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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 2270 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
5 | eluzelz 9535 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
7 | uzss 9546 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | |
8 | 4, 7 | syl 14 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
9 | 8, 1, 3 | 3sstr4g 3198 | . . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑍) |
10 | 9 | sselda 3155 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑍) |
11 | isumrpcl.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
12 | 10, 11 | syldan 282 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) = 𝐴) |
13 | isumrpcl.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ+) | |
14 | 13 | rpred 9694 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) |
15 | 10, 14 | syldan 282 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℝ) |
16 | isumrpcl.6 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) | |
17 | 11, 13 | eqeltrd 2254 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ+) |
18 | 17 | rpcnd 9696 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
19 | 3, 2, 18 | iserex 11342 | . . . 4 ⊢ (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑁( + , 𝐹) ∈ dom ⇝ )) |
20 | 16, 19 | mpbid 147 | . . 3 ⊢ (𝜑 → seq𝑁( + , 𝐹) ∈ dom ⇝ ) |
21 | 1, 6, 12, 15, 20 | isumrecl 11432 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ) |
22 | fveq2 5515 | . . . 4 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁)) | |
23 | 22 | eleq1d 2246 | . . 3 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℝ+ ↔ (𝐹‘𝑁) ∈ ℝ+)) |
24 | 17 | ralrimiva 2550 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ+) |
25 | 23, 24, 2 | rspcdva 2846 | . 2 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ+) |
26 | 8 | sselda 3155 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ (ℤ≥‘𝑀)) |
27 | 26, 3 | eleqtrrdi 2271 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍) |
28 | 27, 17 | syldan 282 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ+) |
29 | rpaddcl 9675 | . . . . 5 ⊢ ((𝑘 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+) → (𝑘 + 𝑦) ∈ ℝ+) | |
30 | 29 | adantl 277 | . . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ ℝ+ ∧ 𝑦 ∈ ℝ+)) → (𝑘 + 𝑦) ∈ ℝ+) |
31 | 6, 28, 30 | seq3-1 10457 | . . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) = (𝐹‘𝑁)) |
32 | uzid 9540 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
33 | 6, 32 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑁)) |
34 | 33, 1 | eleqtrrdi 2271 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑊) |
35 | 15 | recnd 7984 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℂ) |
36 | 1, 6, 12, 35, 20 | isumclim2 11425 | . . . 4 ⊢ (𝜑 → seq𝑁( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑊 𝐴) |
37 | 9 | sseld 3154 | . . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → 𝑚 ∈ 𝑍)) |
38 | fveq2 5515 | . . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) | |
39 | 38 | eleq1d 2246 | . . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℝ+ ↔ (𝐹‘𝑚) ∈ ℝ+)) |
40 | 39 | rspcv 2837 | . . . . . . 7 ⊢ (𝑚 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ+ → (𝐹‘𝑚) ∈ ℝ+)) |
41 | 37, 24, 40 | syl6ci 1445 | . . . . . 6 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → (𝐹‘𝑚) ∈ ℝ+)) |
42 | 41 | imp 124 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ+) |
43 | 42 | rpred 9694 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ) |
44 | 42 | rpge0d 9698 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 0 ≤ (𝐹‘𝑚)) |
45 | 1, 34, 36, 43, 44 | climserle 11348 | . . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴) |
46 | 31, 45 | eqbrtrrd 4027 | . 2 ⊢ (𝜑 → (𝐹‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴) |
47 | 21, 25, 46 | rpgecld 9734 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ+) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ⊆ wss 3129 dom cdm 4626 ‘cfv 5216 (class class class)co 5874 ℝcr 7809 + caddc 7813 ≤ cle 7991 ℤcz 9251 ℤ≥cuz 9526 ℝ+crp 9651 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: effsumlt 11695 eirraplem 11779 |
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