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Mirrors > Home > MPE Home > Th. List > ef4p | Structured version Visualization version GIF version |
Description: Separate out the first four terms of the infinite series expansion of the exponential function. (Contributed by Paul Chapman, 19-Jan-2008.) (Revised by Mario Carneiro, 29-Apr-2014.) |
Ref | Expression |
---|---|
ef4p.1 | ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) |
Ref | Expression |
---|---|
ef4p | ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / 6)) + Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ef4p.1 | . 2 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
2 | df-4 12275 | . 2 ⊢ 4 = (3 + 1) | |
3 | 3nn0 12488 | . 2 ⊢ 3 ∈ ℕ0 | |
4 | id 22 | . 2 ⊢ (𝐴 ∈ ℂ → 𝐴 ∈ ℂ) | |
5 | ax-1cn 11165 | . . . 4 ⊢ 1 ∈ ℂ | |
6 | addcl 11189 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + 𝐴) ∈ ℂ) | |
7 | 5, 6 | mpan 687 | . . 3 ⊢ (𝐴 ∈ ℂ → (1 + 𝐴) ∈ ℂ) |
8 | sqcl 14081 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) ∈ ℂ) | |
9 | 8 | halfcld 12455 | . . 3 ⊢ (𝐴 ∈ ℂ → ((𝐴↑2) / 2) ∈ ℂ) |
10 | 7, 9 | addcld 11231 | . 2 ⊢ (𝐴 ∈ ℂ → ((1 + 𝐴) + ((𝐴↑2) / 2)) ∈ ℂ) |
11 | df-3 12274 | . . 3 ⊢ 3 = (2 + 1) | |
12 | 2nn0 12487 | . . 3 ⊢ 2 ∈ ℕ0 | |
13 | df-2 12273 | . . . 4 ⊢ 2 = (1 + 1) | |
14 | 1nn0 12486 | . . . 4 ⊢ 1 ∈ ℕ0 | |
15 | 5 | a1i 11 | . . . 4 ⊢ (𝐴 ∈ ℂ → 1 ∈ ℂ) |
16 | 1e0p1 12717 | . . . . 5 ⊢ 1 = (0 + 1) | |
17 | 0nn0 12485 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
18 | 0cnd 11205 | . . . . 5 ⊢ (𝐴 ∈ ℂ → 0 ∈ ℂ) | |
19 | 1 | efval2 16026 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 (𝐹‘𝑘)) |
20 | nn0uz 12862 | . . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0) | |
21 | 20 | sumeq1i 15642 | . . . . . . . 8 ⊢ Σ𝑘 ∈ ℕ0 (𝐹‘𝑘) = Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘) |
22 | 19, 21 | eqtr2di 2781 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘) = (exp‘𝐴)) |
23 | 22 | oveq2d 7418 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (0 + Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘)) = (0 + (exp‘𝐴))) |
24 | efcl 16024 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ) | |
25 | 24 | addlidd 11413 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (0 + (exp‘𝐴)) = (exp‘𝐴)) |
26 | 23, 25 | eqtr2d 2765 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (0 + Σ𝑘 ∈ (ℤ≥‘0)(𝐹‘𝑘))) |
27 | eft0val 16054 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → ((𝐴↑0) / (!‘0)) = 1) | |
28 | 27 | oveq2d 7418 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (0 + ((𝐴↑0) / (!‘0))) = (0 + 1)) |
29 | 0p1e1 12332 | . . . . . 6 ⊢ (0 + 1) = 1 | |
30 | 28, 29 | eqtrdi 2780 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (0 + ((𝐴↑0) / (!‘0))) = 1) |
31 | 1, 16, 17, 4, 18, 26, 30 | efsep 16052 | . . . 4 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (1 + Σ𝑘 ∈ (ℤ≥‘1)(𝐹‘𝑘))) |
32 | exp1 14031 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐴↑1) = 𝐴) | |
33 | fac1 14235 | . . . . . . . 8 ⊢ (!‘1) = 1 | |
34 | 33 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (!‘1) = 1) |
35 | 32, 34 | oveq12d 7420 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / (!‘1)) = (𝐴 / 1)) |
36 | div1 11901 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴) | |
37 | 35, 36 | eqtrd 2764 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((𝐴↑1) / (!‘1)) = 𝐴) |
38 | 37 | oveq2d 7418 | . . . 4 ⊢ (𝐴 ∈ ℂ → (1 + ((𝐴↑1) / (!‘1))) = (1 + 𝐴)) |
39 | 1, 13, 14, 4, 15, 31, 38 | efsep 16052 | . . 3 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((1 + 𝐴) + Σ𝑘 ∈ (ℤ≥‘2)(𝐹‘𝑘))) |
40 | fac2 14237 | . . . . . 6 ⊢ (!‘2) = 2 | |
41 | 40 | oveq2i 7413 | . . . . 5 ⊢ ((𝐴↑2) / (!‘2)) = ((𝐴↑2) / 2) |
42 | 41 | oveq2i 7413 | . . . 4 ⊢ ((1 + 𝐴) + ((𝐴↑2) / (!‘2))) = ((1 + 𝐴) + ((𝐴↑2) / 2)) |
43 | 42 | a1i 11 | . . 3 ⊢ (𝐴 ∈ ℂ → ((1 + 𝐴) + ((𝐴↑2) / (!‘2))) = ((1 + 𝐴) + ((𝐴↑2) / 2))) |
44 | 1, 11, 12, 4, 7, 39, 43 | efsep 16052 | . 2 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = (((1 + 𝐴) + ((𝐴↑2) / 2)) + Σ𝑘 ∈ (ℤ≥‘3)(𝐹‘𝑘))) |
45 | fac3 14238 | . . . . 5 ⊢ (!‘3) = 6 | |
46 | 45 | oveq2i 7413 | . . . 4 ⊢ ((𝐴↑3) / (!‘3)) = ((𝐴↑3) / 6) |
47 | 46 | oveq2i 7413 | . . 3 ⊢ (((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / (!‘3))) = (((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / 6)) |
48 | 47 | a1i 11 | . 2 ⊢ (𝐴 ∈ ℂ → (((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / (!‘3))) = (((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / 6))) |
49 | 1, 2, 3, 4, 10, 44, 48 | efsep 16052 | 1 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = ((((1 + 𝐴) + ((𝐴↑2) / 2)) + ((𝐴↑3) / 6)) + Σ𝑘 ∈ (ℤ≥‘4)(𝐹‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ↦ cmpt 5222 ‘cfv 6534 (class class class)co 7402 ℂcc 11105 0cc0 11107 1c1 11108 + caddc 11110 / cdiv 11869 2c2 12265 3c3 12266 4c4 12267 6c6 12269 ℕ0cn0 12470 ℤ≥cuz 12820 ↑cexp 14025 !cfa 14231 Σcsu 15630 expce 16003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-pm 8820 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-n0 12471 df-z 12557 df-uz 12821 df-rp 12973 df-ico 13328 df-fz 13483 df-fzo 13626 df-fl 13755 df-seq 13965 df-exp 14026 df-fac 14232 df-hash 14289 df-shft 15012 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 df-abs 15181 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15631 df-ef 16009 |
This theorem is referenced by: efi4p 16079 |
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