This theorem is used by: esum 16172 efval2 16176 efcvg 16177 reefcl 16179 efaddlem 16185 eflegeo 16215 subfaclim 35775

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