eftval - Metamath Proof Explorer

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Theorem eftval 16041

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 7375	. . 3 ⊢ (𝑛 = 𝑁 → (𝐴↑𝑛) = (𝐴↑𝑁))
2		fveq2 6840	. . 3 ⊢ (𝑛 = 𝑁 → (!‘𝑛) = (!‘𝑁))
3	1, 2	oveq12d 7385	. 2 ⊢ (𝑛 = 𝑁 → ((𝐴↑𝑛) / (!‘𝑛)) = ((𝐴↑𝑁) / (!‘𝑁)))
4		eftval.1	. 2 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))
5		ovex 7400	. 2 ⊢ ((𝐴↑𝑁) / (!‘𝑁)) ∈ V
6	3, 4, 5	fvmpt 6947	1 ⊢ (𝑁 ∈ ℕ₀ → (𝐹‘𝑁) = ((𝐴↑𝑁) / (!‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ↦ cmpt 5166 ‘cfv 6498 (class class class)co 7367 / cdiv 11807 ℕ₀cn0 12437 ↑cexp 14023 !cfa 14235

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-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-iota 6454 df-fun 6500 df-fv 6506 df-ov 7370

This theorem is referenced by: efcllem 16042 ef0lem 16043 eff 16046 efval2 16049 efcvg 16050 efcvgfsum 16051 reefcl 16052 efcj 16057 efaddlem 16058 eftlcvg 16073 eftlcl 16074 reeftlcl 16075 eftlub 16076 efsep 16077 effsumlt 16078 efgt1p2 16081 efgt1p 16082 eflegeo 16088 eirrlem 16171 subfaclim 35370

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