efval - Metamath Proof Explorer

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Theorem efval 15988

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 7359	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥↑𝑘) = (𝐴↑𝑘))
2	1	oveq1d 7367	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥↑𝑘) / (!‘𝑘)) = ((𝐴↑𝑘) / (!‘𝑘)))
3	2	sumeq2sdv 15612	. 2 ⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ ℕ₀ ((𝑥↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ ℕ₀ ((𝐴↑𝑘) / (!‘𝑘)))
4		df-ef 15976	. 2 ⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ₀ ((𝑥↑𝑘) / (!‘𝑘)))
5		sumex 15597	. 2 ⊢ Σ𝑘 ∈ ℕ₀ ((𝐴↑𝑘) / (!‘𝑘)) ∈ V
6	3, 4, 5	fvmpt 6935	1 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ₀ ((𝐴↑𝑘) / (!‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 ‘cfv 6486 (class class class)co 7352 ℂcc 11011 / cdiv 11781 ℕ₀cn0 12388 ↑cexp 13970 !cfa 14182 Σcsu 15595 expce 15970

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5236 ax-nul 5246 ax-pr 5372

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4475 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-br 5094 df-opab 5156 df-mpt 5175 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-iota 6442 df-fun 6488 df-fv 6494 df-ov 7355 df-oprab 7356 df-mpo 7357 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-seq 13911 df-sum 15596 df-ef 15976

This theorem is referenced by: esum 15989 efval2 15993 efcvg 15994 reefcl 15996 efaddlem 16002 eflegeo 16032 subfaclim 35253

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