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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumcvgre | Structured version Visualization version GIF version |
Description: All terms of a converging extended sum shall be finite. (Contributed by Thierry Arnoux, 23-Sep-2019.) |
Ref | Expression |
---|---|
esumcvgre.0 | ⊢ Ⅎ𝑘𝜑 |
esumcvgre.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
esumcvgre.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
esumcvgre.3 | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
esumcvgre | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumcvgre.0 | . . . . . . 7 ⊢ Ⅎ𝑘𝜑 | |
2 | nfre1 3245 | . . . . . . 7 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐴 𝐵 = +∞ | |
3 | 1, 2 | nfan 1862 | . . . . . 6 ⊢ Ⅎ𝑘(𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = +∞) |
4 | esumcvgre.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
5 | 4 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = +∞) → 𝐴 ∈ 𝑉) |
6 | esumcvgre.2 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
7 | 6 | adantlr 702 | . . . . . 6 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = +∞) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
8 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = +∞) → ∃𝑘 ∈ 𝐴 𝐵 = +∞) | |
9 | 3, 5, 7, 8 | esumpinfval 30933 | . . . . 5 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = +∞) → Σ*𝑘 ∈ 𝐴𝐵 = +∞) |
10 | esumcvgre.3 | . . . . . . . . 9 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ) | |
11 | ltpnf 12325 | . . . . . . . . . 10 ⊢ (Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ → Σ*𝑘 ∈ 𝐴𝐵 < +∞) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 < +∞) |
13 | 10, 12 | gtned 10567 | . . . . . . . 8 ⊢ (𝜑 → +∞ ≠ Σ*𝑘 ∈ 𝐴𝐵) |
14 | 13 | adantr 473 | . . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = +∞) → +∞ ≠ Σ*𝑘 ∈ 𝐴𝐵) |
15 | necom 3014 | . . . . . . . 8 ⊢ (+∞ ≠ Σ*𝑘 ∈ 𝐴𝐵 ↔ Σ*𝑘 ∈ 𝐴𝐵 ≠ +∞) | |
16 | 15 | imbi2i 328 | . . . . . . 7 ⊢ (((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = +∞) → +∞ ≠ Σ*𝑘 ∈ 𝐴𝐵) ↔ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = +∞) → Σ*𝑘 ∈ 𝐴𝐵 ≠ +∞)) |
17 | 14, 16 | mpbi 222 | . . . . . 6 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = +∞) → Σ*𝑘 ∈ 𝐴𝐵 ≠ +∞) |
18 | 17 | neneqd 2966 | . . . . 5 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐵 = +∞) → ¬ Σ*𝑘 ∈ 𝐴𝐵 = +∞) |
19 | 9, 18 | pm2.65da 804 | . . . 4 ⊢ (𝜑 → ¬ ∃𝑘 ∈ 𝐴 𝐵 = +∞) |
20 | ralnex 3177 | . . . 4 ⊢ (∀𝑘 ∈ 𝐴 ¬ 𝐵 = +∞ ↔ ¬ ∃𝑘 ∈ 𝐴 𝐵 = +∞) | |
21 | 19, 20 | sylibr 226 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 ¬ 𝐵 = +∞) |
22 | 21 | r19.21bi 3152 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ¬ 𝐵 = +∞) |
23 | eliccxr 12632 | . . . . . 6 ⊢ (𝐵 ∈ (0[,]+∞) → 𝐵 ∈ ℝ*) | |
24 | xrge0neqmnf 12649 | . . . . . 6 ⊢ (𝐵 ∈ (0[,]+∞) → 𝐵 ≠ -∞) | |
25 | xrnemnf 12322 | . . . . . . 7 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞) ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞)) | |
26 | 25 | biimpi 208 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐵 ≠ -∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞)) |
27 | 23, 24, 26 | syl2anc 576 | . . . . 5 ⊢ (𝐵 ∈ (0[,]+∞) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞)) |
28 | 6, 27 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 ∈ ℝ ∨ 𝐵 = +∞)) |
29 | 28 | orcomd 857 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐵 = +∞ ∨ 𝐵 ∈ ℝ)) |
30 | 29 | orcanai 985 | . 2 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐴) ∧ ¬ 𝐵 = +∞) → 𝐵 ∈ ℝ) |
31 | 22, 30 | mpdan 674 | 1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∨ wo 833 = wceq 1507 Ⅎwnf 1746 ∈ wcel 2048 ≠ wne 2961 ∀wral 3082 ∃wrex 3083 class class class wbr 4923 (class class class)co 6970 ℝcr 10326 0cc0 10327 +∞cpnf 10463 -∞cmnf 10464 ℝ*cxr 10465 < clt 10466 [,]cicc 12550 Σ*cesum 30887 |
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 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8890 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 ax-addf 10406 ax-mulf 10407 |
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 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-om 7391 df-1st 7494 df-2nd 7495 df-supp 7627 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-er 8081 df-map 8200 df-pm 8201 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-fsupp 8621 df-fi 8662 df-sup 8693 df-inf 8694 df-oi 8761 df-card 9154 df-cda 9380 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-xnn0 11773 df-z 11787 df-dec 11905 df-uz 12052 df-q 12156 df-rp 12198 df-xneg 12317 df-xadd 12318 df-xmul 12319 df-ioo 12551 df-ioc 12552 df-ico 12553 df-icc 12554 df-fz 12702 df-fzo 12843 df-fl 12970 df-mod 13046 df-seq 13178 df-exp 13238 df-fac 13442 df-bc 13471 df-hash 13499 df-shft 14277 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-limsup 14679 df-clim 14696 df-rlim 14697 df-sum 14894 df-ef 15271 df-sin 15273 df-cos 15274 df-pi 15276 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-starv 16426 df-sca 16427 df-vsca 16428 df-ip 16429 df-tset 16430 df-ple 16431 df-ds 16433 df-unif 16434 df-hom 16435 df-cco 16436 df-rest 16542 df-topn 16543 df-0g 16561 df-gsum 16562 df-topgen 16563 df-pt 16564 df-prds 16567 df-ordt 16620 df-xrs 16621 df-qtop 16626 df-imas 16627 df-xps 16629 df-mre 16705 df-mrc 16706 df-acs 16708 df-ps 17658 df-tsr 17659 df-plusf 17699 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-mhm 17793 df-submnd 17794 df-grp 17884 df-minusg 17885 df-sbg 17886 df-mulg 18002 df-subg 18050 df-cntz 18208 df-cmn 18658 df-abl 18659 df-mgp 18953 df-ur 18965 df-ring 19012 df-cring 19013 df-subrg 19246 df-abv 19300 df-lmod 19348 df-scaf 19349 df-sra 19656 df-rgmod 19657 df-psmet 20229 df-xmet 20230 df-met 20231 df-bl 20232 df-mopn 20233 df-fbas 20234 df-fg 20235 df-cnfld 20238 df-top 21196 df-topon 21213 df-topsp 21235 df-bases 21248 df-cld 21321 df-ntr 21322 df-cls 21323 df-nei 21400 df-lp 21438 df-perf 21439 df-cn 21529 df-cnp 21530 df-haus 21617 df-tx 21864 df-hmeo 22057 df-fil 22148 df-fm 22240 df-flim 22241 df-flf 22242 df-tmd 22374 df-tgp 22375 df-tsms 22428 df-trg 22461 df-xms 22623 df-ms 22624 df-tms 22625 df-nm 22885 df-ngp 22886 df-nrg 22888 df-nlm 22889 df-ii 23178 df-cncf 23179 df-limc 24157 df-dv 24158 df-log 24831 df-esum 30888 |
This theorem is referenced by: omssubadd 31160 |
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