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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 | ⊢ Σ𝑘 ∈ ∅ 𝐴 = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 12282 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1z 12013 | . . . . 5 ⊢ 1 ∈ ℤ | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → 1 ∈ ℤ) |
4 | 0ss 4350 | . . . . 5 ⊢ ∅ ⊆ ℕ | |
5 | 4 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ⊆ ℕ) |
6 | simpr 487 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) | |
7 | 6, 1 | eleqtrdi 2923 | . . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1)) |
8 | c0ex 10635 | . . . . . . 7 ⊢ 0 ∈ V | |
9 | 8 | fvconst2 6966 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘1) → (((ℤ≥‘1) × {0})‘𝑘) = 0) |
10 | 7, 9 | syl 17 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℤ≥‘1) × {0})‘𝑘) = 0) |
11 | noel 4296 | . . . . . 6 ⊢ ¬ 𝑘 ∈ ∅ | |
12 | 11 | iffalsei 4477 | . . . . 5 ⊢ if(𝑘 ∈ ∅, 𝐴, 0) = 0 |
13 | 10, 12 | syl6eqr 2874 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℤ≥‘1) × {0})‘𝑘) = if(𝑘 ∈ ∅, 𝐴, 0)) |
14 | 11 | pm2.21i 119 | . . . . 5 ⊢ (𝑘 ∈ ∅ → 𝐴 ∈ ℂ) |
15 | 14 | adantl 484 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ∅) → 𝐴 ∈ ℂ) |
16 | 1, 3, 5, 13, 15 | zsum 15075 | . . 3 ⊢ (⊤ → Σ𝑘 ∈ ∅ 𝐴 = ( ⇝ ‘seq1( + , ((ℤ≥‘1) × {0})))) |
17 | 16 | mptru 1544 | . 2 ⊢ Σ𝑘 ∈ ∅ 𝐴 = ( ⇝ ‘seq1( + , ((ℤ≥‘1) × {0}))) |
18 | fclim 14910 | . . . 4 ⊢ ⇝ :dom ⇝ ⟶ℂ | |
19 | ffun 6517 | . . . 4 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ ) | |
20 | 18, 19 | ax-mp 5 | . . 3 ⊢ Fun ⇝ |
21 | serclim0 14934 | . . . 4 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0) | |
22 | 2, 21 | ax-mp 5 | . . 3 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0 |
23 | funbrfv 6716 | . . 3 ⊢ (Fun ⇝ → (seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0 → ( ⇝ ‘seq1( + , ((ℤ≥‘1) × {0}))) = 0)) | |
24 | 20, 22, 23 | mp2 9 | . 2 ⊢ ( ⇝ ‘seq1( + , ((ℤ≥‘1) × {0}))) = 0 |
25 | 17, 24 | eqtri 2844 | 1 ⊢ Σ𝑘 ∈ ∅ 𝐴 = 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ⊆ wss 3936 ∅c0 4291 ifcif 4467 {csn 4567 class class class wbr 5066 × cxp 5553 dom cdm 5555 Fun wfun 6349 ⟶wf 6351 ‘cfv 6355 ℂcc 10535 0cc0 10537 1c1 10538 + caddc 10540 ℕcn 11638 ℤcz 11982 ℤ≥cuz 12244 seqcseq 13370 ⇝ cli 14841 Σcsu 15042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 |
This theorem is referenced by: sumz 15079 fsumf1o 15080 fsumcllem 15089 fsumadd 15096 fsum2d 15126 fsumrev2 15137 fsummulc2 15139 fsumconst 15145 modfsummod 15149 fsumabs 15156 telfsumo 15157 fsumparts 15161 fsumrelem 15162 fsumrlim 15166 fsumo1 15167 fsumiun 15176 isumsplit 15195 arisum 15215 arisum2 15216 pwdif 15223 bpoly0 15404 sumeven 15738 sumodd 15739 bitsinv1 15791 bitsinvp1 15798 prmreclem4 16255 prmreclem5 16256 gsumfsum 20612 fsumcn 23478 ovolfiniun 24102 volfiniun 24148 itg10 24289 itgfsum 24427 dvmptfsum 24572 abelthlem6 25024 logfac 25184 log2ublem3 25526 harmonicbnd3 25585 cht1 25742 dchrisumlem1 26065 dchrisumlem3 26067 logdivbnd 26132 pntrsumbnd2 26143 pntrlog2bndlem4 26156 finsumvtxdg2size 27332 esumpcvgval 31337 signsvf0 31850 signsvf1 31851 repr0 31882 breprexplemc 31903 tgoldbachgtda 31932 mettrifi 35047 rrncmslem 35125 mccl 41899 dvmptfprod 42250 dvnprodlem3 42253 sge0rnn0 42670 sge00 42678 sge0sn 42681 |
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