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Mirrors > Home > MPE Home > Th. List > sum0 | Structured version Visualization version GIF version |
Description: Any sum over the empty set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.) |
Ref | Expression |
---|---|
sum0 | β’ Ξ£π β β π΄ = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 12813 | . . . 4 β’ β = (β€β₯β1) | |
2 | 1z 12540 | . . . . 5 β’ 1 β β€ | |
3 | 2 | a1i 11 | . . . 4 β’ (β€ β 1 β β€) |
4 | 0ss 4361 | . . . . 5 β’ β β β | |
5 | 4 | a1i 11 | . . . 4 β’ (β€ β β β β) |
6 | simpr 486 | . . . . . . 7 β’ ((β€ β§ π β β) β π β β) | |
7 | 6, 1 | eleqtrdi 2848 | . . . . . 6 β’ ((β€ β§ π β β) β π β (β€β₯β1)) |
8 | c0ex 11156 | . . . . . . 7 β’ 0 β V | |
9 | 8 | fvconst2 7158 | . . . . . 6 β’ (π β (β€β₯β1) β (((β€β₯β1) Γ {0})βπ) = 0) |
10 | 7, 9 | syl 17 | . . . . 5 β’ ((β€ β§ π β β) β (((β€β₯β1) Γ {0})βπ) = 0) |
11 | noel 4295 | . . . . . 6 β’ Β¬ π β β | |
12 | 11 | iffalsei 4501 | . . . . 5 β’ if(π β β , π΄, 0) = 0 |
13 | 10, 12 | eqtr4di 2795 | . . . 4 β’ ((β€ β§ π β β) β (((β€β₯β1) Γ {0})βπ) = if(π β β , π΄, 0)) |
14 | 11 | pm2.21i 119 | . . . . 5 β’ (π β β β π΄ β β) |
15 | 14 | adantl 483 | . . . 4 β’ ((β€ β§ π β β ) β π΄ β β) |
16 | 1, 3, 5, 13, 15 | zsum 15610 | . . 3 β’ (β€ β Ξ£π β β π΄ = ( β βseq1( + , ((β€β₯β1) Γ {0})))) |
17 | 16 | mptru 1549 | . 2 β’ Ξ£π β β π΄ = ( β βseq1( + , ((β€β₯β1) Γ {0}))) |
18 | fclim 15442 | . . . 4 β’ β :dom β βΆβ | |
19 | ffun 6676 | . . . 4 β’ ( β :dom β βΆβ β Fun β ) | |
20 | 18, 19 | ax-mp 5 | . . 3 β’ Fun β |
21 | serclim0 15466 | . . . 4 β’ (1 β β€ β seq1( + , ((β€β₯β1) Γ {0})) β 0) | |
22 | 2, 21 | ax-mp 5 | . . 3 β’ seq1( + , ((β€β₯β1) Γ {0})) β 0 |
23 | funbrfv 6898 | . . 3 β’ (Fun β β (seq1( + , ((β€β₯β1) Γ {0})) β 0 β ( β βseq1( + , ((β€β₯β1) Γ {0}))) = 0)) | |
24 | 20, 22, 23 | mp2 9 | . 2 β’ ( β βseq1( + , ((β€β₯β1) Γ {0}))) = 0 |
25 | 17, 24 | eqtri 2765 | 1 β’ Ξ£π β β π΄ = 0 |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 397 = wceq 1542 β€wtru 1543 β wcel 2107 β wss 3915 β c0 4287 ifcif 4491 {csn 4591 class class class wbr 5110 Γ cxp 5636 dom cdm 5638 Fun wfun 6495 βΆwf 6497 βcfv 6501 βcc 11056 0cc0 11058 1c1 11059 + caddc 11061 βcn 12160 β€cz 12506 β€β₯cuz 12770 seqcseq 13913 β cli 15373 Ξ£csu 15577 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-rp 12923 df-fz 13432 df-fzo 13575 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 |
This theorem is referenced by: sumz 15614 fsumf1o 15615 fsumcllem 15624 fsumadd 15632 fsum2d 15663 fsumrev2 15674 fsummulc2 15676 fsumconst 15682 modfsummod 15686 fsumabs 15693 telfsumo 15694 fsumparts 15698 fsumrelem 15699 fsumrlim 15703 fsumo1 15704 fsumiun 15713 isumsplit 15732 arisum 15752 arisum2 15753 pwdif 15760 bpoly0 15940 sumeven 16276 sumodd 16277 bitsinv1 16329 bitsinvp1 16336 prmreclem4 16798 prmreclem5 16799 gsumfsum 20880 fsumcn 24249 ovolfiniun 24881 volfiniun 24927 itg10 25068 itgfsum 25207 dvmptfsum 25355 abelthlem6 25811 logfac 25972 log2ublem3 26314 harmonicbnd3 26373 cht1 26530 dchrisumlem1 26853 dchrisumlem3 26855 logdivbnd 26920 pntrsumbnd2 26931 pntrlog2bndlem4 26944 finsumvtxdg2size 28540 esumpcvgval 32717 signsvf0 33232 signsvf1 33233 repr0 33264 breprexplemc 33285 tgoldbachgtda 33314 mettrifi 36245 rrncmslem 36320 mccl 43913 dvmptfprod 44260 dvnprodlem3 44263 sge0rnn0 44683 sge00 44691 sge0sn 44694 |
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