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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 468 | . . . . . . . . 9 |
5 | fsumge0.3 | . . . . . . . . . 10 | |
6 | 5 | adantlr 468 | . . . . . . . . 9 |
7 | snssi 3659 | . . . . . . . . . 10 | |
8 | 7 | adantl 275 | . . . . . . . . 9 |
9 | snfig 6701 | . . . . . . . . . 10 | |
10 | 9 | adantl 275 | . . . . . . . . 9 |
11 | 2, 4, 6, 8, 10 | fsumlessfi 11222 | . . . . . . . 8 |
12 | 11 | adantlr 468 | . . . . . . 7 |
13 | simpr 109 | . . . . . . . 8 | |
14 | 3, 5 | jca 304 | . . . . . . . . . . . . 13 |
15 | 14 | ralrimiva 2503 | . . . . . . . . . . . 12 |
16 | 15 | adantr 274 | . . . . . . . . . . 11 |
17 | nfcsb1v 3030 | . . . . . . . . . . . . . 14 | |
18 | 17 | nfel1 2290 | . . . . . . . . . . . . 13 |
19 | nfcv 2279 | . . . . . . . . . . . . . 14 | |
20 | nfcv 2279 | . . . . . . . . . . . . . 14 | |
21 | 19, 20, 17 | nfbr 3969 | . . . . . . . . . . . . 13 |
22 | 18, 21 | nfan 1544 | . . . . . . . . . . . 12 |
23 | csbeq1a 3007 | . . . . . . . . . . . . . 14 | |
24 | 23 | eleq1d 2206 | . . . . . . . . . . . . 13 |
25 | 23 | breq2d 3936 | . . . . . . . . . . . . 13 |
26 | 24, 25 | anbi12d 464 | . . . . . . . . . . . 12 |
27 | 22, 26 | rspc 2778 | . . . . . . . . . . 11 |
28 | 16, 27 | mpan9 279 | . . . . . . . . . 10 |
29 | 28 | simpld 111 | . . . . . . . . 9 |
30 | 29 | recnd 7787 | . . . . . . . 8 |
31 | sumsns 11177 | . . . . . . . 8 | |
32 | 13, 30, 31 | syl2anc 408 | . . . . . . 7 |
33 | simplr 519 | . . . . . . 7 | |
34 | 12, 32, 33 | 3brtr3d 3954 | . . . . . 6 |
35 | 28 | simprd 113 | . . . . . 6 |
36 | 0re 7759 | . . . . . . 7 | |
37 | letri3 7838 | . . . . . . 7 | |
38 | 29, 36, 37 | sylancl 409 | . . . . . 6 |
39 | 34, 35, 38 | mpbir2and 928 | . . . . 5 |
40 | 39 | ralrimiva 2503 | . . . 4 |
41 | nfv 1508 | . . . . 5 | |
42 | 17 | nfeq1 2289 | . . . . 5 |
43 | 23 | eqeq1d 2146 | . . . . 5 |
44 | 41, 42, 43 | cbvral 2648 | . . . 4 |
45 | 40, 44 | sylibr 133 | . . 3 |
46 | 45 | ex 114 | . 2 |
47 | isumz 11151 | . . . . 5 DECID | |
48 | 47 | olcs 725 | . . . 4 |
49 | sumeq2 11121 | . . . . 5 | |
50 | 49 | eqeq1d 2146 | . . . 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 819 w3a 962 wceq 1331 wcel 1480 wral 2414 csb 2998 wss 3066 csn 3522 class class class wbr 3924 cfv 5118 cfn 6627 cc 7611 cr 7612 cc0 7613 cle 7794 cz 9047 cuz 9319 csu 11115 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 ax-arch 7732 ax-caucvg 7733 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-isom 5127 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-frec 6281 df-1o 6306 df-oadd 6310 df-er 6422 df-en 6628 df-dom 6629 df-fin 6630 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-n0 8971 df-z 9048 df-uz 9320 df-q 9405 df-rp 9435 df-ico 9670 df-fz 9784 df-fzo 9913 df-seqfrec 10212 df-exp 10286 df-ihash 10515 df-cj 10607 df-re 10608 df-im 10609 df-rsqrt 10763 df-abs 10764 df-clim 11041 df-sumdc 11116 |
This theorem is referenced by: (None) |
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