Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2263 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
5 | eluzelz 9496 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
7 | uzss 9507 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | |
8 | 4, 7 | syl 14 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
9 | 8, 1, 3 | 3sstr4g 3190 | . . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑍) |
10 | 9 | sselda 3147 | . . . 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 9653 | . . . 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 2247 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ+) |
18 | 17 | rpcnd 9655 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
19 | 3, 2, 18 | iserex 11302 | . . . 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 11392 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ) |
22 | fveq2 5496 | . . . 4 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁)) | |
23 | 22 | eleq1d 2239 | . . 3 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℝ+ ↔ (𝐹‘𝑁) ∈ ℝ+)) |
24 | 17 | ralrimiva 2543 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ+) |
25 | 23, 24, 2 | rspcdva 2839 | . 2 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ+) |
26 | 8 | sselda 3147 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ (ℤ≥‘𝑀)) |
27 | 26, 3 | eleqtrrdi 2264 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍) |
28 | 27, 17 | syldan 280 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ+) |
29 | rpaddcl 9634 | . . . . 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 10416 | . . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) = (𝐹‘𝑁)) |
32 | uzid 9501 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
33 | 6, 32 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑁)) |
34 | 33, 1 | eleqtrrdi 2264 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑊) |
35 | 15 | recnd 7948 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℂ) |
36 | 1, 6, 12, 35, 20 | isumclim2 11385 | . . . 4 ⊢ (𝜑 → seq𝑁( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑊 𝐴) |
37 | 9 | sseld 3146 | . . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → 𝑚 ∈ 𝑍)) |
38 | fveq2 5496 | . . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) | |
39 | 38 | eleq1d 2239 | . . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℝ+ ↔ (𝐹‘𝑚) ∈ ℝ+)) |
40 | 39 | rspcv 2830 | . . . . . . 7 ⊢ (𝑚 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ+ → (𝐹‘𝑚) ∈ ℝ+)) |
41 | 37, 24, 40 | syl6ci 1438 | . . . . . 6 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → (𝐹‘𝑚) ∈ ℝ+)) |
42 | 41 | imp 123 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ+) |
43 | 42 | rpred 9653 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ) |
44 | 42 | rpge0d 9657 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 0 ≤ (𝐹‘𝑚)) |
45 | 1, 34, 36, 43, 44 | climserle 11308 | . . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴) |
46 | 31, 45 | eqbrtrrd 4013 | . 2 ⊢ (𝜑 → (𝐹‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴) |
47 | 21, 25, 46 | rpgecld 9693 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ+) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 ∀wral 2448 ⊆ wss 3121 dom cdm 4611 ‘cfv 5198 (class class class)co 5853 ℝcr 7773 + caddc 7777 ≤ cle 7955 ℤcz 9212 ℤ≥cuz 9487 ℝ+crp 9610 seqcseq 10401 ⇝ cli 11241 Σcsu 11316 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-seqfrec 10402 df-exp 10476 df-ihash 10710 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-clim 11242 df-sumdc 11317 |
This theorem is referenced by: effsumlt 11655 eirraplem 11739 |
Copyright terms: Public domain | W3C validator |