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Mirrors > Home > MPE Home > Th. List > fsumser | Structured version Unicode version |
Description: A finite sum expressed in terms of a partial sum of an infinite series. The recursive definition of follows as fsum1 13322 and fsump1i 13340, which should make our notation clear and from which, along with closure fsumcl 13314, 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 |
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fsumser.2 |
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fsumser.3 |
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Ref | Expression |
---|---|
fsumser |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2523 |
. . . . . 6
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2 | fveq2 5791 |
. . . . . 6
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3 | 1, 2 | ifbieq1d 3912 |
. . . . 5
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4 | eqid 2451 |
. . . . 5
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5 | fvex 5801 |
. . . . . 6
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6 | c0ex 9483 |
. . . . . 6
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7 | 5, 6 | ifex 3958 |
. . . . 5
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8 | 3, 4, 7 | fvmpt 5875 |
. . . 4
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9 | fsumser.1 |
. . . . 5
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10 | 9 | ifeq1da 3919 |
. . . 4
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11 | 8, 10 | sylan9eqr 2514 |
. . 3
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12 | fsumser.2 |
. . 3
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13 | fsumser.3 |
. . 3
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14 | ssid 3475 |
. . . 4
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15 | 14 | a1i 11 |
. . 3
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16 | 11, 12, 13, 15 | fsumsers 13309 |
. 2
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17 | elfzuz 11552 |
. . . . . 6
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18 | 17, 8 | syl 16 |
. . . . 5
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19 | iftrue 3897 |
. . . . 5
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20 | 18, 19 | eqtrd 2492 |
. . . 4
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21 | 20 | adantl 466 |
. . 3
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22 | 12, 21 | seqfveq 11933 |
. 2
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23 | 16, 22 | eqtrd 2492 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-rep 4503 ax-sep 4513 ax-nul 4521 ax-pow 4570 ax-pr 4631 ax-un 6474 ax-inf2 7950 ax-cnex 9441 ax-resscn 9442 ax-1cn 9443 ax-icn 9444 ax-addcl 9445 ax-addrcl 9446 ax-mulcl 9447 ax-mulrcl 9448 ax-mulcom 9449 ax-addass 9450 ax-mulass 9451 ax-distr 9452 ax-i2m1 9453 ax-1ne0 9454 ax-1rid 9455 ax-rnegex 9456 ax-rrecex 9457 ax-cnre 9458 ax-pre-lttri 9459 ax-pre-lttrn 9460 ax-pre-ltadd 9461 ax-pre-mulgt0 9462 ax-pre-sup 9463 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 df-rab 2804 df-v 3072 df-sbc 3287 df-csb 3389 df-dif 3431 df-un 3433 df-in 3435 df-ss 3442 df-pss 3444 df-nul 3738 df-if 3892 df-pw 3962 df-sn 3978 df-pr 3980 df-tp 3982 df-op 3984 df-uni 4192 df-int 4229 df-iun 4273 df-br 4393 df-opab 4451 df-mpt 4452 df-tr 4486 df-eprel 4732 df-id 4736 df-po 4741 df-so 4742 df-fr 4779 df-se 4780 df-we 4781 df-ord 4822 df-on 4823 df-lim 4824 df-suc 4825 df-xp 4946 df-rel 4947 df-cnv 4948 df-co 4949 df-dm 4950 df-rn 4951 df-res 4952 df-ima 4953 df-iota 5481 df-fun 5520 df-fn 5521 df-f 5522 df-f1 5523 df-fo 5524 df-f1o 5525 df-fv 5526 df-isom 5527 df-riota 6153 df-ov 6195 df-oprab 6196 df-mpt2 6197 df-om 6579 df-1st 6679 df-2nd 6680 df-recs 6934 df-rdg 6968 df-1o 7022 df-oadd 7026 df-er 7203 df-en 7413 df-dom 7414 df-sdom 7415 df-fin 7416 df-sup 7794 df-oi 7827 df-card 8212 df-pnf 9523 df-mnf 9524 df-xr 9525 df-ltxr 9526 df-le 9527 df-sub 9700 df-neg 9701 df-div 10097 df-nn 10426 df-2 10483 df-3 10484 df-n0 10683 df-z 10750 df-uz 10965 df-rp 11095 df-fz 11541 df-fzo 11652 df-seq 11910 df-exp 11969 df-hash 12207 df-cj 12692 df-re 12693 df-im 12694 df-sqr 12828 df-abs 12829 df-clim 13070 df-sum 13268 |
This theorem is referenced by: isumclim3 13330 seqabs 13381 cvgcmpce 13385 isumsplit 13407 climcndslem1 13416 climcndslem2 13417 climcnds 13418 trireciplem 13428 geolim 13434 geo2lim 13439 mertenslem2 13449 mertens 13450 efcvgfsum 13475 effsumlt 13499 prmreclem6 14086 prmrec 14087 ovollb2lem 21089 ovoliunlem1 21103 ovoliun2 21107 ovolscalem1 21114 ovolicc2lem4 21121 uniioovol 21177 uniioombllem3 21183 uniioombllem6 21186 mtest 21987 mtestbdd 21988 psercn2 22006 pserdvlem2 22011 abelthlem6 22019 logfac 22167 emcllem5 22511 basellem8 22543 prmorcht 22634 pclogsum 22672 dchrisumlem2 22857 dchrmusum2 22861 dchrvmasumiflem1 22868 dchrisum0re 22880 dchrisum0lem1b 22882 dchrisum0lem2a 22884 dchrisum0lem2 22885 esumpcvgval 26663 esumcvg 26671 lgamcvg2 27177 |
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