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Mirrors > Home > MPE Home > Th. List > fsumser | Structured version Visualization version GIF version |
Description: A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition follows as fsum1 15092 and fsump1i 15114, which should make our notation clear and from which, along with closure fsumcl 15080, we will derive the basic properties of finite sums. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 21-Apr-2014.) |
Ref | Expression |
---|---|
fsumser.1 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = 𝐴) |
fsumser.2 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
fsumser.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
fsumser | ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( + , 𝐹)‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2895 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (𝑚 ∈ (𝑀...𝑁) ↔ 𝑘 ∈ (𝑀...𝑁))) | |
2 | fveq2 6664 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘)) | |
3 | 1, 2 | ifbieq1d 4488 | . . . . 5 ⊢ (𝑚 = 𝑘 → if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0) = if(𝑘 ∈ (𝑀...𝑁), (𝐹‘𝑘), 0)) |
4 | eqid 2821 | . . . . 5 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0)) = (𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0)) | |
5 | fvex 6677 | . . . . . 6 ⊢ (𝐹‘𝑘) ∈ V | |
6 | c0ex 10624 | . . . . . 6 ⊢ 0 ∈ V | |
7 | 5, 6 | ifex 4513 | . . . . 5 ⊢ if(𝑘 ∈ (𝑀...𝑁), (𝐹‘𝑘), 0) ∈ V |
8 | 3, 4, 7 | fvmpt 6762 | . . . 4 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → ((𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0))‘𝑘) = if(𝑘 ∈ (𝑀...𝑁), (𝐹‘𝑘), 0)) |
9 | fsumser.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) = 𝐴) | |
10 | 9 | ifeq1da 4495 | . . . 4 ⊢ (𝜑 → if(𝑘 ∈ (𝑀...𝑁), (𝐹‘𝑘), 0) = if(𝑘 ∈ (𝑀...𝑁), 𝐴, 0)) |
11 | 8, 10 | sylan9eqr 2878 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0))‘𝑘) = if(𝑘 ∈ (𝑀...𝑁), 𝐴, 0)) |
12 | fsumser.2 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
13 | fsumser.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐴 ∈ ℂ) | |
14 | ssidd 3989 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ⊆ (𝑀...𝑁)) | |
15 | 11, 12, 13, 14 | fsumsers 15075 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( + , (𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0)))‘𝑁)) |
16 | elfzuz 12894 | . . . . . 6 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ≥‘𝑀)) | |
17 | 16, 8 | syl 17 | . . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑁) → ((𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0))‘𝑘) = if(𝑘 ∈ (𝑀...𝑁), (𝐹‘𝑘), 0)) |
18 | iftrue 4471 | . . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑁) → if(𝑘 ∈ (𝑀...𝑁), (𝐹‘𝑘), 0) = (𝐹‘𝑘)) | |
19 | 17, 18 | eqtrd 2856 | . . . 4 ⊢ (𝑘 ∈ (𝑀...𝑁) → ((𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0))‘𝑘) = (𝐹‘𝑘)) |
20 | 19 | adantl 482 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0))‘𝑘) = (𝐹‘𝑘)) |
21 | 12, 20 | seqfveq 13384 | . 2 ⊢ (𝜑 → (seq𝑀( + , (𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0)))‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁)) |
22 | 15, 21 | eqtrd 2856 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( + , 𝐹)‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ifcif 4465 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7145 ℂcc 10524 0cc0 10526 + caddc 10529 ℤ≥cuz 12232 ...cfz 12882 seqcseq 13359 Σcsu 15032 |
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 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-sup 8895 df-oi 8963 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-n0 11887 df-z 11971 df-uz 12233 df-rp 12380 df-fz 12883 df-fzo 13024 df-seq 13360 df-exp 13420 df-hash 13681 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-clim 14835 df-sum 15033 |
This theorem is referenced by: isumclim3 15104 seqabs 15159 cvgcmpce 15163 isumsplit 15185 climcndslem1 15194 climcndslem2 15195 climcnds 15196 trireciplem 15207 geolim 15216 geo2lim 15221 mertenslem2 15231 mertens 15232 efcvgfsum 15429 effsumlt 15454 prmreclem6 16247 prmrec 16248 ovollb2lem 24018 ovoliunlem1 24032 ovoliun2 24036 ovolscalem1 24043 ovolicc2lem4 24050 uniioovol 24109 uniioombllem3 24115 uniioombllem6 24118 mtest 24921 mtestbdd 24922 psercn2 24940 pserdvlem2 24945 abelthlem6 24953 logfac 25111 emcllem5 25505 lgamcvg2 25560 basellem8 25593 prmorcht 25683 pclogsum 25719 dchrisumlem2 25994 dchrmusum2 25998 dchrvmasumiflem1 26005 dchrisum0re 26017 dchrisum0lem1b 26019 dchrisum0lem2a 26021 dchrisum0lem2 26022 esumpcvgval 31237 esumcvg 31245 esumcvgsum 31247 knoppcnlem11 33740 fsumsermpt 41740 sumnnodd 41791 fourierdlem112 42384 sge0isum 42590 sge0seq 42609 |
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