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Mirrors > Home > ILE Home > Th. List > fsum00 | Unicode version |
Description: A sum of nonnegative numbers is zero iff all terms are zero. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
fsumge0.1 | |
fsumge0.2 | |
fsumge0.3 |
Ref | Expression |
---|---|
fsum00 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumge0.1 | . . . . . . . . . 10 | |
2 | 1 | adantr 274 | . . . . . . . . 9 |
3 | fsumge0.2 | . . . . . . . . . 10 | |
4 | 3 | adantlr 469 | . . . . . . . . 9 |
5 | fsumge0.3 | . . . . . . . . . 10 | |
6 | 5 | adantlr 469 | . . . . . . . . 9 |
7 | snssi 3712 | . . . . . . . . . 10 | |
8 | 7 | adantl 275 | . . . . . . . . 9 |
9 | snfig 6772 | . . . . . . . . . 10 | |
10 | 9 | adantl 275 | . . . . . . . . 9 |
11 | 2, 4, 6, 8, 10 | fsumlessfi 11388 | . . . . . . . 8 |
12 | 11 | adantlr 469 | . . . . . . 7 |
13 | simpr 109 | . . . . . . . 8 | |
14 | 3, 5 | jca 304 | . . . . . . . . . . . . 13 |
15 | 14 | ralrimiva 2537 | . . . . . . . . . . . 12 |
16 | 15 | adantr 274 | . . . . . . . . . . 11 |
17 | nfcsb1v 3074 | . . . . . . . . . . . . . 14 | |
18 | 17 | nfel1 2317 | . . . . . . . . . . . . 13 |
19 | nfcv 2306 | . . . . . . . . . . . . . 14 | |
20 | nfcv 2306 | . . . . . . . . . . . . . 14 | |
21 | 19, 20, 17 | nfbr 4023 | . . . . . . . . . . . . 13 |
22 | 18, 21 | nfan 1552 | . . . . . . . . . . . 12 |
23 | csbeq1a 3050 | . . . . . . . . . . . . . 14 | |
24 | 23 | eleq1d 2233 | . . . . . . . . . . . . 13 |
25 | 23 | breq2d 3989 | . . . . . . . . . . . . 13 |
26 | 24, 25 | anbi12d 465 | . . . . . . . . . . . 12 |
27 | 22, 26 | rspc 2820 | . . . . . . . . . . 11 |
28 | 16, 27 | mpan9 279 | . . . . . . . . . 10 |
29 | 28 | simpld 111 | . . . . . . . . 9 |
30 | 29 | recnd 7919 | . . . . . . . 8 |
31 | sumsns 11343 | . . . . . . . 8 | |
32 | 13, 30, 31 | syl2anc 409 | . . . . . . 7 |
33 | simplr 520 | . . . . . . 7 | |
34 | 12, 32, 33 | 3brtr3d 4008 | . . . . . 6 |
35 | 28 | simprd 113 | . . . . . 6 |
36 | 0re 7891 | . . . . . . 7 | |
37 | letri3 7971 | . . . . . . 7 | |
38 | 29, 36, 37 | sylancl 410 | . . . . . 6 |
39 | 34, 35, 38 | mpbir2and 933 | . . . . 5 |
40 | 39 | ralrimiva 2537 | . . . 4 |
41 | nfv 1515 | . . . . 5 | |
42 | 17 | nfeq1 2316 | . . . . 5 |
43 | 23 | eqeq1d 2173 | . . . . 5 |
44 | 41, 42, 43 | cbvral 2686 | . . . 4 |
45 | 40, 44 | sylibr 133 | . . 3 |
46 | 45 | ex 114 | . 2 |
47 | isumz 11317 | . . . . 5 DECID | |
48 | 47 | olcs 726 | . . . 4 |
49 | sumeq2 11287 | . . . . 5 | |
50 | 49 | eqeq1d 2173 | . . . 4 |
51 | 48, 50 | syl5ibrcom 156 | . . 3 |
52 | 1, 51 | syl 14 | . 2 |
53 | 46, 52 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 DECID wdc 824 w3a 967 wceq 1342 wcel 2135 wral 2442 csb 3041 wss 3112 csn 3571 class class class wbr 3977 cfv 5183 cfn 6698 cc 7743 cr 7744 cc0 7745 cle 7926 cz 9183 cuz 9458 csu 11281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4092 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 ax-cnex 7836 ax-resscn 7837 ax-1cn 7838 ax-1re 7839 ax-icn 7840 ax-addcl 7841 ax-addrcl 7842 ax-mulcl 7843 ax-mulrcl 7844 ax-addcom 7845 ax-mulcom 7846 ax-addass 7847 ax-mulass 7848 ax-distr 7849 ax-i2m1 7850 ax-0lt1 7851 ax-1rid 7852 ax-0id 7853 ax-rnegex 7854 ax-precex 7855 ax-cnre 7856 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 ax-pre-apti 7860 ax-pre-ltadd 7861 ax-pre-mulgt0 7862 ax-pre-mulext 7863 ax-arch 7864 ax-caucvg 7865 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2724 df-sbc 2948 df-csb 3042 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-if 3517 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-iun 3863 df-br 3978 df-opab 4039 df-mpt 4040 df-tr 4076 df-id 4266 df-po 4269 df-iso 4270 df-iord 4339 df-on 4341 df-ilim 4342 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-isom 5192 df-riota 5793 df-ov 5840 df-oprab 5841 df-mpo 5842 df-1st 6101 df-2nd 6102 df-recs 6265 df-irdg 6330 df-frec 6351 df-1o 6376 df-oadd 6380 df-er 6493 df-en 6699 df-dom 6700 df-fin 6701 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-sub 8063 df-neg 8064 df-reap 8465 df-ap 8472 df-div 8561 df-inn 8850 df-2 8908 df-3 8909 df-4 8910 df-n0 9107 df-z 9184 df-uz 9459 df-q 9550 df-rp 9582 df-ico 9822 df-fz 9937 df-fzo 10069 df-seqfrec 10372 df-exp 10446 df-ihash 10679 df-cj 10771 df-re 10772 df-im 10773 df-rsqrt 10927 df-abs 10928 df-clim 11207 df-sumdc 11282 |
This theorem is referenced by: (None) |
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