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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 2724 | . . . . . 6 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
3 | efsep.3 | . . . . . . . 8 ⊢ 𝑀 ∈ ℕ0 | |
4 | 3 | nn0zi 12586 | . . . . . . 7 ⊢ 𝑀 ∈ ℤ |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
6 | eqidd 2725 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
7 | eluznn0 12900 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℕ0 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℕ0) | |
8 | 3, 7 | mpan 687 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → 𝑘 ∈ ℕ0) |
9 | efsep.1 | . . . . . . . . . 10 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
10 | 9 | eftval 16022 | . . . . . . . . 9 ⊢ (𝑘 ∈ ℕ0 → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
11 | 10 | adantl 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
12 | efsep.4 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
13 | eftcl 16019 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℂ) | |
14 | 12, 13 | sylan 579 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℂ) |
15 | 11, 14 | eqeltrd 2825 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℂ) |
16 | 8, 15 | sylan2 592 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ) |
17 | 9 | eftlcvg 16052 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
18 | 12, 3, 17 | sylancl 585 | . . . . . 6 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
19 | 2, 5, 6, 16, 18 | isum1p 15789 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) = ((𝐹‘𝑀) + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))(𝐹‘𝑘))) |
20 | 9 | eftval 16022 | . . . . . . 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 2733 | . . . . . . . 8 ⊢ (𝑀 + 1) = 𝑁 |
24 | 23 | fveq2i 6885 | . . . . . . 7 ⊢ (ℤ≥‘(𝑀 + 1)) = (ℤ≥‘𝑁) |
25 | 24 | sumeq1i 15646 | . . . . . 6 ⊢ Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))(𝐹‘𝑘) = Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘) |
26 | 21, 25 | oveq12i 7414 | . . . . 5 ⊢ ((𝐹‘𝑀) + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))(𝐹‘𝑘)) = (((𝐴↑𝑀) / (!‘𝑀)) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘)) |
27 | 19, 26 | eqtrdi 2780 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘𝑀)(𝐹‘𝑘) = (((𝐴↑𝑀) / (!‘𝑀)) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘))) |
28 | 27 | oveq2d 7418 | . . 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 16019 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑀 ∈ ℕ0) → ((𝐴↑𝑀) / (!‘𝑀)) ∈ ℂ) | |
31 | 12, 3, 30 | sylancl 585 | . . . 4 ⊢ (𝜑 → ((𝐴↑𝑀) / (!‘𝑀)) ∈ ℂ) |
32 | peano2nn0 12511 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ0 → (𝑀 + 1) ∈ ℕ0) | |
33 | 3, 32 | ax-mp 5 | . . . . . 6 ⊢ (𝑀 + 1) ∈ ℕ0 |
34 | 22, 33 | eqeltri 2821 | . . . . 5 ⊢ 𝑁 ∈ ℕ0 |
35 | 9 | eftlcl 16053 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘) ∈ ℂ) |
36 | 12, 34, 35 | sylancl 585 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘) ∈ ℂ) |
37 | 29, 31, 36 | addassd 11235 | . . 3 ⊢ (𝜑 → ((𝐵 + ((𝐴↑𝑀) / (!‘𝑀))) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘)) = (𝐵 + (((𝐴↑𝑀) / (!‘𝑀)) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘)))) |
38 | 28, 37 | eqtr4d 2767 | . 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 7417 | . 2 ⊢ (𝜑 → ((𝐵 + ((𝐴↑𝑀) / (!‘𝑀))) + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘)) = (𝐷 + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘))) |
41 | 1, 38, 40 | 3eqtrd 2768 | 1 ⊢ (𝜑 → (exp‘𝐴) = (𝐷 + Σ𝑘 ∈ (ℤ≥‘𝑁)(𝐹‘𝑘))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ↦ cmpt 5222 dom cdm 5667 ‘cfv 6534 (class class class)co 7402 ℂcc 11105 1c1 11108 + caddc 11110 / cdiv 11870 ℕ0cn0 12471 ℤcz 12557 ℤ≥cuz 12821 seqcseq 13967 ↑cexp 14028 !cfa 14234 ⇝ cli 15430 Σcsu 15634 expce 16007 |
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 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12976 df-ico 13331 df-fz 13486 df-fzo 13629 df-fl 13758 df-seq 13968 df-exp 14029 df-fac 14235 df-hash 14292 df-shft 15016 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-limsup 15417 df-clim 15434 df-rlim 15435 df-sum 15635 |
This theorem is referenced by: ef4p 16059 dveflem 25855 |
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