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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 13347 | . . 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 413 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞))) |
6 | 3, 5 | ralrimi 3240 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
7 | nfcv 2907 | . . . . 5 ⊢ Ⅎ𝑘𝐴 | |
8 | 7 | esumcl 32629 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
9 | 2, 6, 8 | syl2anc 584 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
10 | 1, 9 | sselid 3942 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) |
11 | nfrab1 3426 | . . . . 5 ⊢ Ⅎ𝑘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} | |
12 | ssrab2 4037 | . . . . . 6 ⊢ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ⊆ 𝐴 | |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ⊆ 𝐴) |
14 | 0xr 11202 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
15 | pnfxr 11209 | . . . . . . . 8 ⊢ +∞ ∈ ℝ* | |
16 | 0lepnf 13053 | . . . . . . . 8 ⊢ 0 ≤ +∞ | |
17 | ubicc2 13382 | . . . . . . . 8 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 0 ≤ +∞) → +∞ ∈ (0[,]+∞)) | |
18 | 14, 15, 16, 17 | mp3an 1461 | . . . . . . 7 ⊢ +∞ ∈ (0[,]+∞) |
19 | 18 | a1i 11 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ 𝐵 = +∞) → +∞ ∈ (0[,]+∞)) |
20 | 0e0iccpnf 13376 | . . . . . . 7 ⊢ 0 ∈ (0[,]+∞) | |
21 | 20 | a1i 11 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 0 ∈ (0[,]+∞)) |
22 | 19, 21 | ifclda 4521 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝐵 = +∞, +∞, 0) ∈ (0[,]+∞)) |
23 | eldif 3920 | . . . . . . . 8 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ↔ (𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) | |
24 | rabid 3427 | . . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ↔ (𝑘 ∈ 𝐴 ∧ 𝐵 = +∞)) | |
25 | 24 | simplbi2 501 | . . . . . . . . 9 ⊢ (𝑘 ∈ 𝐴 → (𝐵 = +∞ → 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) |
26 | 25 | con3dimp 409 | . . . . . . . 8 ⊢ ((𝑘 ∈ 𝐴 ∧ ¬ 𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → ¬ 𝐵 = +∞) |
27 | 23, 26 | sylbi 216 | . . . . . . 7 ⊢ (𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) → ¬ 𝐵 = +∞) |
28 | 27 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → ¬ 𝐵 = +∞) |
29 | 28 | iffalsed 4497 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → if(𝐵 = +∞, +∞, 0) = 0) |
30 | 3, 11, 7, 13, 2, 22, 29 | esumss 32671 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = Σ*𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0)) |
31 | eqidd 2737 | . . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} = {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) | |
32 | 24 | simprbi 497 | . . . . . . . 8 ⊢ (𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} → 𝐵 = +∞) |
33 | 32 | iftrued 4494 | . . . . . . 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 32630 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞) |
36 | 2, 13 | ssexd 5281 | . . . . . 6 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V) |
37 | nfcv 2907 | . . . . . . 7 ⊢ Ⅎ𝑘+∞ | |
38 | 11, 37 | esumcst 32662 | . . . . . 6 ⊢ (({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V ∧ +∞ ∈ (0[,]+∞)) → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞ = ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·e +∞)) |
39 | 36, 18, 38 | sylancl 586 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}+∞ = ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·e +∞)) |
40 | hashxrcl 14257 | . . . . . . 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 4345 | . . . . . . . 8 ⊢ ({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅ ↔ ∃𝑘 ∈ 𝐴 𝐵 = +∞) | |
44 | 42, 43 | sylibr 233 | . . . . . . 7 ⊢ (𝜑 → {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅) |
45 | hashgt0 14288 | . . . . . . 7 ⊢ (({𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ∈ V ∧ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞} ≠ ∅) → 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) | |
46 | 36, 44, 45 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) |
47 | xmulpnf1 13193 | . . . . . 6 ⊢ (((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ∈ ℝ* ∧ 0 < (♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞})) → ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·e +∞) = +∞) | |
48 | 41, 46, 47 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ((♯‘{𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}) ·e +∞) = +∞) |
49 | 35, 39, 48 | 3eqtrd 2780 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ {𝑘 ∈ 𝐴 ∣ 𝐵 = +∞}if(𝐵 = +∞, +∞, 0) = +∞) |
50 | 30, 49 | eqtr3d 2778 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0) = +∞) |
51 | breq1 5108 | . . . . 5 ⊢ (+∞ = if(𝐵 = +∞, +∞, 0) → (+∞ ≤ 𝐵 ↔ if(𝐵 = +∞, +∞, 0) ≤ 𝐵)) | |
52 | breq1 5108 | . . . . 5 ⊢ (0 = if(𝐵 = +∞, +∞, 0) → (0 ≤ 𝐵 ↔ if(𝐵 = +∞, +∞, 0) ≤ 𝐵)) | |
53 | pnfge 13051 | . . . . . . . 8 ⊢ (+∞ ∈ ℝ* → +∞ ≤ +∞) | |
54 | 15, 53 | ax-mp 5 | . . . . . . 7 ⊢ +∞ ≤ +∞ |
55 | breq2 5109 | . . . . . . 7 ⊢ (𝐵 = +∞ → (+∞ ≤ 𝐵 ↔ +∞ ≤ +∞)) | |
56 | 54, 55 | mpbiri 257 | . . . . . 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 13320 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐵 ∈ (0[,]+∞)) → 0 ≤ 𝐵) | |
60 | 14, 15, 59 | mp3an12 1451 | . . . . . 6 ⊢ (𝐵 ∈ (0[,]+∞) → 0 ≤ 𝐵) |
61 | 58, 60 | syl 17 | . . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 0 ≤ 𝐵) |
62 | 51, 52, 57, 61 | ifbothda 4524 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝐵 = +∞, +∞, 0) ≤ 𝐵) |
63 | 3, 7, 2, 22, 4, 62 | esumlef 32661 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴if(𝐵 = +∞, +∞, 0) ≤ Σ*𝑘 ∈ 𝐴𝐵) |
64 | 50, 63 | eqbrtrrd 5129 | . 2 ⊢ (𝜑 → +∞ ≤ Σ*𝑘 ∈ 𝐴𝐵) |
65 | xgepnf 13084 | . . 3 ⊢ (Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ* → (+∞ ≤ Σ*𝑘 ∈ 𝐴𝐵 ↔ Σ*𝑘 ∈ 𝐴𝐵 = +∞)) | |
66 | 65 | biimpd 228 | . 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 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 {crab 3407 Vcvv 3445 ∖ cdif 3907 ⊆ wss 3910 ∅c0 4282 ifcif 4486 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 0cc0 11051 +∞cpnf 11186 ℝ*cxr 11188 < clt 11189 ≤ cle 11190 ·e cxmu 13032 [,]cicc 13267 ♯chash 14230 Σ*cesum 32626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-oadd 8416 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-xnn0 12486 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-pi 15955 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-ordt 17383 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-ps 18455 df-tsr 18456 df-plusf 18496 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-mhm 18601 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-mulg 18873 df-subg 18925 df-cntz 19097 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-cring 19967 df-subrg 20220 df-abv 20276 df-lmod 20324 df-scaf 20325 df-sra 20633 df-rgmod 20634 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-tmd 23423 df-tgp 23424 df-tsms 23478 df-trg 23511 df-xms 23673 df-ms 23674 df-tms 23675 df-nm 23938 df-ngp 23939 df-nrg 23941 df-nlm 23942 df-ii 24240 df-cncf 24241 df-limc 25230 df-dv 25231 df-log 25912 df-esum 32627 |
This theorem is referenced by: hasheuni 32684 esumcvg 32685 esumcvgre 32690 voliune 32828 volfiniune 32829 |
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