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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 2771 | . . . . 5 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
2 | simpr 471 | . . . . 5 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ) | |
3 | simpl 468 | . . . . 5 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → 𝐴 ⊆ (ℤ≥‘𝑀)) | |
4 | c0ex 10236 | . . . . . . . 8 ⊢ 0 ∈ V | |
5 | 4 | fvconst2 6613 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (((ℤ≥‘𝑀) × {0})‘𝑘) = 0) |
6 | ifid 4264 | . . . . . . 7 ⊢ if(𝑘 ∈ 𝐴, 0, 0) = 0 | |
7 | 5, 6 | syl6eqr 2823 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (((ℤ≥‘𝑀) × {0})‘𝑘) = if(𝑘 ∈ 𝐴, 0, 0)) |
8 | 7 | adantl 467 | . . . . 5 ⊢ (((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (((ℤ≥‘𝑀) × {0})‘𝑘) = if(𝑘 ∈ 𝐴, 0, 0)) |
9 | 0cnd 10235 | . . . . 5 ⊢ (((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ 𝐴) → 0 ∈ ℂ) | |
10 | 1, 2, 3, 8, 9 | zsum 14657 | . . . 4 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 0 = ( ⇝ ‘seq𝑀( + , ((ℤ≥‘𝑀) × {0})))) |
11 | fclim 14492 | . . . . . 6 ⊢ ⇝ :dom ⇝ ⟶ℂ | |
12 | ffun 6188 | . . . . . 6 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ ) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ Fun ⇝ |
14 | serclim0 14516 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , ((ℤ≥‘𝑀) × {0})) ⇝ 0) | |
15 | 14 | adantl 467 | . . . . 5 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → seq𝑀( + , ((ℤ≥‘𝑀) × {0})) ⇝ 0) |
16 | funbrfv 6375 | . . . . 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 2805 | . . 3 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 0 = 0) |
19 | uzf 11891 | . . . . . . . . 9 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
20 | 19 | fdmi 6192 | . . . . . . . 8 ⊢ dom ℤ≥ = ℤ |
21 | 20 | eleq2i 2842 | . . . . . . 7 ⊢ (𝑀 ∈ dom ℤ≥ ↔ 𝑀 ∈ ℤ) |
22 | ndmfv 6359 | . . . . . . 7 ⊢ (¬ 𝑀 ∈ dom ℤ≥ → (ℤ≥‘𝑀) = ∅) | |
23 | 21, 22 | sylnbir 320 | . . . . . 6 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅) |
24 | 23 | sseq2d 3782 | . . . . 5 ⊢ (¬ 𝑀 ∈ ℤ → (𝐴 ⊆ (ℤ≥‘𝑀) ↔ 𝐴 ⊆ ∅)) |
25 | 24 | biimpac 464 | . . . 4 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ ¬ 𝑀 ∈ ℤ) → 𝐴 ⊆ ∅) |
26 | ss0 4118 | . . . 4 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) | |
27 | sumeq1 14627 | . . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 0 = Σ𝑘 ∈ ∅ 0) | |
28 | sum0 14660 | . . . . 5 ⊢ Σ𝑘 ∈ ∅ 0 = 0 | |
29 | 27, 28 | syl6eq 2821 | . . . 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 14649 | . . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))) | |
33 | eqidd 2772 | . . . . . . . . 9 ⊢ (𝑘 = (𝑓‘𝑛) → 0 = 0) | |
34 | simpl 468 | . . . . . . . . 9 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (♯‘𝐴) ∈ ℕ) | |
35 | simpr 471 | . . . . . . . . 9 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) | |
36 | 0cnd 10235 | . . . . . . . . 9 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑘 ∈ 𝐴) → 0 ∈ ℂ) | |
37 | elfznn 12577 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → 𝑛 ∈ ℕ) | |
38 | 4 | fvconst2 6613 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ → ((ℕ × {0})‘𝑛) = 0) |
39 | 37, 38 | syl 17 | . . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → ((ℕ × {0})‘𝑛) = 0) |
40 | 39 | adantl 467 | . . . . . . . . 9 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((ℕ × {0})‘𝑛) = 0) |
41 | 33, 34, 35, 36, 40 | fsum 14659 | . . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 0 = (seq1( + , (ℕ × {0}))‘(♯‘𝐴))) |
42 | nnuz 11925 | . . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1) | |
43 | 42 | ser0 13060 | . . . . . . . . 9 ⊢ ((♯‘𝐴) ∈ ℕ → (seq1( + , (ℕ × {0}))‘(♯‘𝐴)) = 0) |
44 | 43 | adantr 466 | . . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (seq1( + , (ℕ × {0}))‘(♯‘𝐴)) = 0) |
45 | 41, 44 | eqtrd 2805 | . . . . . . 7 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 0 = 0) |
46 | 45 | ex 397 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 0 = 0)) |
47 | 46 | exlimdv 2013 | . . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 0 = 0)) |
48 | 47 | imp 393 | . . . 4 ⊢ (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 0 = 0) |
49 | 29, 48 | jaoi 844 | . . 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 844 | 1 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∨ 𝐴 ∈ Fin) → Σ𝑘 ∈ 𝐴 0 = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 ∨ wo 834 = wceq 1631 ∃wex 1852 ∈ wcel 2145 ⊆ wss 3723 ∅c0 4063 ifcif 4225 𝒫 cpw 4297 {csn 4316 class class class wbr 4786 × cxp 5247 dom cdm 5249 Fun wfun 6025 ⟶wf 6027 –1-1-onto→wf1o 6030 ‘cfv 6031 (class class class)co 6793 Fincfn 8109 ℂcc 10136 0cc0 10138 1c1 10139 + caddc 10141 ℕcn 11222 ℤcz 11579 ℤ≥cuz 11888 ...cfz 12533 seqcseq 13008 ♯chash 13321 ⇝ cli 14423 Σcsu 14624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-sup 8504 df-oi 8571 df-card 8965 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-3 11282 df-n0 11495 df-z 11580 df-uz 11889 df-rp 12036 df-fz 12534 df-fzo 12674 df-seq 13009 df-exp 13068 df-hash 13322 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-clim 14427 df-sum 14625 |
This theorem is referenced by: fsum00 14737 fsumdvds 15239 pwp1fsum 15322 pcfac 15810 ovoliunnul 23495 vitalilem5 23600 itg1addlem5 23687 itg10a 23697 itg0 23766 itgz 23767 plymullem1 24190 coemullem 24226 logtayl 24627 ftalem5 25024 chp1 25114 logexprlim 25171 bposlem2 25231 rpvmasumlem 25397 axcgrid 26017 axlowdimlem16 26058 indsumin 30424 plymulx0 30964 signsplypnf 30967 fsum2dsub 31025 knoppndvlem6 32845 volsupnfl 33787 binomcxplemnn0 39074 binomcxplemnotnn0 39081 sumnnodd 40380 stoweidlem37 40771 fourierdlem103 40943 fourierdlem104 40944 etransclem24 40992 etransclem32 41000 etransclem35 41003 sge0z 41109 aacllem 43078 |
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