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Mirrors > Home > MPE Home > Th. List > efsep | Structured version Visualization version GIF version |
Description: Separate out the next term of the power series expansion of the exponential function. The last hypothesis allows the separated terms to be rearranged as desired. (Contributed by Paul Chapman, 23-Nov-2007.) (Revised by Mario Carneiro, 29-Apr-2014.) |
Ref | Expression |
---|---|
efsep.1 | ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) |
efsep.2 | ⊢ 𝑁 = (𝑀 + 1) |
efsep.3 | ⊢ 𝑀 ∈ ℕ0 |
efsep.4 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
efsep.5 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
efsep.6 | ⊢ (𝜑 → (exp‘𝐴) = (𝐵 + Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘))) |
efsep.7 | ⊢ (𝜑 → (𝐵 + ((𝐴↑𝑀) / (!‘𝑀))) = 𝐷) |
Ref | Expression |
---|---|
efsep | ⊢ (𝜑 → (exp‘𝐴) = (𝐷 + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efsep.6 | . 2 ⊢ (𝜑 → (exp‘𝐴) = (𝐵 + Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘))) | |
2 | eqid 2818 | . . . . . 6 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
3 | efsep.3 | . . . . . . . 8 ⊢ 𝑀 ∈ ℕ0 | |
4 | 3 | nn0zi 11995 | . . . . . . 7 ⊢ 𝑀 ∈ ℤ |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
6 | eqidd 2819 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
7 | eluznn0 12305 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℕ0) | |
8 | 3, 7 | mpan 686 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℕ0) |
9 | efsep.1 | . . . . . . . . . 10 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
10 | 9 | eftval 15418 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
11 | 10 | adantl 482 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
12 | efsep.4 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
13 | eftcl 15415 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℂ) | |
14 | 12, 13 | sylan 580 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℂ) |
15 | 11, 14 | eqeltrd 2910 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℂ) |
16 | 8, 15 | sylan2 592 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ) |
17 | 9 | eftlcvg 15447 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
18 | 12, 3, 17 | sylancl 586 | . . . . . 6 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
19 | 2, 5, 6, 16, 18 | isum1p 15184 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) = ((𝐹‘𝑀) + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))(𝐹‘𝑘))) |
20 | 9 | eftval 15418 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ0 → (𝐹‘𝑀) = ((𝐴↑𝑀) / (!‘𝑀))) |
21 | 3, 20 | ax-mp 5 | . . . . . 6 ⊢ (𝐹‘𝑀) = ((𝐴↑𝑀) / (!‘𝑀)) |
22 | efsep.2 | . . . . . . . . 9 ⊢ 𝑁 = (𝑀 + 1) | |
23 | 22 | eqcomi 2827 | . . . . . . . 8 ⊢ (𝑀 + 1) = 𝑁 |
24 | 23 | fveq2i 6666 | . . . . . . 7 ⊢ (ℤ≥‘(𝑀 + 1)) = (ℤ≥‘𝑁) |
25 | 24 | sumeq1i 15043 | . . . . . 6 ⊢ Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))(𝐹‘𝑘) = Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘) |
26 | 21, 25 | oveq12i 7157 | . . . . 5 ⊢ ((𝐹‘𝑀) + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))(𝐹‘𝑘)) = (((𝐴↑𝑀) / (!‘𝑀)) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘)) |
27 | 19, 26 | syl6eq 2869 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) = (((𝐴↑𝑀) / (!‘𝑀)) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘))) |
28 | 27 | oveq2d 7161 | . . 3 ⊢ (𝜑 → (𝐵 + Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘)) = (𝐵 + (((𝐴↑𝑀) / (!‘𝑀)) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘)))) |
29 | efsep.5 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
30 | eftcl 15415 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) → ((𝐴↑𝑀) / (!‘𝑀)) ∈ ℂ) | |
31 | 12, 3, 30 | sylancl 586 | . . . 4 ⊢ (𝜑 → ((𝐴↑𝑀) / (!‘𝑀)) ∈ ℂ) |
32 | peano2nn0 11925 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ0) | |
33 | 3, 32 | ax-mp 5 | . . . . . 6 ⊢ (𝑀 + 1) ∈ ℕ0 |
34 | 22, 33 | eqeltri 2906 | . . . . 5 ⊢ 𝑁 ∈ ℕ0 |
35 | 9 | eftlcl 15448 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘) ∈ ℂ) |
36 | 12, 34, 35 | sylancl 586 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘) ∈ ℂ) |
37 | 29, 31, 36 | addassd 10651 | . . 3 ⊢ (𝜑 → ((𝐵 + ((𝐴↑𝑀) / (!‘𝑀))) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘)) = (𝐵 + (((𝐴↑𝑀) / (!‘𝑀)) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘)))) |
38 | 28, 37 | eqtr4d 2856 | . 2 ⊢ (𝜑 → (𝐵 + Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘)) = ((𝐵 + ((𝐴↑𝑀) / (!‘𝑀))) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘))) |
39 | efsep.7 | . . 3 ⊢ (𝜑 → (𝐵 + ((𝐴↑𝑀) / (!‘𝑀))) = 𝐷) | |
40 | 39 | oveq1d 7160 | . 2 ⊢ (𝜑 → ((𝐵 + ((𝐴↑𝑀) / (!‘𝑀))) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘)) = (𝐷 + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘))) |
41 | 1, 38, 40 | 3eqtrd 2857 | 1 ⊢ (𝜑 → (exp‘𝐴) = (𝐷 + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ↦ cmpt 5137 dom cdm 5548 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 1c1 10526 + caddc 10528 / cdiv 11285 ℕ0cn0 11885 ℤcz 11969 ℤ≥cuz 12231 seqcseq 13357 ↑cexp 13417 !cfa 13621 ⇝ cli 14829 Σcsu 15030 expce 15403 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-ico 12732 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-fac 13622 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 |
This theorem is referenced by: ef4p 15454 dveflem 24503 |
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