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Mirrors > Home > ILE Home > Th. List > fsum3cvg2 | Unicode version |
Description: The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) (Revised by Jim Kingdon, 2-Dec-2022.) |
Ref | Expression |
---|---|
fsumsers.1 |
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fsumsers.2 |
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fsumsers.3 |
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fsumsers.dc |
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fsumsers.4 |
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Ref | Expression |
---|---|
fsum3cvg2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2229 |
. . . 4
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2 | nfv 1467 |
. . . . 5
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3 | nfcsb1v 2964 |
. . . . 5
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4 | nfcv 2229 |
. . . . 5
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5 | 2, 3, 4 | nfif 3423 |
. . . 4
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6 | eleq1w 2149 |
. . . . 5
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7 | csbeq1a 2942 |
. . . . 5
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8 | 6, 7 | ifbieq1d 3417 |
. . . 4
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9 | 1, 5, 8 | cbvmpt 3939 |
. . 3
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10 | fsumsers.3 |
. . . . 5
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11 | 10 | ralrimiva 2447 |
. . . 4
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12 | 3 | nfel1 2240 |
. . . . 5
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13 | 7 | eleq1d 2157 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 12, 13 | rspc 2717 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 11, 14 | mpan9 276 |
. . 3
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16 | 6 | dcbid 787 |
. . . 4
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17 | fsumsers.dc |
. . . . . 6
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18 | 17 | ralrimiva 2447 |
. . . . 5
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19 | 18 | adantr 271 |
. . . 4
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20 | simpr 109 |
. . . 4
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21 | 16, 19, 20 | rspcdva 2728 |
. . 3
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22 | fsumsers.2 |
. . 3
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23 | fsumsers.4 |
. . 3
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24 | 9, 15, 21, 22, 23 | fsum3cvg 10821 |
. 2
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25 | eluzel2 9078 |
. . . 4
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26 | 22, 25 | syl 14 |
. . 3
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27 | fveq2 5318 |
. . . . 5
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28 | 27 | eleq1d 2157 |
. . . 4
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29 | fsumsers.1 |
. . . . . . 7
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30 | 10 | adantlr 462 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 0cnd 7535 |
. . . . . . . 8
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32 | 30, 31, 17 | ifcldadc 3424 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 29, 32 | eqeltrd 2165 |
. . . . . 6
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34 | 33 | ralrimiva 2447 |
. . . . 5
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35 | 34 | adantr 271 |
. . . 4
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36 | simpr 109 |
. . . 4
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37 | 28, 35, 36 | rspcdva 2728 |
. . 3
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38 | eluzelz 9082 |
. . . . . . 7
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39 | eqid 2089 |
. . . . . . . 8
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40 | 39 | fvmpt2 5399 |
. . . . . . 7
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41 | 38, 32, 40 | syl2an2 562 |
. . . . . 6
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42 | 29, 41 | eqtr4d 2124 |
. . . . 5
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43 | 42 | ralrimiva 2447 |
. . . 4
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44 | nffvmpt1 5329 |
. . . . . 6
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45 | 44 | nfeq2 2241 |
. . . . 5
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46 | fveq2 5318 |
. . . . . 6
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47 | fveq2 5318 |
. . . . . 6
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48 | 46, 47 | eqeq12d 2103 |
. . . . 5
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49 | 45, 48 | rspc 2717 |
. . . 4
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50 | 43, 49 | mpan9 276 |
. . 3
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51 | addcl 7521 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
52 | 51 | adantl 272 |
. . 3
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53 | 26, 37, 50, 52 | seq3feq 9951 |
. 2
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54 | 53 | fveq1d 5320 |
. 2
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55 | 24, 53, 54 | 3brtr4d 3881 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-mulrcl 7498 ax-addcom 7499 ax-mulcom 7500 ax-addass 7501 ax-mulass 7502 ax-distr 7503 ax-i2m1 7504 ax-0lt1 7505 ax-1rid 7506 ax-0id 7507 ax-rnegex 7508 ax-precex 7509 ax-cnre 7510 ax-pre-ltirr 7511 ax-pre-ltwlin 7512 ax-pre-lttrn 7513 ax-pre-apti 7514 ax-pre-ltadd 7515 ax-pre-mulgt0 7516 ax-pre-mulext 7517 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-if 3398 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-po 4132 df-iso 4133 df-iord 4202 df-on 4204 df-ilim 4205 df-suc 4207 df-iom 4419 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-frec 6170 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-sub 7709 df-neg 7710 df-reap 8106 df-ap 8113 df-div 8194 df-inn 8477 df-2 8535 df-n0 8728 df-z 8805 df-uz 9074 df-rp 9189 df-fz 9479 df-iseq 9907 df-seq3 9908 df-exp 10009 df-cj 10330 df-rsqrt 10485 df-abs 10486 df-clim 10721 |
This theorem is referenced by: fsum3cvg3 10843 ef0lem 11004 |
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