|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > sumz | Structured version Visualization version GIF 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 | ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∨ 𝐴 ∈ Fin) → Σ𝑘 ∈ 𝐴 0 = 0) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . . . . 5 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
| 2 | simpr 484 | . . . . 5 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ) | |
| 3 | simpl 482 | . . . . 5 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → 𝐴 ⊆ (ℤ≥‘𝑀)) | |
| 4 | c0ex 11256 | . . . . . . . 8 ⊢ 0 ∈ V | |
| 5 | 4 | fvconst2 7225 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (((ℤ≥‘𝑀) × {0})‘𝑘) = 0) | 
| 6 | ifid 4565 | . . . . . . 7 ⊢ if(𝑘 ∈ 𝐴, 0, 0) = 0 | |
| 7 | 5, 6 | eqtr4di 2794 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (((ℤ≥‘𝑀) × {0})‘𝑘) = if(𝑘 ∈ 𝐴, 0, 0)) | 
| 8 | 7 | adantl 481 | . . . . 5 ⊢ (((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (((ℤ≥‘𝑀) × {0})‘𝑘) = if(𝑘 ∈ 𝐴, 0, 0)) | 
| 9 | 0cnd 11255 | . . . . 5 ⊢ (((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ 𝐴) → 0 ∈ ℂ) | |
| 10 | 1, 2, 3, 8, 9 | zsum 15755 | . . . 4 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 0 = ( ⇝ ‘seq𝑀( + , ((ℤ≥‘𝑀) × {0})))) | 
| 11 | fclim 15590 | . . . . . 6 ⊢ ⇝ :dom ⇝ ⟶ℂ | |
| 12 | ffun 6738 | . . . . . 6 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ ) | |
| 13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ Fun ⇝ | 
| 14 | serclim0 15614 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , ((ℤ≥‘𝑀) × {0})) ⇝ 0) | |
| 15 | 14 | adantl 481 | . . . . 5 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → seq𝑀( + , ((ℤ≥‘𝑀) × {0})) ⇝ 0) | 
| 16 | funbrfv 6956 | . . . . 5 ⊢ (Fun ⇝ → (seq𝑀( + , ((ℤ≥‘𝑀) × {0})) ⇝ 0 → ( ⇝ ‘seq𝑀( + , ((ℤ≥‘𝑀) × {0}))) = 0)) | |
| 17 | 13, 15, 16 | mpsyl 68 | . . . 4 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → ( ⇝ ‘seq𝑀( + , ((ℤ≥‘𝑀) × {0}))) = 0) | 
| 18 | 10, 17 | eqtrd 2776 | . . 3 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 0 = 0) | 
| 19 | uzf 12882 | . . . . . . . . 9 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
| 20 | 19 | fdmi 6746 | . . . . . . . 8 ⊢ dom ℤ≥ = ℤ | 
| 21 | 20 | eleq2i 2832 | . . . . . . 7 ⊢ (𝑀 ∈ dom ℤ≥ ↔ 𝑀 ∈ ℤ) | 
| 22 | ndmfv 6940 | . . . . . . 7 ⊢ (¬ 𝑀 ∈ dom ℤ≥ → (ℤ≥‘𝑀) = ∅) | |
| 23 | 21, 22 | sylnbir 331 | . . . . . 6 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅) | 
| 24 | 23 | sseq2d 4015 | . . . . 5 ⊢ (¬ 𝑀 ∈ ℤ → (𝐴 ⊆ (ℤ≥‘𝑀) ↔ 𝐴 ⊆ ∅)) | 
| 25 | 24 | biimpac 478 | . . . 4 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ ¬ 𝑀 ∈ ℤ) → 𝐴 ⊆ ∅) | 
| 26 | ss0 4401 | . . . 4 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) | |
| 27 | sumeq1 15726 | . . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 0 = Σ𝑘 ∈ ∅ 0) | |
| 28 | sum0 15758 | . . . . 5 ⊢ Σ𝑘 ∈ ∅ 0 = 0 | |
| 29 | 27, 28 | eqtrdi 2792 | . . . 4 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 0 = 0) | 
| 30 | 25, 26, 29 | 3syl 18 | . . 3 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ ¬ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 0 = 0) | 
| 31 | 18, 30 | pm2.61dan 812 | . 2 ⊢ (𝐴 ⊆ (ℤ≥‘𝑀) → Σ𝑘 ∈ 𝐴 0 = 0) | 
| 32 | fz1f1o 15747 | . . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) | |
| 33 | eqidd 2737 | . . . . . . . . 9 ⊢ (𝑘 = (𝑓‘𝑛) → 0 = 0) | |
| 34 | simpl 482 | . . . . . . . . 9 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (♯‘𝐴) ∈ ℕ) | |
| 35 | simpr 484 | . . . . . . . . 9 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) | |
| 36 | 0cnd 11255 | . . . . . . . . 9 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑘 ∈ 𝐴) → 0 ∈ ℂ) | |
| 37 | elfznn 13594 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → 𝑛 ∈ ℕ) | |
| 38 | 4 | fvconst2 7225 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((ℕ × {0})‘𝑛) = 0) | 
| 39 | 37, 38 | syl 17 | . . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → ((ℕ × {0})‘𝑛) = 0) | 
| 40 | 39 | adantl 481 | . . . . . . . . 9 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((ℕ × {0})‘𝑛) = 0) | 
| 41 | 33, 34, 35, 36, 40 | fsum 15757 | . . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 0 = (seq1( + , (ℕ × {0}))‘(♯‘𝐴))) | 
| 42 | nnuz 12922 | . . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1) | |
| 43 | 42 | ser0 14096 | . . . . . . . . 9 ⊢ ((♯‘𝐴) ∈ ℕ → (seq1( + , (ℕ × {0}))‘(♯‘𝐴)) = 0) | 
| 44 | 43 | adantr 480 | . . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (seq1( + , (ℕ × {0}))‘(♯‘𝐴)) = 0) | 
| 45 | 41, 44 | eqtrd 2776 | . . . . . . 7 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 0 = 0) | 
| 46 | 45 | ex 412 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 0 = 0)) | 
| 47 | 46 | exlimdv 1932 | . . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 0 = 0)) | 
| 48 | 47 | imp 406 | . . . 4 ⊢ (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 0 = 0) | 
| 49 | 29, 48 | jaoi 857 | . . 3 ⊢ ((𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → Σ𝑘 ∈ 𝐴 0 = 0) | 
| 50 | 32, 49 | syl 17 | . 2 ⊢ (𝐴 ∈ Fin → Σ𝑘 ∈ 𝐴 0 = 0) | 
| 51 | 31, 50 | jaoi 857 | 1 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∨ 𝐴 ∈ Fin) → Σ𝑘 ∈ 𝐴 0 = 0) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ⊆ wss 3950 ∅c0 4332 ifcif 4524 𝒫 cpw 4599 {csn 4625 class class class wbr 5142 × cxp 5682 dom cdm 5684 Fun wfun 6554 ⟶wf 6556 –1-1-onto→wf1o 6559 ‘cfv 6560 (class class class)co 7432 Fincfn 8986 ℂcc 11154 0cc0 11156 1c1 11157 + caddc 11159 ℕcn 12267 ℤcz 12615 ℤ≥cuz 12879 ...cfz 13548 seqcseq 14043 ♯chash 14370 ⇝ cli 15521 Σcsu 15723 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-fz 13549 df-fzo 13696 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-sum 15724 | 
| This theorem is referenced by: fsum00 15835 fsumdvds 16346 pwp1fsum 16429 pcfac 16938 ovoliunnul 25543 vitalilem5 25648 itg1addlem5 25736 itg10a 25746 itg0 25816 itgz 25817 plymullem1 26254 coemullem 26290 logtayl 26703 ftalem5 27121 chp1 27211 logexprlim 27270 bposlem2 27330 rpvmasumlem 27532 axcgrid 28932 axlowdimlem16 28973 indsumin 32848 elrgspnlem2 33248 plymulx0 34563 signsplypnf 34566 fsum2dsub 34623 knoppndvlem6 36519 volsupnfl 37673 binomcxplemnn0 44373 binomcxplemnotnn0 44380 sumnnodd 45650 stoweidlem37 46057 fourierdlem103 46229 fourierdlem104 46230 etransclem24 46278 etransclem32 46286 etransclem35 46289 sge0z 46395 aacllem 49375 | 
| Copyright terms: Public domain | W3C validator |