![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7366 | . . 3 ⊢ (𝑛 = 𝑁 → (𝐴↑𝑛) = (𝐴↑𝑁)) | |
2 | fveq2 6843 | . . 3 ⊢ (𝑛 = 𝑁 → (!‘𝑛) = (!‘𝑁)) | |
3 | 1, 2 | oveq12d 7376 | . 2 ⊢ (𝑛 = 𝑁 → ((𝐴↑𝑛) / (!‘𝑛)) = ((𝐴↑𝑁) / (!‘𝑁))) |
4 | eftval.1 | . 2 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
5 | ovex 7391 | . 2 ⊢ ((𝐴↑𝑁) / (!‘𝑁)) ∈ V | |
6 | 3, 4, 5 | fvmpt 6949 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝐹‘𝑁) = ((𝐴↑𝑁) / (!‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ↦ cmpt 5189 ‘cfv 6497 (class class class)co 7358 / cdiv 11817 ℕ0cn0 12418 ↑cexp 13973 !cfa 14179 |
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 5257 ax-nul 5264 ax-pr 5385 |
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 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 |
This theorem is referenced by: efcllem 15965 ef0lem 15966 eff 15969 efval2 15971 efcvg 15972 efcvgfsum 15973 reefcl 15974 efcj 15979 efaddlem 15980 eftlcvg 15993 eftlcl 15994 reeftlcl 15995 eftlub 15996 efsep 15997 effsumlt 15998 efgt1p2 16001 efgt1p 16002 eflegeo 16008 eirrlem 16091 subfaclim 33839 |
Copyright terms: Public domain | W3C validator |