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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 2610 | . . . . 5 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀) | |
2 | simpr 476 | . . . . 5 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ) | |
3 | simpl 472 | . . . . 5 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → 𝐴 ⊆ (ℤ≥‘𝑀)) | |
4 | c0ex 9913 | . . . . . . . 8 ⊢ 0 ∈ V | |
5 | 4 | fvconst2 6374 | . . . . . . 7 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (((ℤ≥‘𝑀) × {0})‘𝑘) = 0) |
6 | ifid 4075 | . . . . . . 7 ⊢ if(𝑘 ∈ 𝐴, 0, 0) = 0 | |
7 | 5, 6 | syl6eqr 2662 | . . . . . 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 9912 | . . . . 5 ⊢ (((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) ∧ 𝑘 ∈ 𝐴) → 0 ∈ ℂ) | |
10 | 1, 2, 3, 8, 9 | zsum 14296 | . . . 4 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 0 = ( ⇝ ‘seq𝑀( + , ((ℤ≥‘𝑀) × {0})))) |
11 | fclim 14132 | . . . . . 6 ⊢ ⇝ :dom ⇝ ⟶ℂ | |
12 | ffun 5961 | . . . . . 6 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ ) | |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ Fun ⇝ |
14 | serclim0 14156 | . . . . . 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 6144 | . . . . 5 ⊢ (Fun ⇝ → (seq𝑀( + , ((ℤ≥‘𝑀) × {0})) ⇝ 0 → ( ⇝ ‘seq𝑀( + , ((ℤ≥‘𝑀) × {0}))) = 0)) | |
17 | 13, 15, 16 | mpsyl 66 | . . . 4 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → ( ⇝ ‘seq𝑀( + , ((ℤ≥‘𝑀) × {0}))) = 0) |
18 | 10, 17 | eqtrd 2644 | . . 3 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ 𝑀 ∈ ℤ) → Σ𝑘 ∈ 𝐴 0 = 0) |
19 | uzf 11566 | . . . . . . . . 9 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
20 | 19 | fdmi 5965 | . . . . . . . 8 ⊢ dom ℤ≥ = ℤ |
21 | 20 | eleq2i 2680 | . . . . . . 7 ⊢ (𝑀 ∈ dom ℤ≥ ↔ 𝑀 ∈ ℤ) |
22 | ndmfv 6128 | . . . . . . 7 ⊢ (¬ 𝑀 ∈ dom ℤ≥ → (ℤ≥‘𝑀) = ∅) | |
23 | 21, 22 | sylnbir 320 | . . . . . 6 ⊢ (¬ 𝑀 ∈ ℤ → (ℤ≥‘𝑀) = ∅) |
24 | 23 | sseq2d 3596 | . . . . 5 ⊢ (¬ 𝑀 ∈ ℤ → (𝐴 ⊆ (ℤ≥‘𝑀) ↔ 𝐴 ⊆ ∅)) |
25 | 24 | biimpac 502 | . . . 4 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∧ ¬ 𝑀 ∈ ℤ) → 𝐴 ⊆ ∅) |
26 | ss0 3926 | . . . 4 ⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) | |
27 | sumeq1 14267 | . . . . 5 ⊢ (𝐴 = ∅ → Σ𝑘 ∈ 𝐴 0 = Σ𝑘 ∈ ∅ 0) | |
28 | sum0 14299 | . . . . 5 ⊢ Σ𝑘 ∈ ∅ 0 = 0 | |
29 | 27, 28 | syl6eq 2660 | . . . 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 828 | . 2 ⊢ (𝐴 ⊆ (ℤ≥‘𝑀) → Σ𝑘 ∈ 𝐴 0 = 0) |
32 | fz1f1o 14288 | . . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴))) | |
33 | eqidd 2611 | . . . . . . . . 9 ⊢ (𝑘 = (𝑓‘𝑛) → 0 = 0) | |
34 | simpl 472 | . . . . . . . . 9 ⊢ (((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → (#‘𝐴) ∈ ℕ) | |
35 | simpr 476 | . . . . . . . . 9 ⊢ (((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) | |
36 | 0cnd 9912 | . . . . . . . . 9 ⊢ ((((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) ∧ 𝑘 ∈ 𝐴) → 0 ∈ ℂ) | |
37 | elfznn 12241 | . . . . . . . . . . 11 ⊢ (𝑛 ∈ (1...(#‘𝐴)) → 𝑛 ∈ ℕ) | |
38 | 4 | fvconst2 6374 | . . . . . . . . . . 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 14298 | . . . . . . . 8 ⊢ (((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 0 = (seq1( + , (ℕ × {0}))‘(#‘𝐴))) |
42 | nnuz 11599 | . . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1) | |
43 | 42 | ser0 12715 | . . . . . . . . 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 2644 | . . . . . . 7 ⊢ (((#‘𝐴) ∈ ℕ ∧ 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 0 = 0) |
46 | 45 | ex 449 | . . . . . 6 ⊢ ((#‘𝐴) ∈ ℕ → (𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 0 = 0)) |
47 | 46 | exlimdv 1848 | . . . . 5 ⊢ ((#‘𝐴) ∈ ℕ → (∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴 → Σ𝑘 ∈ 𝐴 0 = 0)) |
48 | 47 | imp 444 | . . . 4 ⊢ (((#‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐴))–1-1-onto→𝐴) → Σ𝑘 ∈ 𝐴 0 = 0) |
49 | 29, 48 | jaoi 393 | . . 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 393 | 1 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑀) ∨ 𝐴 ∈ Fin) → Σ𝑘 ∈ 𝐴 0 = 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 382 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ⊆ wss 3540 ∅c0 3874 ifcif 4036 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 × cxp 5036 dom cdm 5038 Fun wfun 5798 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 ℕcn 10897 ℤcz 11254 ℤ≥cuz 11563 ...cfz 12197 seqcseq 12663 #chash 12979 ⇝ cli 14063 Σcsu 14264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 |
This theorem is referenced by: fsum00 14371 fsumdvds 14868 pwp1fsum 14952 pcfac 15441 ovoliunnul 23082 vitalilem5 23187 itg1addlem5 23273 itg10a 23283 itg0 23352 itgz 23353 plymullem1 23774 coemullem 23810 logtayl 24206 ftalem5 24603 chp1 24693 logexprlim 24750 bposlem2 24810 rpvmasumlem 24976 axcgrid 25596 axlowdimlem16 25637 plymulx0 29950 signsplypnf 29953 knoppndvlem6 31678 volsupnfl 32624 binomcxplemnn0 37570 binomcxplemnotnn0 37577 sumnnodd 38697 stoweidlem37 38930 fourierdlem103 39102 fourierdlem104 39103 etransclem24 39151 etransclem32 39159 etransclem35 39162 sge0z 39268 aacllem 42356 |
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