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Mirrors > Home > ILE Home > Th. List > sum0 | 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 9561 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1zzd 9278 | . . . 4 ⊢ (⊤ → 1 ∈ ℤ) | |
3 | 0ss 3461 | . . . . 5 ⊢ ∅ ⊆ ℕ | |
4 | 3 | a1i 9 | . . . 4 ⊢ (⊤ → ∅ ⊆ ℕ) |
5 | simpr 110 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) | |
6 | 5, 1 | eleqtrdi 2270 | . . . . . 6 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ (ℤ≥‘1)) |
7 | c0ex 7950 | . . . . . . 7 ⊢ 0 ∈ V | |
8 | 7 | fvconst2 5732 | . . . . . 6 ⊢ (𝑘 ∈ (ℤ≥‘1) → (((ℤ≥‘1) × {0})‘𝑘) = 0) |
9 | 6, 8 | syl 14 | . . . . 5 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℤ≥‘1) × {0})‘𝑘) = 0) |
10 | noel 3426 | . . . . . 6 ⊢ ¬ 𝑘 ∈ ∅ | |
11 | 10 | iffalsei 3543 | . . . . 5 ⊢ if(𝑘 ∈ ∅, 𝐴, 0) = 0 |
12 | 9, 11 | eqtr4di 2228 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ℕ) → (((ℤ≥‘1) × {0})‘𝑘) = if(𝑘 ∈ ∅, 𝐴, 0)) |
13 | noel 3426 | . . . . . . . 8 ⊢ ¬ 𝑗 ∈ ∅ | |
14 | 13 | olci 732 | . . . . . . 7 ⊢ (𝑗 ∈ ∅ ∨ ¬ 𝑗 ∈ ∅) |
15 | df-dc 835 | . . . . . . 7 ⊢ (DECID 𝑗 ∈ ∅ ↔ (𝑗 ∈ ∅ ∨ ¬ 𝑗 ∈ ∅)) | |
16 | 14, 15 | mpbir 146 | . . . . . 6 ⊢ DECID 𝑗 ∈ ∅ |
17 | 16 | rgenw 2532 | . . . . 5 ⊢ ∀𝑗 ∈ ℕ DECID 𝑗 ∈ ∅ |
18 | 17 | a1i 9 | . . . 4 ⊢ (⊤ → ∀𝑗 ∈ ℕ DECID 𝑗 ∈ ∅) |
19 | 10 | pm2.21i 646 | . . . . 5 ⊢ (𝑘 ∈ ∅ → 𝐴 ∈ ℂ) |
20 | 19 | adantl 277 | . . . 4 ⊢ ((⊤ ∧ 𝑘 ∈ ∅) → 𝐴 ∈ ℂ) |
21 | 1, 2, 4, 12, 18, 20 | zsumdc 11387 | . . 3 ⊢ (⊤ → Σ𝑘 ∈ ∅ 𝐴 = ( ⇝ ‘seq1( + , ((ℤ≥‘1) × {0})))) |
22 | 21 | mptru 1362 | . 2 ⊢ Σ𝑘 ∈ ∅ 𝐴 = ( ⇝ ‘seq1( + , ((ℤ≥‘1) × {0}))) |
23 | fclim 11297 | . . . 4 ⊢ ⇝ :dom ⇝ ⟶ℂ | |
24 | ffun 5368 | . . . 4 ⊢ ( ⇝ :dom ⇝ ⟶ℂ → Fun ⇝ ) | |
25 | 23, 24 | ax-mp 5 | . . 3 ⊢ Fun ⇝ |
26 | 1z 9277 | . . . 4 ⊢ 1 ∈ ℤ | |
27 | serclim0 11308 | . . . 4 ⊢ (1 ∈ ℤ → seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0) | |
28 | 26, 27 | ax-mp 5 | . . 3 ⊢ seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0 |
29 | funbrfv 5554 | . . 3 ⊢ (Fun ⇝ → (seq1( + , ((ℤ≥‘1) × {0})) ⇝ 0 → ( ⇝ ‘seq1( + , ((ℤ≥‘1) × {0}))) = 0)) | |
30 | 25, 28, 29 | mp2 16 | . 2 ⊢ ( ⇝ ‘seq1( + , ((ℤ≥‘1) × {0}))) = 0 |
31 | 22, 30 | eqtri 2198 | 1 ⊢ Σ𝑘 ∈ ∅ 𝐴 = 0 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 104 ∨ wo 708 DECID wdc 834 = wceq 1353 ⊤wtru 1354 ∈ wcel 2148 ∀wral 2455 ⊆ wss 3129 ∅c0 3422 ifcif 3534 {csn 3592 class class class wbr 4003 × cxp 4624 dom cdm 4626 Fun wfun 5210 ⟶wf 5212 ‘cfv 5216 ℂcc 7808 0cc0 7810 1c1 7811 + caddc 7813 ℕcn 8917 ℤcz 9251 ℤ≥cuz 9526 seqcseq 10442 ⇝ cli 11281 Σcsu 11356 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-isom 5225 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-frec 6391 df-1o 6416 df-oadd 6420 df-er 6534 df-en 6740 df-dom 6741 df-fin 6742 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-2 8976 df-3 8977 df-4 8978 df-n0 9175 df-z 9252 df-uz 9527 df-q 9618 df-rp 9652 df-fz 10007 df-fzo 10140 df-seqfrec 10443 df-exp 10517 df-ihash 10751 df-cj 10846 df-re 10847 df-im 10848 df-rsqrt 11002 df-abs 11003 df-clim 11282 df-sumdc 11357 |
This theorem is referenced by: isumz 11392 fsumf1o 11393 fsumcllem 11402 fsumadd 11409 fsum2d 11438 fisumrev2 11449 fsummulc2 11451 fsumconst 11457 modfsummod 11461 fsumabs 11468 telfsumo 11469 fsumparts 11473 fsumrelem 11474 fsumiun 11480 isumsplit 11494 arisum 11501 arisum2 11502 cvgratnnlemseq 11529 fsumcncntop 13949 |
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