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Mirrors > Home > ILE Home > Th. List > isum1p | GIF version |
Description: The infinite sum of a converging infinite series equals the first term plus the infinite sum of the rest of it. (Contributed by NM, 2-Jan-2006.) (Revised by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
isum1p.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
isum1p.3 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
isum1p.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) |
isum1p.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
isum1p.6 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
Ref | Expression |
---|---|
isum1p | ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ((𝐹‘𝑀) + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isum1p.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | eqid 2177 | . . 3 ⊢ (ℤ≥‘(𝑀 + 1)) = (ℤ≥‘(𝑀 + 1)) | |
3 | isum1p.3 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
4 | uzid 9540 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
5 | 3, 4 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
6 | peano2uz 9581 | . . . . 5 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (𝑀 + 1) ∈ (ℤ≥‘𝑀)) | |
7 | 5, 6 | syl 14 | . . . 4 ⊢ (𝜑 → (𝑀 + 1) ∈ (ℤ≥‘𝑀)) |
8 | 7, 1 | eleqtrrdi 2271 | . . 3 ⊢ (𝜑 → (𝑀 + 1) ∈ 𝑍) |
9 | isum1p.4 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
10 | isum1p.5 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) | |
11 | isum1p.6 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) | |
12 | 1, 2, 8, 9, 10, 11 | isumsplit 11494 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = (Σ𝑘 ∈ (𝑀...((𝑀 + 1) − 1))𝐴 + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))𝐴)) |
13 | 3 | zcnd 9374 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
14 | ax-1cn 7903 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
15 | pncan 8161 | . . . . . . 7 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) − 1) = 𝑀) | |
16 | 13, 14, 15 | sylancl 413 | . . . . . 6 ⊢ (𝜑 → ((𝑀 + 1) − 1) = 𝑀) |
17 | 16 | oveq2d 5890 | . . . . 5 ⊢ (𝜑 → (𝑀...((𝑀 + 1) − 1)) = (𝑀...𝑀)) |
18 | 17 | sumeq1d 11369 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...((𝑀 + 1) − 1))𝐴 = Σ𝑘 ∈ (𝑀...𝑀)𝐴) |
19 | elfzuz 10018 | . . . . . . 7 ⊢ (𝑘 ∈ (𝑀...𝑀) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
20 | 19, 1 | eleqtrrdi 2271 | . . . . . 6 ⊢ (𝑘 ∈ (𝑀...𝑀) → 𝑘 ∈ 𝑍) |
21 | 20, 9 | sylan2 286 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑀)) → (𝐹‘𝑘) = 𝐴) |
22 | 21 | sumeq2dv 11371 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑀)(𝐹‘𝑘) = Σ𝑘 ∈ (𝑀...𝑀)𝐴) |
23 | fveq2 5515 | . . . . . . 7 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀)) | |
24 | 23 | eleq1d 2246 | . . . . . 6 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑀) ∈ ℂ)) |
25 | 9, 10 | eqeltrd 2254 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
26 | 25 | ralrimiva 2550 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) |
27 | 5, 1 | eleqtrrdi 2271 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
28 | 24, 26, 27 | rspcdva 2846 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ) |
29 | 23 | fsum1 11415 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝐹‘𝑀) ∈ ℂ) → Σ𝑘 ∈ (𝑀...𝑀)(𝐹‘𝑘) = (𝐹‘𝑀)) |
30 | 3, 28, 29 | syl2anc 411 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑀)(𝐹‘𝑘) = (𝐹‘𝑀)) |
31 | 18, 22, 30 | 3eqtr2d 2216 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...((𝑀 + 1) − 1))𝐴 = (𝐹‘𝑀)) |
32 | 31 | oveq1d 5889 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...((𝑀 + 1) − 1))𝐴 + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))𝐴) = ((𝐹‘𝑀) + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))𝐴)) |
33 | 12, 32 | eqtrd 2210 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ((𝐹‘𝑀) + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 dom cdm 4626 ‘cfv 5216 (class class class)co 5874 ℂcc 7808 1c1 7811 + caddc 7813 − cmin 8126 ℤcz 9251 ℤ≥cuz 9526 ...cfz 10006 seqcseq 10442 ⇝ cli 11281 Σcsu 11356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-frec 6391 df-1o 6416 df-oadd 6420 df-er 6534 df-en 6740 df-dom 6741 df-fin 6742 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-n0 9175 df-z 9252 df-uz 9527 df-q 9618 df-rp 9652 df-fz 10007 df-fzo 10140 df-seqfrec 10443 df-exp 10517 df-ihash 10751 df-cj 10846 df-re 10847 df-im 10848 df-rsqrt 11002 df-abs 11003 df-clim 11282 df-sumdc 11357 |
This theorem is referenced by: isumnn0nn 11496 efsep 11694 |
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