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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 2332 |
. . . 4
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2 | nfv 1539 |
. . . . 5
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3 | nfcsb1v 3105 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
4 | nfcv 2332 |
. . . . 5
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5 | 2, 3, 4 | nfif 3577 |
. . . 4
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6 | eleq1w 2250 |
. . . . 5
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7 | csbeq1a 3081 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
8 | 6, 7 | ifbieq1d 3571 |
. . . 4
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9 | 1, 5, 8 | cbvmpt 4113 |
. . 3
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10 | fsumsers.3 |
. . . . 5
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11 | 10 | ralrimiva 2563 |
. . . 4
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12 | 3 | nfel1 2343 |
. . . . 5
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13 | 7 | eleq1d 2258 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 12, 13 | rspc 2850 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
15 | 11, 14 | mpan9 281 |
. . 3
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16 | 6 | dcbid 839 |
. . . 4
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17 | fsumsers.dc |
. . . . . 6
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18 | 17 | ralrimiva 2563 |
. . . . 5
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19 | 18 | adantr 276 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | simpr 110 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 16, 19, 20 | rspcdva 2861 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | fsumsers.2 |
. . 3
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23 | fsumsers.4 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 9, 15, 21, 22, 23 | fsum3cvg 11413 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | eluzel2 9558 |
. . . 4
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26 | 22, 25 | syl 14 |
. . 3
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27 | fveq2 5531 |
. . . . 5
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28 | 27 | eleq1d 2258 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | fsumsers.1 |
. . . . . . 7
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30 | 10 | adantlr 477 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 0cnd 7975 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
32 | 30, 31, 17 | ifcldadc 3578 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
33 | 29, 32 | eqeltrd 2266 |
. . . . . 6
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34 | 33 | ralrimiva 2563 |
. . . . 5
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35 | 34 | adantr 276 |
. . . 4
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36 | simpr 110 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
37 | 28, 35, 36 | rspcdva 2861 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
38 | eluzelz 9562 |
. . . . . . 7
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39 | eqid 2189 |
. . . . . . . 8
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40 | 39 | fvmpt2 5616 |
. . . . . . 7
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41 | 38, 32, 40 | syl2an2 594 |
. . . . . 6
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42 | 29, 41 | eqtr4d 2225 |
. . . . 5
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43 | 42 | ralrimiva 2563 |
. . . 4
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44 | nffvmpt1 5542 |
. . . . . 6
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45 | 44 | nfeq2 2344 |
. . . . 5
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46 | fveq2 5531 |
. . . . . 6
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47 | fveq2 5531 |
. . . . . 6
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48 | 46, 47 | eqeq12d 2204 |
. . . . 5
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49 | 45, 48 | rspc 2850 |
. . . 4
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50 | 43, 49 | mpan9 281 |
. . 3
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51 | addcl 7961 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
52 | 51 | adantl 277 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
53 | 26, 37, 50, 52 | seq3feq 10498 |
. 2
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54 | 53 | fveq1d 5533 |
. 2
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55 | 24, 53, 54 | 3brtr4d 4050 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-mulrcl 7935 ax-addcom 7936 ax-mulcom 7937 ax-addass 7938 ax-mulass 7939 ax-distr 7940 ax-i2m1 7941 ax-0lt1 7942 ax-1rid 7943 ax-0id 7944 ax-rnegex 7945 ax-precex 7946 ax-cnre 7947 ax-pre-ltirr 7948 ax-pre-ltwlin 7949 ax-pre-lttrn 7950 ax-pre-apti 7951 ax-pre-ltadd 7952 ax-pre-mulgt0 7953 ax-pre-mulext 7954 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-1st 6160 df-2nd 6161 df-recs 6325 df-frec 6411 df-pnf 8019 df-mnf 8020 df-xr 8021 df-ltxr 8022 df-le 8023 df-sub 8155 df-neg 8156 df-reap 8557 df-ap 8564 df-div 8655 df-inn 8945 df-2 9003 df-n0 9202 df-z 9279 df-uz 9554 df-rp 9679 df-fz 10034 df-seqfrec 10472 df-exp 10546 df-cj 10878 df-rsqrt 11034 df-abs 11035 df-clim 11314 |
This theorem is referenced by: fsumsersdc 11430 fsum3cvg3 11431 ef0lem 11695 |
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