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Mirrors > Home > ILE Home > Th. List > isumge0 | GIF version |
Description: An infinite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 28-Apr-2014.) |
Ref | Expression |
---|---|
isumrecl.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
isumrecl.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
isumrecl.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) |
isumrecl.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) |
isumrecl.5 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
isumge0.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐴) |
Ref | Expression |
---|---|
isumge0 | ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝑍 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumrecl.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | isumrecl.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | isumrecl.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
4 | isumrecl.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) | |
5 | 4 | recnd 7711 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
6 | isumrecl.5 | . . . . 5 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) | |
7 | 1, 2, 3, 5, 6 | isumclim2 11076 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑍 𝐴) |
8 | fveq2 5373 | . . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘)) | |
9 | 8 | cbvsumv 11015 | . . . . 5 ⊢ Σ𝑗 ∈ 𝑍 (𝐹‘𝑗) = Σ𝑘 ∈ 𝑍 (𝐹‘𝑘) |
10 | 3 | sumeq2dv 11022 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 (𝐹‘𝑘) = Σ𝑘 ∈ 𝑍 𝐴) |
11 | 9, 10 | syl5eq 2157 | . . . 4 ⊢ (𝜑 → Σ𝑗 ∈ 𝑍 (𝐹‘𝑗) = Σ𝑘 ∈ 𝑍 𝐴) |
12 | 7, 11 | breqtrrd 3919 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ Σ𝑗 ∈ 𝑍 (𝐹‘𝑗)) |
13 | 3, 4 | eqeltrd 2189 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
14 | isumge0.6 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐴) | |
15 | 14, 3 | breqtrrd 3919 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) |
16 | 1, 2, 12, 13, 15 | iserge0 10997 | . 2 ⊢ (𝜑 → 0 ≤ Σ𝑗 ∈ 𝑍 (𝐹‘𝑗)) |
17 | 16, 11 | breqtrd 3917 | 1 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝑍 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1312 ∈ wcel 1461 class class class wbr 3893 dom cdm 4497 ‘cfv 5079 ℝcr 7539 0cc0 7540 + caddc 7543 ≤ cle 7718 ℤcz 8951 ℤ≥cuz 9221 seqcseq 10104 ⇝ cli 10932 Σcsu 11007 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-iinf 4460 ax-cnex 7629 ax-resscn 7630 ax-1cn 7631 ax-1re 7632 ax-icn 7633 ax-addcl 7634 ax-addrcl 7635 ax-mulcl 7636 ax-mulrcl 7637 ax-addcom 7638 ax-mulcom 7639 ax-addass 7640 ax-mulass 7641 ax-distr 7642 ax-i2m1 7643 ax-0lt1 7644 ax-1rid 7645 ax-0id 7646 ax-rnegex 7647 ax-precex 7648 ax-cnre 7649 ax-pre-ltirr 7650 ax-pre-ltwlin 7651 ax-pre-lttrn 7652 ax-pre-apti 7653 ax-pre-ltadd 7654 ax-pre-mulgt0 7655 ax-pre-mulext 7656 ax-arch 7657 ax-caucvg 7658 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rmo 2396 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-if 3439 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-tr 3985 df-id 4173 df-po 4176 df-iso 4177 df-iord 4246 df-on 4248 df-ilim 4249 df-suc 4251 df-iom 4463 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-isom 5088 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5989 df-2nd 5990 df-recs 6153 df-irdg 6218 df-frec 6239 df-1o 6264 df-oadd 6268 df-er 6380 df-en 6586 df-dom 6587 df-fin 6588 df-pnf 7719 df-mnf 7720 df-xr 7721 df-ltxr 7722 df-le 7723 df-sub 7851 df-neg 7852 df-reap 8248 df-ap 8255 df-div 8339 df-inn 8624 df-2 8682 df-3 8683 df-4 8684 df-n0 8875 df-z 8952 df-uz 9222 df-q 9307 df-rp 9337 df-fz 9677 df-fzo 9806 df-seqfrec 10105 df-exp 10179 df-ihash 10408 df-cj 10500 df-re 10501 df-im 10502 df-rsqrt 10655 df-abs 10656 df-clim 10933 df-sumdc 11008 |
This theorem is referenced by: mertenslemi1 11189 mertenslem2 11190 trilpolemlt1 12911 |
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