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Mirrors > Home > MPE Home > Th. List > sumz | Structured version Visualization version Unicode version |
Description: Any sum of zero over a summable set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.) |
Ref | Expression |
---|---|
sumz |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . . 5 | |
2 | simpr 477 | . . . . 5 | |
3 | simpl 473 | . . . . 5 | |
4 | c0ex 10034 | . . . . . . . 8 | |
5 | 4 | fvconst2 6469 | . . . . . . 7 |
6 | ifid 4125 | . . . . . . 7 | |
7 | 5, 6 | syl6eqr 2674 | . . . . . 6 |
8 | 7 | adantl 482 | . . . . 5 |
9 | 0cnd 10033 | . . . . 5 | |
10 | 1, 2, 3, 8, 9 | zsum 14449 | . . . 4 |
11 | fclim 14284 | . . . . . 6 | |
12 | ffun 6048 | . . . . . 6 | |
13 | 11, 12 | ax-mp 5 | . . . . 5 |
14 | serclim0 14308 | . . . . . 6 | |
15 | 14 | adantl 482 | . . . . 5 |
16 | funbrfv 6234 | . . . . 5 | |
17 | 13, 15, 16 | mpsyl 68 | . . . 4 |
18 | 10, 17 | eqtrd 2656 | . . 3 |
19 | uzf 11690 | . . . . . . . . 9 | |
20 | 19 | fdmi 6052 | . . . . . . . 8 |
21 | 20 | eleq2i 2693 | . . . . . . 7 |
22 | ndmfv 6218 | . . . . . . 7 | |
23 | 21, 22 | sylnbir 321 | . . . . . 6 |
24 | 23 | sseq2d 3633 | . . . . 5 |
25 | 24 | biimpac 503 | . . . 4 |
26 | ss0 3974 | . . . 4 | |
27 | sumeq1 14419 | . . . . 5 | |
28 | sum0 14452 | . . . . 5 | |
29 | 27, 28 | syl6eq 2672 | . . . 4 |
30 | 25, 26, 29 | 3syl 18 | . . 3 |
31 | 18, 30 | pm2.61dan 832 | . 2 |
32 | fz1f1o 14441 | . . 3 | |
33 | eqidd 2623 | . . . . . . . . 9 | |
34 | simpl 473 | . . . . . . . . 9 | |
35 | simpr 477 | . . . . . . . . 9 | |
36 | 0cnd 10033 | . . . . . . . . 9 | |
37 | elfznn 12370 | . . . . . . . . . . 11 | |
38 | 4 | fvconst2 6469 | . . . . . . . . . . 11 |
39 | 37, 38 | syl 17 | . . . . . . . . . 10 |
40 | 39 | adantl 482 | . . . . . . . . 9 |
41 | 33, 34, 35, 36, 40 | fsum 14451 | . . . . . . . 8 |
42 | nnuz 11723 | . . . . . . . . . 10 | |
43 | 42 | ser0 12853 | . . . . . . . . 9 |
44 | 43 | adantr 481 | . . . . . . . 8 |
45 | 41, 44 | eqtrd 2656 | . . . . . . 7 |
46 | 45 | ex 450 | . . . . . 6 |
47 | 46 | exlimdv 1861 | . . . . 5 |
48 | 47 | imp 445 | . . . 4 |
49 | 29, 48 | jaoi 394 | . . 3 |
50 | 32, 49 | syl 17 | . 2 |
51 | 31, 50 | jaoi 394 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wo 383 wa 384 wceq 1483 wex 1704 wcel 1990 wss 3574 c0 3915 cif 4086 cpw 4158 csn 4177 class class class wbr 4653 cxp 5112 cdm 5114 wfun 5882 wf 5884 wf1o 5887 cfv 5888 (class class class)co 6650 cfn 7955 cc 9934 cc0 9936 c1 9937 caddc 9939 cn 11020 cz 11377 cuz 11687 cfz 12326 cseq 12801 chash 13117 cli 14215 csu 14416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 |
This theorem is referenced by: fsum00 14530 fsumdvds 15030 pwp1fsum 15114 pcfac 15603 ovoliunnul 23275 vitalilem5 23381 itg1addlem5 23467 itg10a 23477 itg0 23546 itgz 23547 plymullem1 23970 coemullem 24006 logtayl 24406 ftalem5 24803 chp1 24893 logexprlim 24950 bposlem2 25010 rpvmasumlem 25176 axcgrid 25796 axlowdimlem16 25837 indsumin 30084 plymulx0 30624 signsplypnf 30627 fsum2dsub 30685 knoppndvlem6 32508 volsupnfl 33454 binomcxplemnn0 38548 binomcxplemnotnn0 38555 sumnnodd 39862 stoweidlem37 40254 fourierdlem103 40426 fourierdlem104 40427 etransclem24 40475 etransclem32 40483 etransclem35 40486 sge0z 40592 aacllem 42547 |
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