eftval - Metamath Proof Explorer

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Theorem eftval 16020

Description: The value of a term in the series expansion of the exponential function. (Contributed by Paul Chapman, 21-Aug-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)

**Hypothesis**
Ref	Expression
eftval.1	⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))

**Assertion**
Ref	Expression
eftval	⊢ (𝑁 ∈ ℕ₀ → (𝐹‘𝑁) = ((𝐴↑𝑁) / (!‘𝑁)))

Distinct variable groups: 𝐴,𝑛 𝑛,𝑁 Allowed substitution hint: 𝐹(𝑛)

**Proof of Theorem eftval**
Step	Hyp	Ref	Expression
1		oveq2 7417	. . 3 ⊢ (𝑛 = 𝑁 → (𝐴↑𝑛) = (𝐴↑𝑁))
2		fveq2 6892	. . 3 ⊢ (𝑛 = 𝑁 → (!‘𝑛) = (!‘𝑁))
3	1, 2	oveq12d 7427	. 2 ⊢ (𝑛 = 𝑁 → ((𝐴↑𝑛) / (!‘𝑛)) = ((𝐴↑𝑁) / (!‘𝑁)))
4		eftval.1	. 2 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))
5		ovex 7442	. 2 ⊢ ((𝐴↑𝑁) / (!‘𝑁)) ∈ V
6	3, 4, 5	fvmpt 6999	1 ⊢ (𝑁 ∈ ℕ₀ → (𝐹‘𝑁) = ((𝐴↑𝑁) / (!‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7409 / cdiv 11871 ℕ₀cn0 12472 ↑cexp 14027 !cfa 14233

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412

This theorem is referenced by: efcllem 16021 ef0lem 16022 eff 16025 efval2 16027 efcvg 16028 efcvgfsum 16029 reefcl 16030 efcj 16035 efaddlem 16036 eftlcvg 16049 eftlcl 16050 reeftlcl 16051 eftlub 16052 efsep 16053 effsumlt 16054 efgt1p2 16057 efgt1p 16058 eflegeo 16064 eirrlem 16147 subfaclim 34179

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