efval - Metamath Proof Explorer

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Theorem efval 16044

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 7374	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥↑𝑘) = (𝐴↑𝑘))
2	1	oveq1d 7382	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥↑𝑘) / (!‘𝑘)) = ((𝐴↑𝑘) / (!‘𝑘)))
3	2	sumeq2sdv 15665	. 2 ⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ ℕ₀ ((𝑥↑𝑘) / (!‘𝑘)) = Σ𝑘 ∈ ℕ₀ ((𝐴↑𝑘) / (!‘𝑘)))
4		df-ef 16032	. 2 ⊢ exp = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ ℕ₀ ((𝑥↑𝑘) / (!‘𝑘)))
5		sumex 15650	. 2 ⊢ Σ𝑘 ∈ ℕ₀ ((𝐴↑𝑘) / (!‘𝑘)) ∈ V
6	3, 4, 5	fvmpt 6947	1 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ₀ ((𝐴↑𝑘) / (!‘𝑘)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ‘cfv 6498 (class class class)co 7367 ℂcc 11036 / cdiv 11807 ℕ₀cn0 12437 ↑cexp 14023 !cfa 14235 Σcsu 15648 expce 16026

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pr 5375

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-iota 6454 df-fun 6500 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-seq 13964 df-sum 15649 df-ef 16032

This theorem is referenced by: esum 16045 efval2 16049 efcvg 16050 reefcl 16052 efaddlem 16058 eflegeo 16088 subfaclim 35370

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