Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > sum0 | Unicode 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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 9492 | . . . 4 | |
2 | 1zzd 9209 | . . . 4 | |
3 | 0ss 3442 | . . . . 5 | |
4 | 3 | a1i 9 | . . . 4 |
5 | simpr 109 | . . . . . . 7 | |
6 | 5, 1 | eleqtrdi 2257 | . . . . . 6 |
7 | c0ex 7884 | . . . . . . 7 | |
8 | 7 | fvconst2 5695 | . . . . . 6 |
9 | 6, 8 | syl 14 | . . . . 5 |
10 | noel 3408 | . . . . . 6 | |
11 | 10 | iffalsei 3524 | . . . . 5 |
12 | 9, 11 | eqtr4di 2215 | . . . 4 |
13 | noel 3408 | . . . . . . . 8 | |
14 | 13 | olci 722 | . . . . . . 7 |
15 | df-dc 825 | . . . . . . 7 DECID | |
16 | 14, 15 | mpbir 145 | . . . . . 6 DECID |
17 | 16 | rgenw 2519 | . . . . 5 DECID |
18 | 17 | a1i 9 | . . . 4 DECID |
19 | 10 | pm2.21i 636 | . . . . 5 |
20 | 19 | adantl 275 | . . . 4 |
21 | 1, 2, 4, 12, 18, 20 | zsumdc 11311 | . . 3 |
22 | 21 | mptru 1351 | . 2 |
23 | fclim 11221 | . . . 4 | |
24 | ffun 5334 | . . . 4 | |
25 | 23, 24 | ax-mp 5 | . . 3 |
26 | 1z 9208 | . . . 4 | |
27 | serclim0 11232 | . . . 4 | |
28 | 26, 27 | ax-mp 5 | . . 3 |
29 | funbrfv 5519 | . . 3 | |
30 | 25, 28, 29 | mp2 16 | . 2 |
31 | 22, 30 | eqtri 2185 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wo 698 DECID wdc 824 wceq 1342 wtru 1343 wcel 2135 wral 2442 wss 3111 c0 3404 cif 3515 csn 3570 class class class wbr 3976 cxp 4596 cdm 4598 wfun 5176 wf 5178 cfv 5182 cc 7742 cc0 7744 c1 7745 caddc 7747 cn 8848 cz 9182 cuz 9457 cseq 10370 cli 11205 csu 11280 |
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 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
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 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-frec 6350 df-1o 6375 df-oadd 6379 df-er 6492 df-en 6698 df-dom 6699 df-fin 6700 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fz 9936 df-fzo 10068 df-seqfrec 10371 df-exp 10445 df-ihash 10678 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-clim 11206 df-sumdc 11281 |
This theorem is referenced by: isumz 11316 fsumf1o 11317 fsumcllem 11326 fsumadd 11333 fsum2d 11362 fisumrev2 11373 fsummulc2 11375 fsumconst 11381 modfsummod 11385 fsumabs 11392 telfsumo 11393 fsumparts 11397 fsumrelem 11398 fsumiun 11404 isumsplit 11418 arisum 11425 arisum2 11426 cvgratnnlemseq 11453 fsumcncntop 13103 |
Copyright terms: Public domain | W3C validator |