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Mirrors > Home > ILE Home > Th. List > isumlessdc | Unicode version |
Description: A finite sum of nonnegative numbers is less than or equal to its limit. (Contributed by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
isumless.1 |
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isumless.2 |
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isumless.3 |
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isumless.4 |
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isumless.5 |
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isumless.dc |
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isumless.6 |
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isumless.7 |
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isumless.8 |
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Ref | Expression |
---|---|
isumlessdc |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumless.4 |
. . 3
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2 | isumless.dc |
. . 3
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3 | 1 | sselda 3170 |
. . . . 5
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4 | isumless.6 |
. . . . . 6
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5 | 4 | recnd 8005 |
. . . . 5
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6 | 3, 5 | syldan 282 |
. . . 4
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7 | 6 | ralrimiva 2563 |
. . 3
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8 | isumless.2 |
. . . . 5
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9 | isumless.1 |
. . . . . . 7
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10 | 9 | eqimssi 3226 |
. . . . . 6
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11 | 10 | a1i 9 |
. . . . 5
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12 | 9 | eleq2i 2256 |
. . . . . . . . . 10
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13 | 12 | biimpri 133 |
. . . . . . . . 9
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14 | 13 | orcd 734 |
. . . . . . . 8
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15 | df-dc 836 |
. . . . . . . 8
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16 | 14, 15 | sylibr 134 |
. . . . . . 7
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17 | 16 | rgen 2543 |
. . . . . 6
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18 | 17 | a1i 9 |
. . . . 5
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19 | 8, 11, 18 | 3jca 1179 |
. . . 4
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20 | 19 | orcd 734 |
. . 3
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21 | 1, 2, 7, 20 | isumss2 11420 |
. 2
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22 | simpr 110 |
. . . . 5
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23 | isumless.5 |
. . . . . . . 8
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24 | 23, 4 | eqeltrd 2266 |
. . . . . . 7
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25 | 24 | adantr 276 |
. . . . . 6
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26 | 0red 7977 |
. . . . . 6
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27 | 2 | r19.21bi 2578 |
. . . . . 6
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28 | 25, 26, 27 | ifcldadc 3578 |
. . . . 5
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29 | eleq1w 2250 |
. . . . . . 7
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30 | fveq2 5530 |
. . . . . . 7
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31 | 29, 30 | ifbieq1d 3571 |
. . . . . 6
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32 | eqid 2189 |
. . . . . 6
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33 | 31, 32 | fvmptg 5608 |
. . . . 5
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34 | 22, 28, 33 | syl2anc 411 |
. . . 4
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35 | 23 | ifeq1d 3566 |
. . . 4
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36 | 34, 35 | eqtrd 2222 |
. . 3
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37 | 35, 28 | eqeltrrd 2267 |
. . 3
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38 | 4 | leidd 8490 |
. . . 4
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39 | isumless.7 |
. . . 4
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40 | breq1 4021 |
. . . . 5
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41 | breq1 4021 |
. . . . 5
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42 | 40, 41 | ifbothdc 3582 |
. . . 4
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43 | 38, 39, 27, 42 | syl3anc 1249 |
. . 3
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44 | isumless.3 |
. . . 4
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45 | 13, 27 | sylan2 286 |
. . . 4
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46 | 9, 8, 44, 1, 45, 36, 6 | fsum3cvg3 11423 |
. . 3
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47 | isumless.8 |
. . 3
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48 | 9, 8, 36, 37, 23, 4, 43, 46, 47 | isumle 11522 |
. 2
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49 | 21, 48 | eqbrtrd 4040 |
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 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-mulrcl 7929 ax-addcom 7930 ax-mulcom 7931 ax-addass 7932 ax-mulass 7933 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-1rid 7937 ax-0id 7938 ax-rnegex 7939 ax-precex 7940 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 ax-pre-mulgt0 7947 ax-pre-mulext 7948 ax-arch 7949 ax-caucvg 7950 |
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 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-isom 5240 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-irdg 6389 df-frec 6410 df-1o 6435 df-oadd 6439 df-er 6553 df-en 6759 df-dom 6760 df-fin 6761 df-sup 7002 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-reap 8551 df-ap 8558 df-div 8649 df-inn 8939 df-2 8997 df-3 8998 df-4 8999 df-n0 9196 df-z 9273 df-uz 9548 df-q 9639 df-rp 9673 df-fz 10028 df-fzo 10162 df-seqfrec 10465 df-exp 10539 df-ihash 10775 df-cj 10870 df-re 10871 df-im 10872 df-rsqrt 11026 df-abs 11027 df-clim 11306 df-sumdc 11381 |
This theorem is referenced by: mertenslemi1 11562 |
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