eftval - Metamath Proof Explorer

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Theorem eftval 15522

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 7178	. . 3 ⊢ (𝑛 = 𝑁 → (𝐴↑𝑛) = (𝐴↑𝑁))
2		fveq2 6674	. . 3 ⊢ (𝑛 = 𝑁 → (!‘𝑛) = (!‘𝑁))
3	1, 2	oveq12d 7188	. 2 ⊢ (𝑛 = 𝑁 → ((𝐴↑𝑛) / (!‘𝑛)) = ((𝐴↑𝑁) / (!‘𝑁)))
4		eftval.1	. 2 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))
5		ovex 7203	. 2 ⊢ ((𝐴↑𝑁) / (!‘𝑁)) ∈ V
6	3, 4, 5	fvmpt 6775	1 ⊢ (𝑁 ∈ ℕ₀ → (𝐹‘𝑁) = ((𝐴↑𝑁) / (!‘𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ↦ cmpt 5110 ‘cfv 6339 (class class class)co 7170 / cdiv 11375 ℕ₀cn0 11976 ↑cexp 13521 !cfa 13725

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-iota 6297 df-fun 6341 df-fv 6347 df-ov 7173

This theorem is referenced by: efcllem 15523 ef0lem 15524 eff 15527 efval2 15529 efcvg 15530 efcvgfsum 15531 reefcl 15532 efcj 15537 efaddlem 15538 eftlcvg 15551 eftlcl 15552 reeftlcl 15553 eftlub 15554 efsep 15555 effsumlt 15556 efgt1p2 15559 efgt1p 15560 eflegeo 15566 eirrlem 15649 subfaclim 32721

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