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Mirrors > Home > ILE Home > Th. List > isumrpcl | GIF version |
Description: The infinite sum of positive reals is positive. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
isumrpcl.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
isumrpcl.2 | ⊢ 𝑊 = (ℤ≥‘𝑁) |
isumrpcl.3 | ⊢ (𝜑 → 𝑁 ∈ 𝑍) |
isumrpcl.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) |
isumrpcl.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ+) |
isumrpcl.6 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
Ref | Expression |
---|---|
isumrpcl | ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumrpcl.2 | . . 3 ⊢ 𝑊 = (ℤ≥‘𝑁) | |
2 | isumrpcl.3 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ 𝑍) | |
3 | isumrpcl.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
4 | 2, 3 | eleqtrdi 2232 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
5 | eluzelz 9335 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
7 | uzss 9346 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | |
8 | 4, 7 | syl 14 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
9 | 8, 1, 3 | 3sstr4g 3140 | . . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑍) |
10 | 9 | sselda 3097 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑍) |
11 | isumrpcl.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
12 | 10, 11 | syldan 280 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) = 𝐴) |
13 | isumrpcl.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ+) | |
14 | 13 | rpred 9483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) |
15 | 10, 14 | syldan 280 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℝ) |
16 | isumrpcl.6 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) | |
17 | 11, 13 | eqeltrd 2216 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ+) |
18 | 17 | rpcnd 9485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
19 | 3, 2, 18 | iserex 11108 | . . . 4 ⊢ (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑁( + , 𝐹) ∈ dom ⇝ )) |
20 | 16, 19 | mpbid 146 | . . 3 ⊢ (𝜑 → seq𝑁( + , 𝐹) ∈ dom ⇝ ) |
21 | 1, 6, 12, 15, 20 | isumrecl 11198 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ) |
22 | fveq2 5421 | . . . 4 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁)) | |
23 | 22 | eleq1d 2208 | . . 3 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℝ+ ↔ (𝐹‘𝑁) ∈ ℝ+)) |
24 | 17 | ralrimiva 2505 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ+) |
25 | 23, 24, 2 | rspcdva 2794 | . 2 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ+) |
26 | 8 | sselda 3097 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ (ℤ≥‘𝑀)) |
27 | 26, 3 | eleqtrrdi 2233 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍) |
28 | 27, 17 | syldan 280 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ+) |
29 | rpaddcl 9465 | . . . . 5 ⊢ ((𝑘 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+) → (𝑘 + 𝑦) ∈ ℝ+) | |
30 | 29 | adantl 275 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) → (𝑘 + 𝑦) ∈ ℝ+) |
31 | 6, 28, 30 | seq3-1 10233 | . . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) = (𝐹‘𝑁)) |
32 | uzid 9340 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
33 | 6, 32 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑁)) |
34 | 33, 1 | eleqtrrdi 2233 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑊) |
35 | 15 | recnd 7794 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℂ) |
36 | 1, 6, 12, 35, 20 | isumclim2 11191 | . . . 4 ⊢ (𝜑 → seq𝑁( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑊 𝐴) |
37 | 9 | sseld 3096 | . . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → 𝑚 ∈ 𝑍)) |
38 | fveq2 5421 | . . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) | |
39 | 38 | eleq1d 2208 | . . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℝ+ ↔ (𝐹‘𝑚) ∈ ℝ+)) |
40 | 39 | rspcv 2785 | . . . . . . 7 ⊢ (𝑚 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ+ → (𝐹‘𝑚) ∈ ℝ+)) |
41 | 37, 24, 40 | syl6ci 1421 | . . . . . 6 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → (𝐹‘𝑚) ∈ ℝ+)) |
42 | 41 | imp 123 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ+) |
43 | 42 | rpred 9483 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ) |
44 | 42 | rpge0d 9487 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 0 ≤ (𝐹‘𝑚)) |
45 | 1, 34, 36, 43, 44 | climserle 11114 | . . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴) |
46 | 31, 45 | eqbrtrrd 3952 | . 2 ⊢ (𝜑 → (𝐹‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴) |
47 | 21, 25, 46 | rpgecld 9523 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ+) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ⊆ wss 3071 dom cdm 4539 ‘cfv 5123 (class class class)co 5774 ℝcr 7619 + caddc 7623 ≤ cle 7801 ℤcz 9054 ℤ≥cuz 9326 ℝ+crp 9441 seqcseq 10218 ⇝ cli 11047 Σcsu 11122 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-frec 6288 df-1o 6313 df-oadd 6317 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-fz 9791 df-fzo 9920 df-seqfrec 10219 df-exp 10293 df-ihash 10522 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-clim 11048 df-sumdc 11123 |
This theorem is referenced by: effsumlt 11398 eirraplem 11483 |
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