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Mirrors > Home > ILE Home > Th. List > fisumcvg | Unicode version |
Description: The sequence of partial sums of a finite sum converges to the whole sum. (Contributed by Mario Carneiro, 20-Apr-2014.) |
Ref | Expression |
---|---|
isummo.1 |
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isummo.2 |
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isummo.dc |
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isumrb.3 |
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fisumcvg.4 |
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Ref | Expression |
---|---|
fisumcvg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2083 |
. 2
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2 | isumrb.3 |
. . 3
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3 | eluzelz 8921 |
. . 3
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4 | 2, 3 | syl 14 |
. 2
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5 | iseqex 9740 |
. . 3
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6 | 5 | a1i 9 |
. 2
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7 | eqid 2083 |
. . . 4
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8 | eluzel2 8917 |
. . . . 5
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9 | 2, 8 | syl 14 |
. . . 4
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10 | eluzelz 8921 |
. . . . . . 7
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11 | 10 | adantl 271 |
. . . . . 6
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12 | iftrue 3378 |
. . . . . . . . . . 11
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13 | 12 | adantl 271 |
. . . . . . . . . 10
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14 | isummo.2 |
. . . . . . . . . 10
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15 | 13, 14 | eqeltrd 2159 |
. . . . . . . . 9
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16 | 15 | ex 113 |
. . . . . . . 8
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17 | 16 | adantr 270 |
. . . . . . 7
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18 | iffalse 3381 |
. . . . . . . . 9
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19 | 0cn 7381 |
. . . . . . . . 9
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20 | 18, 19 | syl6eqel 2173 |
. . . . . . . 8
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21 | 20 | a1i 9 |
. . . . . . 7
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22 | isummo.dc |
. . . . . . . 8
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23 | exmiddc 778 |
. . . . . . . 8
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24 | 22, 23 | syl 14 |
. . . . . . 7
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25 | 17, 21, 24 | mpjaod 671 |
. . . . . 6
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26 | isummo.1 |
. . . . . . 7
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27 | 26 | fvmpt2 5329 |
. . . . . 6
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28 | 11, 25, 27 | syl2anc 403 |
. . . . 5
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29 | 28, 25 | eqeltrd 2159 |
. . . 4
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30 | 7, 9, 29 | iserf 9766 |
. . 3
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31 | 30, 2 | ffvelrnd 5378 |
. 2
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32 | addid1 7521 |
. . . . 5
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33 | 32 | adantl 271 |
. . . 4
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34 | 2 | adantr 270 |
. . . 4
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35 | simpr 108 |
. . . 4
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36 | 31 | adantr 270 |
. . . 4
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37 | elfzuz 9329 |
. . . . . 6
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38 | eluzelz 8921 |
. . . . . . . . 9
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39 | 38 | adantl 271 |
. . . . . . . 8
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40 | fisumcvg.4 |
. . . . . . . . . . . 12
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41 | 40 | sseld 3009 |
. . . . . . . . . . 11
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42 | fznuz 9407 |
. . . . . . . . . . 11
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43 | 41, 42 | syl6 33 |
. . . . . . . . . 10
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44 | 43 | con2d 587 |
. . . . . . . . 9
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45 | 44 | imp 122 |
. . . . . . . 8
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46 | 39, 45 | eldifd 2994 |
. . . . . . 7
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47 | fveq2 5251 |
. . . . . . . . 9
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48 | 47 | eqeq1d 2091 |
. . . . . . . 8
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49 | eldifi 3106 |
. . . . . . . . . 10
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50 | eldifn 3107 |
. . . . . . . . . . . 12
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51 | 50, 18 | syl 14 |
. . . . . . . . . . 11
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52 | 51, 19 | syl6eqel 2173 |
. . . . . . . . . 10
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53 | 49, 52, 27 | syl2anc 403 |
. . . . . . . . 9
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54 | 53, 51 | eqtrd 2115 |
. . . . . . . 8
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55 | 48, 54 | vtoclga 2675 |
. . . . . . 7
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56 | 46, 55 | syl 14 |
. . . . . 6
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57 | 37, 56 | sylan2 280 |
. . . . 5
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58 | 57 | adantlr 461 |
. . . 4
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59 | cnex 7367 |
. . . . 5
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60 | 59 | a1i 9 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
61 | 47 | eleq1d 2151 |
. . . . 5
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62 | 29 | ralrimiva 2440 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
63 | 62 | ad2antrr 472 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
64 | simpr 108 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
65 | 61, 63, 64 | rspcdva 2717 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
66 | addcl 7368 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
67 | 66 | adantl 271 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
68 | 33, 34, 35, 36, 58, 60, 65, 67 | iseqid2 9781 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
69 | 68 | eqcomd 2088 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
70 | 1, 4, 6, 31, 69 | climconst 10501 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 ax-cnex 7337 ax-resscn 7338 ax-1cn 7339 ax-1re 7340 ax-icn 7341 ax-addcl 7342 ax-addrcl 7343 ax-mulcl 7344 ax-mulrcl 7345 ax-addcom 7346 ax-mulcom 7347 ax-addass 7348 ax-mulass 7349 ax-distr 7350 ax-i2m1 7351 ax-0lt1 7352 ax-1rid 7353 ax-0id 7354 ax-rnegex 7355 ax-precex 7356 ax-cnre 7357 ax-pre-ltirr 7358 ax-pre-ltwlin 7359 ax-pre-lttrn 7360 ax-pre-apti 7361 ax-pre-ltadd 7362 ax-pre-mulgt0 7363 ax-pre-mulext 7364 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-if 3374 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4083 df-po 4086 df-iso 4087 df-iord 4156 df-on 4158 df-ilim 4159 df-suc 4161 df-iom 4368 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-res 4411 df-ima 4412 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-f1 4972 df-fo 4973 df-f1o 4974 df-fv 4975 df-riota 5545 df-ov 5592 df-oprab 5593 df-mpt2 5594 df-1st 5844 df-2nd 5845 df-recs 6000 df-frec 6086 df-pnf 7425 df-mnf 7426 df-xr 7427 df-ltxr 7428 df-le 7429 df-sub 7556 df-neg 7557 df-reap 7950 df-ap 7957 df-div 8036 df-inn 8315 df-2 8373 df-n0 8564 df-z 8645 df-uz 8913 df-rp 9028 df-fz 9318 df-iseq 9739 df-iexp 9790 df-cj 10101 df-rsqrt 10256 df-abs 10257 df-clim 10490 |
This theorem is referenced by: (None) |
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