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Mirrors > Home > ILE Home > Th. List > fsumge1 | Unicode version |
Description: A sum of nonnegative numbers is greater than or equal to any one of its terms. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 4-Jun-2014.) |
Ref | Expression |
---|---|
fsumge0.1 |
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fsumge0.2 |
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fsumge0.3 |
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fsumge1.4 |
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fsumge1.5 |
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Ref | Expression |
---|---|
fsumge1 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumge1.5 |
. . 3
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2 | fsumge1.4 |
. . . . 5
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3 | 2 | eleq1d 2246 |
. . . 4
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4 | fsumge0.2 |
. . . . . 6
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5 | 4 | recnd 7981 |
. . . . 5
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6 | 5 | ralrimiva 2550 |
. . . 4
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7 | 3, 6, 1 | rspcdva 2846 |
. . 3
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8 | 2 | sumsn 11411 |
. . 3
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9 | 1, 7, 8 | syl2anc 411 |
. 2
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10 | fsumge0.1 |
. . 3
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11 | fsumge0.3 |
. . 3
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12 | 1 | snssd 3737 |
. . 3
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13 | snfig 6810 |
. . . 4
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14 | 1, 13 | syl 14 |
. . 3
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15 | 10, 4, 11, 12, 14 | fsumlessfi 11460 |
. 2
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16 | 9, 15 | eqbrtrrd 4026 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 ax-cnex 7898 ax-resscn 7899 ax-1cn 7900 ax-1re 7901 ax-icn 7902 ax-addcl 7903 ax-addrcl 7904 ax-mulcl 7905 ax-mulrcl 7906 ax-addcom 7907 ax-mulcom 7908 ax-addass 7909 ax-mulass 7910 ax-distr 7911 ax-i2m1 7912 ax-0lt1 7913 ax-1rid 7914 ax-0id 7915 ax-rnegex 7916 ax-precex 7917 ax-cnre 7918 ax-pre-ltirr 7919 ax-pre-ltwlin 7920 ax-pre-lttrn 7921 ax-pre-apti 7922 ax-pre-ltadd 7923 ax-pre-mulgt0 7924 ax-pre-mulext 7925 ax-arch 7926 ax-caucvg 7927 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-po 4295 df-iso 4296 df-iord 4365 df-on 4367 df-ilim 4368 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5176 df-fun 5216 df-fn 5217 df-f 5218 df-f1 5219 df-fo 5220 df-f1o 5221 df-fv 5222 df-isom 5223 df-riota 5827 df-ov 5874 df-oprab 5875 df-mpo 5876 df-1st 6137 df-2nd 6138 df-recs 6302 df-irdg 6367 df-frec 6388 df-1o 6413 df-oadd 6417 df-er 6531 df-en 6737 df-dom 6738 df-fin 6739 df-pnf 7989 df-mnf 7990 df-xr 7991 df-ltxr 7992 df-le 7993 df-sub 8125 df-neg 8126 df-reap 8527 df-ap 8534 df-div 8625 df-inn 8915 df-2 8973 df-3 8974 df-4 8975 df-n0 9172 df-z 9249 df-uz 9524 df-q 9615 df-rp 9649 df-ico 9889 df-fz 10004 df-fzo 10137 df-seqfrec 10440 df-exp 10514 df-ihash 10748 df-cj 10843 df-re 10844 df-im 10845 df-rsqrt 10999 df-abs 11000 df-clim 11279 df-sumdc 11354 |
This theorem is referenced by: mertenslemub 11534 |
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