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Mirrors > Home > MPE Home > Th. List > sum0 | Structured version Visualization version GIF version |
Description: Any sum over the empty set is zero. (Contributed by Mario Carneiro, 12-Aug-2013.) (Revised by Mario Carneiro, 20-Apr-2014.) |
Ref | Expression |
---|---|
sum0 | ⊢ Σ𝑘 ∈ ∅ 𝐴 = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 11599 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1z 11284 | . . . . 5 ⊢ 1 ∈ ℤ | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → 1 ∈ ℤ) |
4 | 0ss 3924 | . . . . 5 ⊢ ∅ ⊆ ℕ | |
5 | 4 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ⊆ ℕ) |
6 | simpr 476 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) | |
7 | 6, 1 | syl6eleq 2698 | . . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1)) |
8 | c0ex 9913 | . . . . . . 7 ⊢ 0 ∈ V | |
9 | 8 | fvconst2 6374 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘1) → (((ℤ≥‘1) × {0})‘𝑘) = 0) |
10 | 7, 9 | syl 17 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℤ≥‘1) × {0})‘𝑘) = 0) |
11 | noel 3878 | . . . . . 6 ⊢ ¬ 𝑘 ∈ ∅ | |
12 | 11 | iffalsei 4046 | . . . . 5 ⊢ if(𝑘 ∈ ∅, 𝐴, 0) = 0 |
13 | 10, 12 | syl6eqr 2662 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℤ≥‘1) × {0})‘𝑘) = if(𝑘 ∈ ∅, 𝐴, 0)) |
14 | 11 | pm2.21i 115 | . . . . 5 ⊢ (𝑘 ∈ ∅ → 𝐴 ∈ ℂ) |
15 | 14 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ∅) → 𝐴 ∈ ℂ) |
16 | 1, 3, 5, 13, 15 | zsum 14296 | . . 3 ⊢ (⊤ → Σ𝑘 ∈ ∅ 𝐴 = ( ⇝ ‘seq1( + , ((ℤ≥‘1) × {0})))) |
17 | 16 | trud 1484 | . 2 ⊢ Σ𝑘 ∈ ∅ 𝐴 = ( ⇝ ‘seq1( + , ((ℤ≥‘1) × {0}))) |
18 | fclim 14132 | . . . 4 ⊢ ⇝ :dom ⇝ ⟶ℂ | |
19 | ffun 5961 | . . . 4 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ ) | |
20 | 18, 19 | ax-mp 5 | . . 3 ⊢ Fun ⇝ |
21 | serclim0 14156 | . . . 4 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0) | |
22 | 2, 21 | ax-mp 5 | . . 3 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0 |
23 | funbrfv 6144 | . . 3 ⊢ (Fun ⇝ → (seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0 → ( ⇝ ‘seq1( + , ((ℤ≥‘1) × {0}))) = 0)) | |
24 | 20, 22, 23 | mp2 9 | . 2 ⊢ ( ⇝ ‘seq1( + , ((ℤ≥‘1) × {0}))) = 0 |
25 | 17, 24 | eqtri 2632 | 1 ⊢ Σ𝑘 ∈ ∅ 𝐴 = 0 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 ⊆ wss 3540 ∅c0 3874 ifcif 4036 {csn 4125 class class class wbr 4583 × cxp 5036 dom cdm 5038 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 ℕcn 10897 ℤcz 11254 ℤ≥cuz 11563 seqcseq 12663 ⇝ 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: sumz 14300 fsumf1o 14301 fsumcllem 14310 fsumadd 14317 fsum2d 14344 fsumrev2 14356 fsummulc2 14358 fsumconst 14364 modfsummod 14367 fsumabs 14374 telfsumo 14375 fsumparts 14379 fsumrelem 14380 fsumrlim 14384 fsumo1 14385 fsumiun 14394 isumsplit 14411 arisum 14431 arisum2 14432 bpoly0 14620 sumeven 14948 sumodd 14949 bitsinv1 15002 bitsinvp1 15009 prmreclem4 15461 prmreclem5 15462 gsumfsum 19632 fsumcn 22481 ovolfiniun 23076 volfiniun 23122 itg10 23261 itgfsum 23399 dvmptfsum 23542 abelthlem6 23994 logfac 24151 log2ublem3 24475 harmonicbnd3 24534 cht1 24691 dchrisumlem1 24978 dchrisumlem3 24980 logdivbnd 25045 pntrsumbnd2 25056 pntrlog2bndlem4 25069 esumpcvgval 29467 signsvf0 29983 signsvf1 29984 mettrifi 32723 rrncmslem 32801 mccl 38665 dvmptfprod 38835 dvnprodlem3 38838 sge0rnn0 39261 sge00 39269 sge0sn 39272 pwdif 40039 |
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