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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 12911 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1z 12638 | . . . . 5 ⊢ 1 ∈ ℤ | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → 1 ∈ ℤ) |
4 | 0ss 4394 | . . . . 5 ⊢ ∅ ⊆ ℕ | |
5 | 4 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ⊆ ℕ) |
6 | simpr 483 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) | |
7 | 6, 1 | eleqtrdi 2836 | . . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1)) |
8 | c0ex 11249 | . . . . . . 7 ⊢ 0 ∈ V | |
9 | 8 | fvconst2 7213 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘1) → (((ℤ≥‘1) × {0})‘𝑘) = 0) |
10 | 7, 9 | syl 17 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℤ≥‘1) × {0})‘𝑘) = 0) |
11 | noel 4330 | . . . . . 6 ⊢ ¬ 𝑘 ∈ ∅ | |
12 | 11 | iffalsei 4533 | . . . . 5 ⊢ if(𝑘 ∈ ∅, 𝐴, 0) = 0 |
13 | 10, 12 | eqtr4di 2784 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℤ≥‘1) × {0})‘𝑘) = if(𝑘 ∈ ∅, 𝐴, 0)) |
14 | 11 | pm2.21i 119 | . . . . 5 ⊢ (𝑘 ∈ ∅ → 𝐴 ∈ ℂ) |
15 | 14 | adantl 480 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ∅) → 𝐴 ∈ ℂ) |
16 | 1, 3, 5, 13, 15 | zsum 15717 | . . 3 ⊢ (⊤ → Σ𝑘 ∈ ∅ 𝐴 = ( ⇝ ‘seq1( + , ((ℤ≥‘1) × {0})))) |
17 | 16 | mptru 1541 | . 2 ⊢ Σ𝑘 ∈ ∅ 𝐴 = ( ⇝ ‘seq1( + , ((ℤ≥‘1) × {0}))) |
18 | fclim 15550 | . . . 4 ⊢ ⇝ :dom ⇝ ⟶ℂ | |
19 | ffun 6723 | . . . 4 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ ) | |
20 | 18, 19 | ax-mp 5 | . . 3 ⊢ Fun ⇝ |
21 | serclim0 15574 | . . . 4 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0) | |
22 | 2, 21 | ax-mp 5 | . . 3 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0 |
23 | funbrfv 6944 | . . 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 2754 | 1 ⊢ Σ𝑘 ∈ ∅ 𝐴 = 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1534 ⊤wtru 1535 ∈ wcel 2099 ⊆ wss 3946 ∅c0 4322 ifcif 4523 {csn 4623 class class class wbr 5145 × cxp 5672 dom cdm 5674 Fun wfun 6540 ⟶wf 6542 ‘cfv 6546 ℂcc 11147 0cc0 11149 1c1 11150 + caddc 11152 ℕcn 12258 ℤcz 12604 ℤ≥cuz 12868 seqcseq 14015 ⇝ cli 15481 Σcsu 15685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-inf2 9677 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-isom 6555 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9478 df-oi 9546 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-n0 12519 df-z 12605 df-uz 12869 df-rp 13023 df-fz 13533 df-fzo 13676 df-seq 14016 df-exp 14076 df-hash 14343 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-clim 15485 df-sum 15686 |
This theorem is referenced by: sumz 15721 fsumf1o 15722 fsumcllem 15731 fsumadd 15739 fsum2d 15770 fsumrev2 15781 fsummulc2 15783 fsumconst 15789 modfsummod 15793 fsumabs 15800 telfsumo 15801 fsumparts 15805 fsumrelem 15806 fsumrlim 15810 fsumo1 15811 fsumiun 15820 isumsplit 15839 arisum 15859 arisum2 15860 pwdif 15867 bpoly0 16047 sumeven 16384 sumodd 16385 bitsinv1 16437 bitsinvp1 16444 prmreclem4 16916 prmreclem5 16917 gsumfsum 21427 fsumcn 24876 ovolfiniun 25518 volfiniun 25564 itg10 25705 itgfsum 25844 dvmptfsum 25995 abelthlem6 26463 logfac 26625 log2ublem3 26973 harmonicbnd3 27033 cht1 27190 dchrisumlem1 27515 dchrisumlem3 27517 logdivbnd 27582 pntrsumbnd2 27593 pntrlog2bndlem4 27606 finsumvtxdg2size 29484 esumpcvgval 33924 signsvf0 34439 signsvf1 34440 repr0 34470 breprexplemc 34491 tgoldbachgtda 34520 mettrifi 37471 rrncmslem 37546 deg1gprod 41852 sumcubes 42040 mccl 45255 dvmptfprod 45602 dvnprodlem3 45605 sge0rnn0 46025 sge00 46033 sge0sn 46036 |
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