eftval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > eftval		Structured version Visualization version GIF version

Theorem eftval 16134

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 7418	. . 3 ⊢ (𝑛 = 𝑁 → (𝐴↑𝑛) = (𝐴↑𝑁))
2		fveq2 6881	. . 3 ⊢ (𝑛 = 𝑁 → (!‘𝑛) = (!‘𝑁))
3	1, 2	oveq12d 7428	. 2 ⊢ (𝑛 = 𝑁 → ((𝐴↑𝑛) / (!‘𝑛)) = ((𝐴↑𝑁) / (!‘𝑁)))
4		eftval.1	. 2 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((𝐴↑𝑛) / (!‘𝑛)))
5		ovex 7443	. 2 ⊢ ((𝐴↑𝑁) / (!‘𝑁)) ∈ V
6	3, 4, 5	fvmpt 6989	1 ⊢ (𝑁 ∈ ℕ₀ → (𝐹‘𝑁) = ((𝐴↑𝑁) / (!‘𝑁)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2143 ↦ cmpt 5192 ‘cfv 6536 (class class class)co 7410 / cdiv 11875 ℕ₀cn0 12508 ↑cexp 14102 !cfa 14314

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pr 5404

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-iota 6492 df-fun 6538 df-fv 6544 df-ov 7413

This theorem is used by: efcllem 16135 ef0lem 16136 eff 16139 efval2 16142 efcvg 16143 efcvgfsum 16144 reefcl 16145 efcj 16150 efaddlem 16151 eftlcvg 16166 eftlcl 16167 reeftlcl 16168 eftlub 16169 efsep 16170 effsumlt 16171 efgt1p2 16174 efgt1p 16175 eflegeo 16181 eirrlem 16264 subfaclim 35688

W3C validator