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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 2735 | . . . . 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 11253 | . . . . . . . 8 ⊢ 0 ∈ V | |
5 | 4 | fvconst2 7224 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (((ℤ≥‘𝑀) × {0})‘𝑘) = 0) |
6 | ifid 4571 | . . . . . . 7 ⊢ if(𝑘 ∈ 𝐴, 0, 0) = 0 | |
7 | 5, 6 | eqtr4di 2793 | . . . . . 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 11252 | . . . . 5 ⊢ (((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ 𝐴) → 0 ∈ ℂ) | |
10 | 1, 2, 3, 8, 9 | zsum 15751 | . . . 4 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 0 = ( ⇝ ‘seq𝑀( + , ((ℤ≥‘𝑀) × {0})))) |
11 | fclim 15586 | . . . . . 6 ⊢ ⇝ :dom ⇝ ⟶ℂ | |
12 | ffun 6740 | . . . . . 6 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ ) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ Fun ⇝ |
14 | serclim0 15610 | . . . . . 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 6958 | . . . . 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 2775 | . . 3 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 0 = 0) |
19 | uzf 12879 | . . . . . . . . 9 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
20 | 19 | fdmi 6748 | . . . . . . . 8 ⊢ dom ℤ≥ = ℤ |
21 | 20 | eleq2i 2831 | . . . . . . 7 ⊢ (𝑀 ∈ dom ℤ≥ ↔ 𝑀 ∈ ℤ) |
22 | ndmfv 6942 | . . . . . . 7 ⊢ (¬ 𝑀 ∈ dom ℤ≥ → (ℤ≥‘𝑀) = ∅) | |
23 | 21, 22 | sylnbir 331 | . . . . . 6 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅) |
24 | 23 | sseq2d 4028 | . . . . 5 ⊢ (¬ 𝑀 ∈ ℤ → (𝐴 ⊆ (ℤ≥‘𝑀) ↔ 𝐴 ⊆ ∅)) |
25 | 24 | biimpac 478 | . . . 4 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ ¬ 𝑀 ∈ ℤ) → 𝐴 ⊆ ∅) |
26 | ss0 4408 | . . . 4 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) | |
27 | sumeq1 15722 | . . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 0 = Σ𝑘 ∈ ∅ 0) | |
28 | sum0 15754 | . . . . 5 ⊢ Σ𝑘 ∈ ∅ 0 = 0 | |
29 | 27, 28 | eqtrdi 2791 | . . . 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 813 | . 2 ⊢ (𝐴 ⊆ (ℤ≥‘𝑀) → Σ𝑘 ∈ 𝐴 0 = 0) |
32 | fz1f1o 15743 | . . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) | |
33 | eqidd 2736 | . . . . . . . . 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 11252 | . . . . . . . . 9 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑘 ∈ 𝐴) → 0 ∈ ℂ) | |
37 | elfznn 13590 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → 𝑛 ∈ ℕ) | |
38 | 4 | fvconst2 7224 | . . . . . . . . . . 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 15753 | . . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 0 = (seq1( + , (ℕ × {0}))‘(♯‘𝐴))) |
42 | nnuz 12919 | . . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1) | |
43 | 42 | ser0 14092 | . . . . . . . . 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 2775 | . . . . . . 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 1931 | . . . . 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 1537 ∃wex 1776 ∈ wcel 2106 ⊆ wss 3963 ∅c0 4339 ifcif 4531 𝒫 cpw 4605 {csn 4631 class class class wbr 5148 × cxp 5687 dom cdm 5689 Fun wfun 6557 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 ℕcn 12264 ℤcz 12611 ℤ≥cuz 12876 ...cfz 13544 seqcseq 14039 ♯chash 14366 ⇝ cli 15517 Σcsu 15719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 |
This theorem is referenced by: fsum00 15831 fsumdvds 16342 pwp1fsum 16425 pcfac 16933 ovoliunnul 25556 vitalilem5 25661 itg1addlem5 25750 itg10a 25760 itg0 25830 itgz 25831 plymullem1 26268 coemullem 26304 logtayl 26717 ftalem5 27135 chp1 27225 logexprlim 27284 bposlem2 27344 rpvmasumlem 27546 axcgrid 28946 axlowdimlem16 28987 elrgspnlem2 33233 indsumin 34003 plymulx0 34541 signsplypnf 34544 fsum2dsub 34601 knoppndvlem6 36500 volsupnfl 37652 binomcxplemnn0 44345 binomcxplemnotnn0 44352 sumnnodd 45586 stoweidlem37 45993 fourierdlem103 46165 fourierdlem104 46166 etransclem24 46214 etransclem32 46222 etransclem35 46225 sge0z 46331 aacllem 49032 |
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