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Mirrors > Home > MPE Home > Th. List > isumrpcl | Structured version Visualization version 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 2837 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
5 | eluzelz 12836 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
7 | uzss 12849 | . . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) | |
8 | 4, 7 | syl 17 | . . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝑀)) |
9 | 8, 1, 3 | 3sstr4g 4022 | . . . . 5 ⊢ (𝜑 → 𝑊 ⊆ 𝑍) |
10 | 9 | sselda 3977 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑍) |
11 | isumrpcl.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
12 | 10, 11 | syldan 590 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) = 𝐴) |
13 | isumrpcl.5 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ+) | |
14 | 13 | rpred 13022 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) |
15 | 10, 14 | syldan 590 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℝ) |
16 | isumrpcl.6 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) | |
17 | 11, 13 | eqeltrd 2827 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ+) |
18 | 17 | rpcnd 13024 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
19 | 3, 2, 18 | iserex 15609 | . . . 4 ⊢ (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑁( + , 𝐹) ∈ dom ⇝ )) |
20 | 16, 19 | mpbid 231 | . . 3 ⊢ (𝜑 → seq𝑁( + , 𝐹) ∈ dom ⇝ ) |
21 | 1, 6, 12, 15, 20 | isumrecl 15717 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ) |
22 | fveq2 6885 | . . . 4 ⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁)) | |
23 | 22 | eleq1d 2812 | . . 3 ⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℝ+ ↔ (𝐹‘𝑁) ∈ ℝ+)) |
24 | 17 | ralrimiva 3140 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ+) |
25 | 23, 24, 2 | rspcdva 3607 | . 2 ⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ+) |
26 | seq1 13985 | . . . 4 ⊢ (𝑁 ∈ ℤ → (seq𝑁( + , 𝐹)‘𝑁) = (𝐹‘𝑁)) | |
27 | 6, 26 | syl 17 | . . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) = (𝐹‘𝑁)) |
28 | uzid 12841 | . . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁)) | |
29 | 6, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑁)) |
30 | 29, 1 | eleqtrrdi 2838 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ 𝑊) |
31 | 15 | recnd 11246 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℂ) |
32 | 1, 6, 12, 31, 20 | isumclim2 15710 | . . . 4 ⊢ (𝜑 → seq𝑁( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑊 𝐴) |
33 | 9 | sseld 3976 | . . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → 𝑚 ∈ 𝑍)) |
34 | fveq2 6885 | . . . . . . . . 9 ⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) | |
35 | 34 | eleq1d 2812 | . . . . . . . 8 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) ∈ ℝ+ ↔ (𝐹‘𝑚) ∈ ℝ+)) |
36 | 35 | rspcv 3602 | . . . . . . 7 ⊢ (𝑚 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℝ+ → (𝐹‘𝑚) ∈ ℝ+)) |
37 | 33, 24, 36 | syl6ci 71 | . . . . . 6 ⊢ (𝜑 → (𝑚 ∈ 𝑊 → (𝐹‘𝑚) ∈ ℝ+)) |
38 | 37 | imp 406 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ+) |
39 | 38 | rpred 13022 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → (𝐹‘𝑚) ∈ ℝ) |
40 | 38 | rpge0d 13026 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑊) → 0 ≤ (𝐹‘𝑚)) |
41 | 1, 30, 32, 39, 40 | climserle 15615 | . . 3 ⊢ (𝜑 → (seq𝑁( + , 𝐹)‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴) |
42 | 27, 41 | eqbrtrrd 5165 | . 2 ⊢ (𝜑 → (𝐹‘𝑁) ≤ Σ𝑘 ∈ 𝑊 𝐴) |
43 | 21, 25, 42 | rpgecld 13061 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝑊 𝐴 ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ⊆ wss 3943 dom cdm 5669 ‘cfv 6537 ℝcr 11111 + caddc 11115 ≤ cle 11253 ℤcz 12562 ℤ≥cuz 12826 ℝ+crp 12980 seqcseq 13972 ⇝ cli 15434 Σcsu 15638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-fl 13763 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-rlim 15439 df-sum 15639 |
This theorem is referenced by: effsumlt 16061 eirrlem 16154 aaliou3lem3 26234 |
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