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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 14892 and fsump1i 14914, which should make our notation clear and from which, along with closure fsumcl 14880, 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 2842 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (𝑚 ∈ (𝑀...𝑁) ↔ 𝑘 ∈ (𝑀...𝑁))) | |
2 | fveq2 6448 | . . . . . 6 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘)) | |
3 | 1, 2 | ifbieq1d 4330 | . . . . 5 ⊢ (𝑚 = 𝑘 → if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0) = if(𝑘 ∈ (𝑀...𝑁), (𝐹‘𝑘), 0)) |
4 | eqid 2778 | . . . . 5 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0)) = (𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0)) | |
5 | fvex 6461 | . . . . . 6 ⊢ (𝐹‘𝑘) ∈ V | |
6 | c0ex 10372 | . . . . . 6 ⊢ 0 ∈ V | |
7 | 5, 6 | ifex 4355 | . . . . 5 ⊢ if(𝑘 ∈ (𝑀...𝑁), (𝐹‘𝑘), 0) ∈ V |
8 | 3, 4, 7 | fvmpt 6544 | . . . 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 4337 | . . . 4 ⊢ (𝜑 → if(𝑘 ∈ (𝑀...𝑁), (𝐹‘𝑘), 0) = if(𝑘 ∈ (𝑀...𝑁), 𝐴, 0)) |
11 | 8, 10 | sylan9eqr 2836 | . . 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 3843 | . . 3 ⊢ (𝜑 → (𝑀...𝑁) ⊆ (𝑀...𝑁)) | |
15 | 11, 12, 13, 14 | fsumsers 14875 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( + , (𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0)))‘𝑁)) |
16 | elfzuz 12660 | . . . . . 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 4313 | . . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑁) → if(𝑘 ∈ (𝑀...𝑁), (𝐹‘𝑘), 0) = (𝐹‘𝑘)) | |
19 | 17, 18 | eqtrd 2814 | . . . 4 ⊢ (𝑘 ∈ (𝑀...𝑁) → ((𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0))‘𝑘) = (𝐹‘𝑘)) |
20 | 19 | adantl 475 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0))‘𝑘) = (𝐹‘𝑘)) |
21 | 12, 20 | seqfveq 13148 | . 2 ⊢ (𝜑 → (seq𝑀( + , (𝑚 ∈ (ℤ≥‘𝑀) ↦ if(𝑚 ∈ (𝑀...𝑁), (𝐹‘𝑚), 0)))‘𝑁) = (seq𝑀( + , 𝐹)‘𝑁)) |
22 | 15, 21 | eqtrd 2814 | 1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀...𝑁)𝐴 = (seq𝑀( + , 𝐹)‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ifcif 4307 ↦ cmpt 4967 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 0cc0 10274 + caddc 10277 ℤ≥cuz 11997 ...cfz 12648 seqcseq 13124 Σcsu 14833 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-n0 11648 df-z 11734 df-uz 11998 df-rp 12143 df-fz 12649 df-fzo 12790 df-seq 13125 df-exp 13184 df-hash 13442 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-clim 14636 df-sum 14834 |
This theorem is referenced by: isumclim3 14904 seqabs 14959 cvgcmpce 14963 isumsplit 14985 climcndslem1 14994 climcndslem2 14995 climcnds 14996 trireciplem 15007 geolim 15014 geo2lim 15019 mertenslem2 15029 mertens 15030 efcvgfsum 15227 effsumlt 15252 prmreclem6 16040 prmrec 16041 ovollb2lem 23703 ovoliunlem1 23717 ovoliun2 23721 ovolscalem1 23728 ovolicc2lem4 23735 uniioovol 23794 uniioombllem3 23800 uniioombllem6 23803 mtest 24606 mtestbdd 24607 psercn2 24625 pserdvlem2 24630 abelthlem6 24638 logfac 24795 emcllem5 25189 lgamcvg2 25244 basellem8 25277 prmorcht 25367 pclogsum 25403 dchrisumlem2 25648 dchrmusum2 25652 dchrvmasumiflem1 25659 dchrisum0re 25671 dchrisum0lem1b 25673 dchrisum0lem2a 25675 dchrisum0lem2 25676 esumpcvgval 30746 esumcvg 30754 esumcvgsum 30756 knoppcnlem11 33084 fsumsermpt 40733 sumnnodd 40784 fourierdlem112 41376 sge0isum 41582 sge0seq 41601 |
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