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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 12256 | . . 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 450 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞))) |
6 | 3, 5 | ralrimi 2957 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
7 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑘𝐴 | |
8 | 7 | esumcl 30092 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
9 | 2, 6, 8 | syl2anc 693 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
10 | 1, 9 | sseldi 3601 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) |
11 | nfrab1 3122 | . . . . 5 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} | |
12 | ssrab2 3687 | . . . . . 6 ⊢ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ⊆ 𝐴 | |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ⊆ 𝐴) |
14 | 0xr 10086 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
15 | pnfxr 10092 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
16 | 0lepnf 11966 | . . . . . . . 8 ⊢ 0 ≤ +∞ | |
17 | ubicc2 12289 | . . . . . . . 8 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞)) | |
18 | 14, 15, 16, 17 | mp3an 1424 | . . . . . . 7 ⊢ +∞ ∈ (0[,]+∞) |
19 | 18 | a1i 11 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = +∞) → +∞ ∈ (0[,]+∞)) |
20 | 0e0iccpnf 12283 | . . . . . . 7 ⊢ 0 ∈ (0[,]+∞) | |
21 | 20 | a1i 11 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 0 ∈ (0[,]+∞)) |
22 | 19, 21 | ifclda 4120 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝐵 = +∞, +∞, 0) ∈ (0[,]+∞)) |
23 | eldif 3584 | . . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) | |
24 | rabid 3116 | . . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ↔ (𝑘 ∈ 𝐴 ∧ 𝐵 = +∞)) | |
25 | 24 | simplbi2 655 | . . . . . . . . 9 ⊢ (𝑘 ∈ 𝐴 → (𝐵 = +∞ → 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) |
26 | 25 | con3dimp 457 | . . . . . . . 8 ⊢ ((𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → ¬ 𝐵 = +∞) |
27 | 23, 26 | sylbi 207 | . . . . . . 7 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → ¬ 𝐵 = +∞) |
28 | 27 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → ¬ 𝐵 = +∞) |
29 | 28 | iffalsed 4097 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → if(𝐵 = +∞, +∞, 0) = 0) |
30 | 3, 11, 7, 13, 2, 22, 29 | esumss 30134 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = Σ*𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0)) |
31 | eqidd 2623 | . . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} = {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) | |
32 | 24 | simprbi 480 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} → 𝐵 = +∞) |
33 | 32 | iftrued 4094 | . . . . . . 7 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} → if(𝐵 = +∞, +∞, 0) = +∞) |
34 | 33 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → if(𝐵 = +∞, +∞, 0) = +∞) |
35 | 3, 31, 34 | esumeq12dvaf 30093 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞) |
36 | 2, 13 | ssexd 4805 | . . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V) |
37 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑘+∞ | |
38 | 11, 37 | esumcst 30125 | . . . . . 6 ⊢ (({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V ∧ +∞ ∈ (0[,]+∞)) → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞ = ((#‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·e +∞)) |
39 | 36, 18, 38 | sylancl 694 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞ = ((#‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·e +∞)) |
40 | hashxrcl 13148 | . . . . . . 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 3958 | . . . . . . . 8 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅ ↔ ∃𝑘 ∈ 𝐴 𝐵 = +∞) | |
44 | 42, 43 | sylibr 224 | . . . . . . 7 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅) |
45 | hashgt0 13177 | . . . . . . 7 ⊢ (({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V ∧ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅) → 0 < (#‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) | |
46 | 36, 44, 45 | syl2anc 693 | . . . . . 6 ⊢ (𝜑 → 0 < (#‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) |
47 | xmulpnf1 12104 | . . . . . 6 ⊢ (((#‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ∈ ℝ* ∧ 0 < (#‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → ((#‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·e +∞) = +∞) | |
48 | 41, 46, 47 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → ((#‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·e +∞) = +∞) |
49 | 35, 39, 48 | 3eqtrd 2660 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = +∞) |
50 | 30, 49 | eqtr3d 2658 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0) = +∞) |
51 | breq1 4656 | . . . . 5 ⊢ (+∞ = if(𝐵 = +∞, +∞, 0) → (+∞ ≤ 𝐵 ↔ if(𝐵 = +∞, +∞, 0) ≤ 𝐵)) | |
52 | breq1 4656 | . . . . 5 ⊢ (0 = if(𝐵 = +∞, +∞, 0) → (0 ≤ 𝐵 ↔ if(𝐵 = +∞, +∞, 0) ≤ 𝐵)) | |
53 | pnfge 11964 | . . . . . . . 8 ⊢ (+∞ ∈ ℝ* → +∞ ≤ +∞) | |
54 | 15, 53 | ax-mp 5 | . . . . . . 7 ⊢ +∞ ≤ +∞ |
55 | breq2 4657 | . . . . . . 7 ⊢ (𝐵 = +∞ → (+∞ ≤ 𝐵 ↔ +∞ ≤ +∞)) | |
56 | 54, 55 | mpbiri 248 | . . . . . 6 ⊢ (𝐵 = +∞ → +∞ ≤ 𝐵) |
57 | 56 | adantl 482 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = +∞) → +∞ ≤ 𝐵) |
58 | 4 | adantr 481 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 𝐵 ∈ (0[,]+∞)) |
59 | iccgelb 12230 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐵 ∈ (0[,]+∞)) → 0 ≤ 𝐵) | |
60 | 14, 15, 59 | mp3an12 1414 | . . . . . 6 ⊢ (𝐵 ∈ (0[,]+∞) → 0 ≤ 𝐵) |
61 | 58, 60 | syl 17 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 0 ≤ 𝐵) |
62 | 51, 52, 57, 61 | ifbothda 4123 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝐵 = +∞, +∞, 0) ≤ 𝐵) |
63 | 3, 7, 2, 22, 4, 62 | esumlef 30124 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0) ≤ Σ*𝑘 ∈ 𝐴𝐵) |
64 | 50, 63 | eqbrtrrd 4677 | . 2 ⊢ (𝜑 → +∞ ≤ Σ*𝑘 ∈ 𝐴𝐵) |
65 | xgepnf 11996 | . . 3 ⊢ (Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ* → (+∞ ≤ Σ*𝑘 ∈ 𝐴𝐵 ↔ Σ*𝑘 ∈ 𝐴𝐵 = +∞)) | |
66 | 65 | biimpd 219 | . 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 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∖ cdif 3571 ⊆ wss 3574 ∅c0 3915 ifcif 4086 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 0cc0 9936 +∞cpnf 10071 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 ·e cxmu 11945 [,]cicc 12178 #chash 13117 Σ*cesum 30089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-ordt 16161 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-ps 17200 df-tsr 17201 df-plusf 17241 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-subrg 18778 df-abv 18817 df-lmod 18865 df-scaf 18866 df-sra 19172 df-rgmod 19173 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-tmd 21876 df-tgp 21877 df-tsms 21930 df-trg 21963 df-xms 22125 df-ms 22126 df-tms 22127 df-nm 22387 df-ngp 22388 df-nrg 22390 df-nlm 22391 df-ii 22680 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-esum 30090 |
This theorem is referenced by: hasheuni 30147 esumcvg 30148 esumcvgre 30153 voliune 30292 volfiniune 30293 |
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