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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 15090 and fsump1i 15112, which should make our notation clear and from which, along with closure fsumcl 15078, 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 2892 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (𝑚 ∈ (𝑀...𝑁) ↔ 𝑘 ∈ (𝑀...𝑁))) | |
2 | fveq2 6663 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘)) | |
3 | 1, 2 | ifbieq1d 4486 | . . . . 5 ⊢ (𝑚 = 𝑘 → if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0) = if(𝑘 ∈ (𝑀...𝑁), (𝐹‘𝑘), 0)) |
4 | eqid 2818 | . . . . 5 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0)) = (𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0)) | |
5 | fvex 6676 | . . . . . 6 ⊢ (𝐹‘𝑘) ∈ V | |
6 | c0ex 10623 | . . . . . 6 ⊢ 0 ∈ V | |
7 | 5, 6 | ifex 4511 | . . . . 5 ⊢ if(𝑘 ∈ (𝑀...𝑁), (𝐹‘𝑘), 0) ∈ V |
8 | 3, 4, 7 | fvmpt 6761 | . . . 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 4493 | . . . 4 ⊢ (𝜑 → if(𝑘 ∈ (𝑀...𝑁), (𝐹‘𝑘), 0) = if(𝑘 ∈ (𝑀...𝑁), 𝐴, 0)) |
11 | 8, 10 | sylan9eqr 2875 | . . 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 3987 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ⊆ (𝑀...𝑁)) | |
15 | 11, 12, 13, 14 | fsumsers 15073 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( + , (𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0)))‘𝑁)) |
16 | elfzuz 12892 | . . . . . 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 4469 | . . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑁) → if(𝑘 ∈ (𝑀...𝑁), (𝐹‘𝑘), 0) = (𝐹‘𝑘)) | |
19 | 17, 18 | eqtrd 2853 | . . . 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 13382 | . 2 ⊢ (𝜑 → (seq𝑀( + , (𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0)))‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁)) |
22 | 15, 21 | eqtrd 2853 | 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 4463 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 + caddc 10528 ℤ≥cuz 12231 ...cfz 12880 seqcseq 13357 Σcsu 15030 |
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 |
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 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-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 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-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 |
This theorem is referenced by: isumclim3 15102 seqabs 15157 cvgcmpce 15161 isumsplit 15183 climcndslem1 15192 climcndslem2 15193 climcnds 15194 trireciplem 15205 geolim 15214 geo2lim 15219 mertenslem2 15229 mertens 15230 efcvgfsum 15427 effsumlt 15452 prmreclem6 16245 prmrec 16246 ovollb2lem 24016 ovoliunlem1 24030 ovoliun2 24034 ovolscalem1 24041 ovolicc2lem4 24048 uniioovol 24107 uniioombllem3 24113 uniioombllem6 24116 mtest 24919 mtestbdd 24920 psercn2 24938 pserdvlem2 24943 abelthlem6 24951 logfac 25111 emcllem5 25504 lgamcvg2 25559 basellem8 25592 prmorcht 25682 pclogsum 25718 dchrisumlem2 25993 dchrmusum2 25997 dchrvmasumiflem1 26004 dchrisum0re 26016 dchrisum0lem1b 26018 dchrisum0lem2a 26020 dchrisum0lem2 26021 esumpcvgval 31236 esumcvg 31244 esumcvgsum 31246 knoppcnlem11 33739 fsumsermpt 41736 sumnnodd 41787 fourierdlem112 42380 sge0isum 42586 sge0seq 42605 |
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