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Mirrors > Home > MPE Home > Th. List > eftval | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
eftval.1 | ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) |
Ref | Expression |
---|---|
eftval | ⊢ (𝑁 ∈ ℕ0 → (𝐹‘𝑁) = ((𝐴↑𝑁) / (!‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7163 | . . 3 ⊢ (𝑛 = 𝑁 → (𝐴↑𝑛) = (𝐴↑𝑁)) | |
2 | fveq2 6669 | . . 3 ⊢ (𝑛 = 𝑁 → (!‘𝑛) = (!‘𝑁)) | |
3 | 1, 2 | oveq12d 7173 | . 2 ⊢ (𝑛 = 𝑁 → ((𝐴↑𝑛) / (!‘𝑛)) = ((𝐴↑𝑁) / (!‘𝑁))) |
4 | eftval.1 | . 2 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
5 | ovex 7188 | . 2 ⊢ ((𝐴↑𝑁) / (!‘𝑁)) ∈ V | |
6 | 3, 4, 5 | fvmpt 6767 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝐹‘𝑁) = ((𝐴↑𝑁) / (!‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ↦ cmpt 5145 ‘cfv 6354 (class class class)co 7155 / cdiv 11296 ℕ0cn0 11896 ↑cexp 13428 !cfa 13632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 |
This theorem is referenced by: efcllem 15430 ef0lem 15431 eff 15434 efval2 15436 efcvg 15437 efcvgfsum 15438 reefcl 15439 efcj 15444 efaddlem 15445 eftlcvg 15458 eftlcl 15459 reeftlcl 15460 eftlub 15461 efsep 15462 effsumlt 15463 efgt1p2 15466 efgt1p 15467 eflegeo 15473 eirrlem 15556 subfaclim 32435 |
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