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Mirrors > Home > ILE Home > Th. List > isumrpcl | Unicode version |
Description: The infinite sum of positive reals is positive. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
isumrpcl.1 |
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isumrpcl.2 |
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isumrpcl.3 |
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isumrpcl.4 |
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isumrpcl.5 |
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isumrpcl.6 |
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Ref | Expression |
---|---|
isumrpcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumrpcl.2 |
. . 3
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2 | isumrpcl.3 |
. . . . 5
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3 | isumrpcl.1 |
. . . . 5
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4 | 2, 3 | eleqtrdi 2280 |
. . . 4
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5 | eluzelz 9550 |
. . . 4
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6 | 4, 5 | syl 14 |
. . 3
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7 | uzss 9561 |
. . . . . . 7
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8 | 4, 7 | syl 14 |
. . . . . 6
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9 | 8, 1, 3 | 3sstr4g 3210 |
. . . . 5
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10 | 9 | sselda 3167 |
. . . 4
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11 | isumrpcl.4 |
. . . 4
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12 | 10, 11 | syldan 282 |
. . 3
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13 | isumrpcl.5 |
. . . . 5
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14 | 13 | rpred 9709 |
. . . 4
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15 | 10, 14 | syldan 282 |
. . 3
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16 | isumrpcl.6 |
. . . 4
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17 | 11, 13 | eqeltrd 2264 |
. . . . . 6
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18 | 17 | rpcnd 9711 |
. . . . 5
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19 | 3, 2, 18 | iserex 11360 |
. . . 4
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20 | 16, 19 | mpbid 147 |
. . 3
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21 | 1, 6, 12, 15, 20 | isumrecl 11450 |
. 2
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22 | fveq2 5527 |
. . . 4
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23 | 22 | eleq1d 2256 |
. . 3
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24 | 17 | ralrimiva 2560 |
. . 3
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25 | 23, 24, 2 | rspcdva 2858 |
. 2
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26 | 8 | sselda 3167 |
. . . . . 6
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27 | 26, 3 | eleqtrrdi 2281 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 27, 17 | syldan 282 |
. . . 4
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29 | rpaddcl 9690 |
. . . . 5
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30 | 29 | adantl 277 |
. . . 4
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31 | 6, 28, 30 | seq3-1 10473 |
. . 3
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32 | uzid 9555 |
. . . . . 6
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33 | 6, 32 | syl 14 |
. . . . 5
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34 | 33, 1 | eleqtrrdi 2281 |
. . . 4
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35 | 15 | recnd 7999 |
. . . . 5
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36 | 1, 6, 12, 35, 20 | isumclim2 11443 |
. . . 4
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37 | 9 | sseld 3166 |
. . . . . . 7
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38 | fveq2 5527 |
. . . . . . . . 9
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39 | 38 | eleq1d 2256 |
. . . . . . . 8
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40 | 39 | rspcv 2849 |
. . . . . . 7
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41 | 37, 24, 40 | syl6ci 1455 |
. . . . . 6
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42 | 41 | imp 124 |
. . . . 5
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43 | 42 | rpred 9709 |
. . . 4
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44 | 42 | rpge0d 9713 |
. . . 4
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45 | 1, 34, 36, 43, 44 | climserle 11366 |
. . 3
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46 | 31, 45 | eqbrtrrd 4039 |
. 2
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47 | 21, 25, 46 | rpgecld 9749 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-iinf 4599 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-mulrcl 7923 ax-addcom 7924 ax-mulcom 7925 ax-addass 7926 ax-mulass 7927 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-1rid 7931 ax-0id 7932 ax-rnegex 7933 ax-precex 7934 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-apti 7939 ax-pre-ltadd 7940 ax-pre-mulgt0 7941 ax-pre-mulext 7942 ax-arch 7943 ax-caucvg 7944 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rmo 2473 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-if 3547 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-po 4308 df-iso 4309 df-iord 4378 df-on 4380 df-ilim 4381 df-suc 4383 df-iom 4602 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-isom 5237 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-1st 6154 df-2nd 6155 df-recs 6319 df-irdg 6384 df-frec 6405 df-1o 6430 df-oadd 6434 df-er 6548 df-en 6754 df-dom 6755 df-fin 6756 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-reap 8545 df-ap 8552 df-div 8643 df-inn 8933 df-2 8991 df-3 8992 df-4 8993 df-n0 9190 df-z 9267 df-uz 9542 df-q 9633 df-rp 9667 df-fz 10022 df-fzo 10156 df-seqfrec 10459 df-exp 10533 df-ihash 10769 df-cj 10864 df-re 10865 df-im 10866 df-rsqrt 11020 df-abs 11021 df-clim 11300 df-sumdc 11375 |
This theorem is referenced by: effsumlt 11713 eirraplem 11797 |
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