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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumpinfval | Structured version Visualization version GIF version |
Description: The value of the extended sum of nonnegative terms, with at least one infinite term. (Contributed by Thierry Arnoux, 19-Jun-2017.) |
Ref | Expression |
---|---|
esumpinfval.0 | ⊢ Ⅎ𝑘𝜑 |
esumpinfval.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
esumpinfval.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
esumpinfval.3 | ⊢ (𝜑 → ∃𝑘 ∈ 𝐴 𝐵 = +∞) |
Ref | Expression |
---|---|
esumpinfval | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssxr 12638 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | esumpinfval.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | esumpinfval.0 | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
4 | esumpinfval.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
5 | 4 | ex 405 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞))) |
6 | 3, 5 | ralrimi 3166 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
7 | nfcv 2932 | . . . . 5 ⊢ Ⅎ𝑘𝐴 | |
8 | 7 | esumcl 30933 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
9 | 2, 6, 8 | syl2anc 576 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
10 | 1, 9 | sseldi 3858 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) |
11 | nfrab1 3324 | . . . . 5 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} | |
12 | ssrab2 3948 | . . . . . 6 ⊢ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ⊆ 𝐴 | |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ⊆ 𝐴) |
14 | 0xr 10489 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
15 | pnfxr 10496 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
16 | 0lepnf 12347 | . . . . . . . 8 ⊢ 0 ≤ +∞ | |
17 | ubicc2 12672 | . . . . . . . 8 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞)) | |
18 | 14, 15, 16, 17 | mp3an 1440 | . . . . . . 7 ⊢ +∞ ∈ (0[,]+∞) |
19 | 18 | a1i 11 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = +∞) → +∞ ∈ (0[,]+∞)) |
20 | 0e0iccpnf 12666 | . . . . . . 7 ⊢ 0 ∈ (0[,]+∞) | |
21 | 20 | a1i 11 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 0 ∈ (0[,]+∞)) |
22 | 19, 21 | ifclda 4385 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝐵 = +∞, +∞, 0) ∈ (0[,]+∞)) |
23 | eldif 3841 | . . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) | |
24 | rabid 3317 | . . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ↔ (𝑘 ∈ 𝐴 ∧ 𝐵 = +∞)) | |
25 | 24 | simplbi2 493 | . . . . . . . . 9 ⊢ (𝑘 ∈ 𝐴 → (𝐵 = +∞ → 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) |
26 | 25 | con3dimp 400 | . . . . . . . 8 ⊢ ((𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → ¬ 𝐵 = +∞) |
27 | 23, 26 | sylbi 209 | . . . . . . 7 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → ¬ 𝐵 = +∞) |
28 | 27 | adantl 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → ¬ 𝐵 = +∞) |
29 | 28 | iffalsed 4362 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → if(𝐵 = +∞, +∞, 0) = 0) |
30 | 3, 11, 7, 13, 2, 22, 29 | esumss 30975 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = Σ*𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0)) |
31 | eqidd 2779 | . . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} = {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) | |
32 | 24 | simprbi 489 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} → 𝐵 = +∞) |
33 | 32 | iftrued 4359 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} → if(𝐵 = +∞, +∞, 0) = +∞) |
34 | 33 | adantl 474 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → if(𝐵 = +∞, +∞, 0) = +∞) |
35 | 3, 31, 34 | esumeq12dvaf 30934 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞) |
36 | 2, 13 | ssexd 5085 | . . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V) |
37 | nfcv 2932 | . . . . . . 7 ⊢ Ⅎ𝑘+∞ | |
38 | 11, 37 | esumcst 30966 | . . . . . 6 ⊢ (({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V ∧ +∞ ∈ (0[,]+∞)) → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞ = ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·e +∞)) |
39 | 36, 18, 38 | sylancl 577 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞ = ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·e +∞)) |
40 | hashxrcl 13536 | . . . . . . 7 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V → (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ∈ ℝ*) | |
41 | 36, 40 | syl 17 | . . . . . 6 ⊢ (𝜑 → (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ∈ ℝ*) |
42 | esumpinfval.3 | . . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ 𝐴 𝐵 = +∞) | |
43 | rabn0 4227 | . . . . . . . 8 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅ ↔ ∃𝑘 ∈ 𝐴 𝐵 = +∞) | |
44 | 42, 43 | sylibr 226 | . . . . . . 7 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅) |
45 | hashgt0 13565 | . . . . . . 7 ⊢ (({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V ∧ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅) → 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) | |
46 | 36, 44, 45 | syl2anc 576 | . . . . . 6 ⊢ (𝜑 → 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) |
47 | xmulpnf1 12486 | . . . . . 6 ⊢ (((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ∈ ℝ* ∧ 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·e +∞) = +∞) | |
48 | 41, 46, 47 | syl2anc 576 | . . . . 5 ⊢ (𝜑 → ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·e +∞) = +∞) |
49 | 35, 39, 48 | 3eqtrd 2818 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = +∞) |
50 | 30, 49 | eqtr3d 2816 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0) = +∞) |
51 | breq1 4933 | . . . . 5 ⊢ (+∞ = if(𝐵 = +∞, +∞, 0) → (+∞ ≤ 𝐵 ↔ if(𝐵 = +∞, +∞, 0) ≤ 𝐵)) | |
52 | breq1 4933 | . . . . 5 ⊢ (0 = if(𝐵 = +∞, +∞, 0) → (0 ≤ 𝐵 ↔ if(𝐵 = +∞, +∞, 0) ≤ 𝐵)) | |
53 | pnfge 12345 | . . . . . . . 8 ⊢ (+∞ ∈ ℝ* → +∞ ≤ +∞) | |
54 | 15, 53 | ax-mp 5 | . . . . . . 7 ⊢ +∞ ≤ +∞ |
55 | breq2 4934 | . . . . . . 7 ⊢ (𝐵 = +∞ → (+∞ ≤ 𝐵 ↔ +∞ ≤ +∞)) | |
56 | 54, 55 | mpbiri 250 | . . . . . 6 ⊢ (𝐵 = +∞ → +∞ ≤ 𝐵) |
57 | 56 | adantl 474 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = +∞) → +∞ ≤ 𝐵) |
58 | 4 | adantr 473 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 𝐵 ∈ (0[,]+∞)) |
59 | iccgelb 12612 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐵 ∈ (0[,]+∞)) → 0 ≤ 𝐵) | |
60 | 14, 15, 59 | mp3an12 1430 | . . . . . 6 ⊢ (𝐵 ∈ (0[,]+∞) → 0 ≤ 𝐵) |
61 | 58, 60 | syl 17 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 0 ≤ 𝐵) |
62 | 51, 52, 57, 61 | ifbothda 4388 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝐵 = +∞, +∞, 0) ≤ 𝐵) |
63 | 3, 7, 2, 22, 4, 62 | esumlef 30965 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0) ≤ Σ*𝑘 ∈ 𝐴𝐵) |
64 | 50, 63 | eqbrtrrd 4954 | . 2 ⊢ (𝜑 → +∞ ≤ Σ*𝑘 ∈ 𝐴𝐵) |
65 | xgepnf 12378 | . . 3 ⊢ (Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ* → (+∞ ≤ Σ*𝑘 ∈ 𝐴𝐵 ↔ Σ*𝑘 ∈ 𝐴𝐵 = +∞)) | |
66 | 65 | biimpd 221 | . 2 ⊢ (Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ* → (+∞ ≤ Σ*𝑘 ∈ 𝐴𝐵 → Σ*𝑘 ∈ 𝐴𝐵 = +∞)) |
67 | 10, 64, 66 | sylc 65 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = +∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1507 Ⅎwnf 1746 ∈ wcel 2050 ≠ wne 2967 ∀wral 3088 ∃wrex 3089 {crab 3092 Vcvv 3415 ∖ cdif 3828 ⊆ wss 3831 ∅c0 4180 ifcif 4351 class class class wbr 4930 ‘cfv 6190 (class class class)co 6978 0cc0 10337 +∞cpnf 10473 ℝ*cxr 10475 < clt 10476 ≤ cle 10477 ·e cxmu 12326 [,]cicc 12560 ♯chash 13508 Σ*cesum 30930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-inf2 8900 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415 ax-addf 10416 ax-mulf 10417 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-iin 4796 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-of 7229 df-om 7399 df-1st 7503 df-2nd 7504 df-supp 7636 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-2o 7908 df-oadd 7911 df-er 8091 df-map 8210 df-pm 8211 df-ixp 8262 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-fsupp 8631 df-fi 8672 df-sup 8703 df-inf 8704 df-oi 8771 df-card 9164 df-cda 9390 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-7 11511 df-8 11512 df-9 11513 df-n0 11711 df-xnn0 11783 df-z 11797 df-dec 11915 df-uz 12062 df-q 12166 df-rp 12208 df-xneg 12327 df-xadd 12328 df-xmul 12329 df-ioo 12561 df-ioc 12562 df-ico 12563 df-icc 12564 df-fz 12712 df-fzo 12853 df-fl 12980 df-mod 13056 df-seq 13188 df-exp 13248 df-fac 13452 df-bc 13481 df-hash 13509 df-shft 14290 df-cj 14322 df-re 14323 df-im 14324 df-sqrt 14458 df-abs 14459 df-limsup 14692 df-clim 14709 df-rlim 14710 df-sum 14907 df-ef 15284 df-sin 15286 df-cos 15287 df-pi 15289 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-starv 16439 df-sca 16440 df-vsca 16441 df-ip 16442 df-tset 16443 df-ple 16444 df-ds 16446 df-unif 16447 df-hom 16448 df-cco 16449 df-rest 16555 df-topn 16556 df-0g 16574 df-gsum 16575 df-topgen 16576 df-pt 16577 df-prds 16580 df-ordt 16633 df-xrs 16634 df-qtop 16639 df-imas 16640 df-xps 16642 df-mre 16718 df-mrc 16719 df-acs 16721 df-ps 17671 df-tsr 17672 df-plusf 17712 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-mhm 17806 df-submnd 17807 df-grp 17897 df-minusg 17898 df-sbg 17899 df-mulg 18015 df-subg 18063 df-cntz 18221 df-cmn 18671 df-abl 18672 df-mgp 18966 df-ur 18978 df-ring 19025 df-cring 19026 df-subrg 19259 df-abv 19313 df-lmod 19361 df-scaf 19362 df-sra 19669 df-rgmod 19670 df-psmet 20242 df-xmet 20243 df-met 20244 df-bl 20245 df-mopn 20246 df-fbas 20247 df-fg 20248 df-cnfld 20251 df-top 21209 df-topon 21226 df-topsp 21248 df-bases 21261 df-cld 21334 df-ntr 21335 df-cls 21336 df-nei 21413 df-lp 21451 df-perf 21452 df-cn 21542 df-cnp 21543 df-haus 21630 df-tx 21877 df-hmeo 22070 df-fil 22161 df-fm 22253 df-flim 22254 df-flf 22255 df-tmd 22387 df-tgp 22388 df-tsms 22441 df-trg 22474 df-xms 22636 df-ms 22637 df-tms 22638 df-nm 22898 df-ngp 22899 df-nrg 22901 df-nlm 22902 df-ii 23191 df-cncf 23192 df-limc 24170 df-dv 24171 df-log 24844 df-esum 30931 |
This theorem is referenced by: hasheuni 30988 esumcvg 30989 esumcvgre 30994 voliune 31133 volfiniune 31134 |
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