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Mirrors > Home > ILE Home > Th. List > climserle | Unicode version |
Description: The partial sums of a converging infinite series with nonnegative terms are bounded by its limit. (Contributed by NM, 27-Dec-2005.) (Revised by Mario Carneiro, 9-Feb-2014.) |
Ref | Expression |
---|---|
clim2iser.1 |
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climserle.2 |
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climserle.3 |
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climserle.4 |
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climserle.5 |
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Ref | Expression |
---|---|
climserle |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clim2iser.1 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | climserle.2 |
. 2
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3 | climserle.3 |
. 2
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4 | 2, 1 | syl6eleq 2180 |
. . . . 5
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5 | eluzel2 9024 |
. . . . 5
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6 | 4, 5 | syl 14 |
. . . 4
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7 | climserle.4 |
. . . 4
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8 | 1, 6, 7 | serfre 9901 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | 8 | ffvelrnda 5434 |
. 2
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10 | 1 | peano2uzs 9072 |
. . . . 5
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11 | fveq2 5305 |
. . . . . . . . 9
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12 | 11 | breq2d 3857 |
. . . . . . . 8
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13 | 12 | imbi2d 228 |
. . . . . . 7
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14 | climserle.5 |
. . . . . . . 8
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15 | 14 | expcom 114 |
. . . . . . 7
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16 | 13, 15 | vtoclga 2685 |
. . . . . 6
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17 | 16 | impcom 123 |
. . . . 5
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18 | 10, 17 | sylan2 280 |
. . . 4
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19 | 11 | eleq1d 2156 |
. . . . . . . . 9
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20 | 19 | imbi2d 228 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 7 | expcom 114 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 20, 21 | vtoclga 2685 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | 22 | impcom 123 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
24 | 10, 23 | sylan2 280 |
. . . . 5
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25 | 9, 24 | addge01d 8010 |
. . . 4
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26 | 18, 25 | mpbid 145 |
. . 3
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27 | 1 | eleq2i 2154 |
. . . . . 6
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28 | 27 | biimpi 118 |
. . . . 5
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29 | 28 | adantl 271 |
. . . 4
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30 | simpll 496 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
31 | 1 | eleq2i 2154 |
. . . . . . 7
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32 | 31 | biimpri 131 |
. . . . . 6
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33 | 32 | adantl 271 |
. . . . 5
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34 | 30, 33, 7 | syl2anc 403 |
. . . 4
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35 | readdcl 7468 |
. . . . 5
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36 | 35 | adantl 271 |
. . . 4
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37 | 29, 34, 36 | seq3p1 9884 |
. . 3
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38 | 26, 37 | breqtrrd 3871 |
. 2
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39 | 1, 2, 3, 9, 38 | climub 10733 |
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 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-iinf 4403 ax-cnex 7436 ax-resscn 7437 ax-1cn 7438 ax-1re 7439 ax-icn 7440 ax-addcl 7441 ax-addrcl 7442 ax-mulcl 7443 ax-mulrcl 7444 ax-addcom 7445 ax-mulcom 7446 ax-addass 7447 ax-mulass 7448 ax-distr 7449 ax-i2m1 7450 ax-0lt1 7451 ax-1rid 7452 ax-0id 7453 ax-rnegex 7454 ax-precex 7455 ax-cnre 7456 ax-pre-ltirr 7457 ax-pre-ltwlin 7458 ax-pre-lttrn 7459 ax-pre-apti 7460 ax-pre-ltadd 7461 ax-pre-mulgt0 7462 ax-pre-mulext 7463 ax-arch 7464 ax-caucvg 7465 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-nel 2351 df-ral 2364 df-rex 2365 df-reu 2366 df-rmo 2367 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-if 3394 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-id 4120 df-po 4123 df-iso 4124 df-iord 4193 df-on 4195 df-ilim 4196 df-suc 4198 df-iom 4406 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-recs 6070 df-frec 6156 df-pnf 7524 df-mnf 7525 df-xr 7526 df-ltxr 7527 df-le 7528 df-sub 7655 df-neg 7656 df-reap 8052 df-ap 8059 df-div 8140 df-inn 8423 df-2 8481 df-3 8482 df-4 8483 df-n0 8674 df-z 8751 df-uz 9020 df-rp 9135 df-fz 9425 df-iseq 9853 df-seq3 9854 df-exp 9955 df-cj 10276 df-re 10277 df-im 10278 df-rsqrt 10431 df-abs 10432 df-clim 10667 |
This theorem is referenced by: isumrpcl 10888 ege2le3 10961 |
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