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Mirrors > Home > ILE Home > Th. List > sumnul | GIF version |
Description: The sum of a non-convergent infinite series evaluates to the empty set. (Contributed by Paul Chapman, 4-Nov-2007.) (Revised by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
isumcl.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
isumcl.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
isumcl.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) |
isumcl.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
sumnul.5 | ⊢ (𝜑 → ¬ seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
Ref | Expression |
---|---|
sumnul | ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumcl.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | isumcl.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | isumcl.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
4 | isumcl.4 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) | |
5 | 1, 2, 3, 4 | isum 11122 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘seq𝑀( + , 𝐹))) |
6 | seqex 10188 | . . 3 ⊢ seq𝑀( + , 𝐹) ∈ V | |
7 | sumnul.5 | . . 3 ⊢ (𝜑 → ¬ seq𝑀( + , 𝐹) ∈ dom ⇝ ) | |
8 | ndmfvg 5420 | . . 3 ⊢ ((seq𝑀( + , 𝐹) ∈ V ∧ ¬ seq𝑀( + , 𝐹) ∈ dom ⇝ ) → ( ⇝ ‘seq𝑀( + , 𝐹)) = ∅) | |
9 | 6, 7, 8 | sylancr 410 | . 2 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( + , 𝐹)) = ∅) |
10 | 5, 9 | eqtrd 2150 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 Vcvv 2660 ∅c0 3333 dom cdm 4509 ‘cfv 5093 ℂcc 7586 + caddc 7591 ℤcz 9022 ℤ≥cuz 9294 seqcseq 10186 ⇝ cli 11015 Σcsu 11090 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-isom 5102 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-frec 6256 df-1o 6281 df-oadd 6285 df-er 6397 df-en 6603 df-dom 6604 df-fin 6605 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 df-inn 8689 df-2 8747 df-n0 8946 df-z 9023 df-uz 9295 df-q 9380 df-rp 9410 df-fz 9759 df-fzo 9888 df-seqfrec 10187 df-exp 10261 df-ihash 10490 df-cj 10582 df-rsqrt 10738 df-abs 10739 df-clim 11016 df-sumdc 11091 |
This theorem is referenced by: (None) |
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