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Mirrors > Home > ILE Home > Th. List > ser3ge0 | Unicode version |
Description: A finite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 8-Feb-2014.) (Revised by Mario Carneiro, 27-May-2014.) |
Ref | Expression |
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ser3ge0.1 | |
ser3ge0.2 | |
ser3ge0.3 |
Ref | Expression |
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ser3ge0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ser3ge0.1 | . . 3 | |
2 | eluzfz2 9805 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | fveq2 5414 | . . . . 5 | |
5 | 4 | breq2d 3936 | . . . 4 |
6 | 5 | imbi2d 229 | . . 3 |
7 | fveq2 5414 | . . . . 5 | |
8 | 7 | breq2d 3936 | . . . 4 |
9 | 8 | imbi2d 229 | . . 3 |
10 | fveq2 5414 | . . . . 5 | |
11 | 10 | breq2d 3936 | . . . 4 |
12 | 11 | imbi2d 229 | . . 3 |
13 | fveq2 5414 | . . . . 5 | |
14 | 13 | breq2d 3936 | . . . 4 |
15 | 14 | imbi2d 229 | . . 3 |
16 | fveq2 5414 | . . . . . . 7 | |
17 | 16 | breq2d 3936 | . . . . . 6 |
18 | ser3ge0.3 | . . . . . . 7 | |
19 | 18 | ralrimiva 2503 | . . . . . 6 |
20 | eluzfz1 9804 | . . . . . . 7 | |
21 | 1, 20 | syl 14 | . . . . . 6 |
22 | 17, 19, 21 | rspcdva 2789 | . . . . 5 |
23 | eluzel2 9324 | . . . . . . 7 | |
24 | 1, 23 | syl 14 | . . . . . 6 |
25 | ser3ge0.2 | . . . . . 6 | |
26 | readdcl 7739 | . . . . . . 7 | |
27 | 26 | adantl 275 | . . . . . 6 |
28 | 24, 25, 27 | seq3-1 10226 | . . . . 5 |
29 | 22, 28 | breqtrrd 3951 | . . . 4 |
30 | 29 | a1i 9 | . . 3 |
31 | eqid 2137 | . . . . . . . . . . 11 | |
32 | 31, 24, 25, 27 | seqf 10227 | . . . . . . . . . 10 |
33 | 32 | ad2antrr 479 | . . . . . . . . 9 ..^ |
34 | elfzouz 9921 | . . . . . . . . . 10 ..^ | |
35 | 34 | ad2antlr 480 | . . . . . . . . 9 ..^ |
36 | 33, 35 | ffvelrnd 5549 | . . . . . . . 8 ..^ |
37 | fveq2 5414 | . . . . . . . . . . 11 | |
38 | 37 | eleq1d 2206 | . . . . . . . . . 10 |
39 | 25 | ralrimiva 2503 | . . . . . . . . . . 11 |
40 | 39 | adantr 274 | . . . . . . . . . 10 ..^ |
41 | peano2uz 9371 | . . . . . . . . . . . 12 | |
42 | 34, 41 | syl 14 | . . . . . . . . . . 11 ..^ |
43 | 42 | adantl 275 | . . . . . . . . . 10 ..^ |
44 | 38, 40, 43 | rspcdva 2789 | . . . . . . . . 9 ..^ |
45 | 44 | adantr 274 | . . . . . . . 8 ..^ |
46 | simpr 109 | . . . . . . . 8 ..^ | |
47 | 37 | breq2d 3936 | . . . . . . . . 9 |
48 | 19 | ad2antrr 479 | . . . . . . . . 9 ..^ |
49 | fzofzp1 9997 | . . . . . . . . . 10 ..^ | |
50 | 49 | ad2antlr 480 | . . . . . . . . 9 ..^ |
51 | 47, 48, 50 | rspcdva 2789 | . . . . . . . 8 ..^ |
52 | 36, 45, 46, 51 | addge0d 8277 | . . . . . . 7 ..^ |
53 | 25 | adantlr 468 | . . . . . . . . 9 ..^ |
54 | 53 | adantlr 468 | . . . . . . . 8 ..^ |
55 | 26 | adantl 275 | . . . . . . . 8 ..^ |
56 | 35, 54, 55 | seq3p1 10228 | . . . . . . 7 ..^ |
57 | 52, 56 | breqtrrd 3951 | . . . . . 6 ..^ |
58 | 57 | ex 114 | . . . . 5 ..^ |
59 | 58 | expcom 115 | . . . 4 ..^ |
60 | 59 | a2d 26 | . . 3 ..^ |
61 | 6, 9, 12, 15, 30, 60 | fzind2 10009 | . 2 |
62 | 3, 61 | mpcom 36 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2414 class class class wbr 3924 wf 5114 cfv 5118 (class class class)co 5767 cr 7612 cc0 7613 c1 7614 caddc 7616 cle 7794 cz 9047 cuz 9319 cfz 9783 ..^cfzo 9912 cseq 10211 |
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-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 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-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-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-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-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-fz 9784 df-fzo 9913 df-seqfrec 10212 |
This theorem is referenced by: ser3le 10284 |
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