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Mirrors > Home > MPE Home > Th. List > isum1p | Structured version Visualization version 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 2737 | . . 3 ⊢ (ℤ≥‘(𝑀 + 1)) = (ℤ≥‘(𝑀 + 1)) | |
3 | isum1p.3 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
4 | uzid 12785 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
5 | 3, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
6 | peano2uz 12833 | . . . . 5 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (𝑀 + 1) ∈ (ℤ≥‘𝑀)) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀 + 1) ∈ (ℤ≥‘𝑀)) |
8 | 7, 1 | eleqtrrdi 2849 | . . 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 15732 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = (Σ𝑘 ∈ (𝑀...((𝑀 + 1) − 1))𝐴 + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))𝐴)) |
13 | 3 | zcnd 12615 | . . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
14 | ax-1cn 11116 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
15 | pncan 11414 | . . . . . . 7 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑀 + 1) − 1) = 𝑀) | |
16 | 13, 14, 15 | sylancl 587 | . . . . . 6 ⊢ (𝜑 → ((𝑀 + 1) − 1) = 𝑀) |
17 | 16 | oveq2d 7378 | . . . . 5 ⊢ (𝜑 → (𝑀...((𝑀 + 1) − 1)) = (𝑀...𝑀)) |
18 | 17 | sumeq1d 15593 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...((𝑀 + 1) − 1))𝐴 = Σ𝑘 ∈ (𝑀...𝑀)𝐴) |
19 | elfzuz 13444 | . . . . . . 7 ⊢ (𝑘 ∈ (𝑀...𝑀) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
20 | 19, 1 | eleqtrrdi 2849 | . . . . . 6 ⊢ (𝑘 ∈ (𝑀...𝑀) → 𝑘 ∈ 𝑍) |
21 | 20, 9 | sylan2 594 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑀)) → (𝐹‘𝑘) = 𝐴) |
22 | 21 | sumeq2dv 15595 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑀)(𝐹‘𝑘) = Σ𝑘 ∈ (𝑀...𝑀)𝐴) |
23 | fveq2 6847 | . . . . . . 7 ⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀)) | |
24 | 23 | eleq1d 2823 | . . . . . 6 ⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑀) ∈ ℂ)) |
25 | 9, 10 | eqeltrd 2838 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
26 | 25 | ralrimiva 3144 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) |
27 | 5, 1 | eleqtrrdi 2849 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
28 | 24, 26, 27 | rspcdva 3585 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ) |
29 | 23 | fsum1 15639 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝐹‘𝑀) ∈ ℂ) → Σ𝑘 ∈ (𝑀...𝑀)(𝐹‘𝑘) = (𝐹‘𝑀)) |
30 | 3, 28, 29 | syl2anc 585 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑀)(𝐹‘𝑘) = (𝐹‘𝑀)) |
31 | 18, 22, 30 | 3eqtr2d 2783 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...((𝑀 + 1) − 1))𝐴 = (𝐹‘𝑀)) |
32 | 31 | oveq1d 7377 | . 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝑀...((𝑀 + 1) − 1))𝐴 + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))𝐴) = ((𝐹‘𝑀) + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))𝐴)) |
33 | 12, 32 | eqtrd 2777 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ((𝐹‘𝑀) + Σ𝑘 ∈ (ℤ≥‘(𝑀 + 1))𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 dom cdm 5638 ‘cfv 6501 (class class class)co 7362 ℂcc 11056 1c1 11059 + caddc 11061 − cmin 11392 ℤcz 12506 ℤ≥cuz 12770 ...cfz 13431 seqcseq 13913 ⇝ cli 15373 Σcsu 15577 |
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 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-fz 13432 df-fzo 13575 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 |
This theorem is referenced by: isumnn0nn 15734 efsep 15999 rpnnen2lem9 16111 binomcxplemnotnn0 42710 |
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