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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 2818 | . . . . 5 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
2 | simpr 485 | . . . . 5 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ) | |
3 | simpl 483 | . . . . 5 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → 𝐴 ⊆ (ℤ≥‘𝑀)) | |
4 | c0ex 10623 | . . . . . . . 8 ⊢ 0 ∈ V | |
5 | 4 | fvconst2 6958 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (((ℤ≥‘𝑀) × {0})‘𝑘) = 0) |
6 | ifid 4502 | . . . . . . 7 ⊢ if(𝑘 ∈ 𝐴, 0, 0) = 0 | |
7 | 5, 6 | syl6eqr 2871 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (((ℤ≥‘𝑀) × {0})‘𝑘) = if(𝑘 ∈ 𝐴, 0, 0)) |
8 | 7 | adantl 482 | . . . . 5 ⊢ (((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (((ℤ≥‘𝑀) × {0})‘𝑘) = if(𝑘 ∈ 𝐴, 0, 0)) |
9 | 0cnd 10622 | . . . . 5 ⊢ (((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ 𝐴) → 0 ∈ ℂ) | |
10 | 1, 2, 3, 8, 9 | zsum 15063 | . . . 4 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 0 = ( ⇝ ‘seq𝑀( + , ((ℤ≥‘𝑀) × {0})))) |
11 | fclim 14898 | . . . . . 6 ⊢ ⇝ :dom ⇝ ⟶ℂ | |
12 | ffun 6510 | . . . . . 6 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ ) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ Fun ⇝ |
14 | serclim0 14922 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , ((ℤ≥‘𝑀) × {0})) ⇝ 0) | |
15 | 14 | adantl 482 | . . . . 5 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → seq𝑀( + , ((ℤ≥‘𝑀) × {0})) ⇝ 0) |
16 | funbrfv 6709 | . . . . 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 2853 | . . 3 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 0 = 0) |
19 | uzf 12234 | . . . . . . . . 9 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
20 | 19 | fdmi 6517 | . . . . . . . 8 ⊢ dom ℤ≥ = ℤ |
21 | 20 | eleq2i 2901 | . . . . . . 7 ⊢ (𝑀 ∈ dom ℤ≥ ↔ 𝑀 ∈ ℤ) |
22 | ndmfv 6693 | . . . . . . 7 ⊢ (¬ 𝑀 ∈ dom ℤ≥ → (ℤ≥‘𝑀) = ∅) | |
23 | 21, 22 | sylnbir 332 | . . . . . 6 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅) |
24 | 23 | sseq2d 3996 | . . . . 5 ⊢ (¬ 𝑀 ∈ ℤ → (𝐴 ⊆ (ℤ≥‘𝑀) ↔ 𝐴 ⊆ ∅)) |
25 | 24 | biimpac 479 | . . . 4 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ ¬ 𝑀 ∈ ℤ) → 𝐴 ⊆ ∅) |
26 | ss0 4349 | . . . 4 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) | |
27 | sumeq1 15033 | . . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 0 = Σ𝑘 ∈ ∅ 0) | |
28 | sum0 15066 | . . . . 5 ⊢ Σ𝑘 ∈ ∅ 0 = 0 | |
29 | 27, 28 | syl6eq 2869 | . . . 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 809 | . 2 ⊢ (𝐴 ⊆ (ℤ≥‘𝑀) → Σ𝑘 ∈ 𝐴 0 = 0) |
32 | fz1f1o 15055 | . . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) | |
33 | eqidd 2819 | . . . . . . . . 9 ⊢ (𝑘 = (𝑓‘𝑛) → 0 = 0) | |
34 | simpl 483 | . . . . . . . . 9 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (♯‘𝐴) ∈ ℕ) | |
35 | simpr 485 | . . . . . . . . 9 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) | |
36 | 0cnd 10622 | . . . . . . . . 9 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑘 ∈ 𝐴) → 0 ∈ ℂ) | |
37 | elfznn 12924 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → 𝑛 ∈ ℕ) | |
38 | 4 | fvconst2 6958 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((ℕ × {0})‘𝑛) = 0) |
39 | 37, 38 | syl 17 | . . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → ((ℕ × {0})‘𝑛) = 0) |
40 | 39 | adantl 482 | . . . . . . . . 9 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((ℕ × {0})‘𝑛) = 0) |
41 | 33, 34, 35, 36, 40 | fsum 15065 | . . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 0 = (seq1( + , (ℕ × {0}))‘(♯‘𝐴))) |
42 | nnuz 12269 | . . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1) | |
43 | 42 | ser0 13410 | . . . . . . . . 9 ⊢ ((♯‘𝐴) ∈ ℕ → (seq1( + , (ℕ × {0}))‘(♯‘𝐴)) = 0) |
44 | 43 | adantr 481 | . . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (seq1( + , (ℕ × {0}))‘(♯‘𝐴)) = 0) |
45 | 41, 44 | eqtrd 2853 | . . . . . . 7 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 0 = 0) |
46 | 45 | ex 413 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 0 = 0)) |
47 | 46 | exlimdv 1925 | . . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 0 = 0)) |
48 | 47 | imp 407 | . . . 4 ⊢ (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 0 = 0) |
49 | 29, 48 | jaoi 851 | . . 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 851 | 1 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∨ 𝐴 ∈ Fin) → Σ𝑘 ∈ 𝐴 0 = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 841 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ⊆ wss 3933 ∅c0 4288 ifcif 4463 𝒫 cpw 4535 {csn 4557 class class class wbr 5057 × cxp 5546 dom cdm 5548 Fun wfun 6342 ⟶wf 6344 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 Fincfn 8497 ℂcc 10523 0cc0 10525 1c1 10526 + caddc 10528 ℕcn 11626 ℤcz 11969 ℤ≥cuz 12231 ...cfz 12880 seqcseq 13357 ♯chash 13678 ⇝ cli 14829 Σcsu 15030 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 |
This theorem is referenced by: fsum00 15141 fsumdvds 15646 pwp1fsum 15730 pcfac 16223 ovoliunnul 24035 vitalilem5 24140 itg1addlem5 24228 itg10a 24238 itg0 24307 itgz 24308 plymullem1 24731 coemullem 24767 logtayl 25170 ftalem5 25581 chp1 25671 logexprlim 25728 bposlem2 25788 rpvmasumlem 25990 axcgrid 26629 axlowdimlem16 26670 indsumin 31180 plymulx0 31716 signsplypnf 31719 fsum2dsub 31777 knoppndvlem6 33753 volsupnfl 34818 binomcxplemnn0 40558 binomcxplemnotnn0 40565 sumnnodd 41787 stoweidlem37 42199 fourierdlem103 42371 fourierdlem104 42372 etransclem24 42420 etransclem32 42428 etransclem35 42431 sge0z 42534 aacllem 44830 |
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