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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumpinfsum | Structured version Visualization version GIF version |
Description: The value of the extended sum of infinitely many terms greater than one. (Contributed by Thierry Arnoux, 29-Jun-2017.) |
Ref | Expression |
---|---|
esumpinfsum.p | ⊢ Ⅎ𝑘𝜑 |
esumpinfsum.a | ⊢ Ⅎ𝑘𝐴 |
esumpinfsum.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
esumpinfsum.2 | ⊢ (𝜑 → ¬ 𝐴 ∈ Fin) |
esumpinfsum.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
esumpinfsum.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ≤ 𝐵) |
esumpinfsum.5 | ⊢ (𝜑 → 𝑀 ∈ ℝ*) |
esumpinfsum.6 | ⊢ (𝜑 → 0 < 𝑀) |
Ref | Expression |
---|---|
esumpinfsum | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssxr 12635 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | esumpinfsum.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | esumpinfsum.p | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
4 | esumpinfsum.3 | . . . . . 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 | esumpinfsum.a | . . . . 5 ⊢ Ⅎ𝑘𝐴 | |
8 | 7 | esumcl 30939 | . . . 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 3856 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) |
11 | esumpinfsum.5 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ*) | |
12 | esumpinfsum.6 | . . . . . . 7 ⊢ (𝜑 → 0 < 𝑀) | |
13 | 0xr 10487 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
14 | xrltle 12359 | . . . . . . . 8 ⊢ ((0 ∈ ℝ* ∧ 𝑀 ∈ ℝ*) → (0 < 𝑀 → 0 ≤ 𝑀)) | |
15 | 13, 11, 14 | sylancr 578 | . . . . . . 7 ⊢ (𝜑 → (0 < 𝑀 → 0 ≤ 𝑀)) |
16 | 12, 15 | mpd 15 | . . . . . 6 ⊢ (𝜑 → 0 ≤ 𝑀) |
17 | pnfge 12342 | . . . . . . 7 ⊢ (𝑀 ∈ ℝ* → 𝑀 ≤ +∞) | |
18 | 11, 17 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑀 ≤ +∞) |
19 | pnfxr 10494 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
20 | elicc1 12598 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑀 ∈ (0[,]+∞) ↔ (𝑀 ∈ ℝ* ∧ 0 ≤ 𝑀 ∧ 𝑀 ≤ +∞))) | |
21 | 13, 19, 20 | mp2an 679 | . . . . . 6 ⊢ (𝑀 ∈ (0[,]+∞) ↔ (𝑀 ∈ ℝ* ∧ 0 ≤ 𝑀 ∧ 𝑀 ≤ +∞)) |
22 | 11, 16, 18, 21 | syl3anbrc 1323 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (0[,]+∞)) |
23 | nfcv 2932 | . . . . . 6 ⊢ Ⅎ𝑘𝑀 | |
24 | 7, 23 | esumcst 30972 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑀 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝑀 = ((♯‘𝐴) ·e 𝑀)) |
25 | 2, 22, 24 | syl2anc 576 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝑀 = ((♯‘𝐴) ·e 𝑀)) |
26 | esumpinfsum.2 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐴 ∈ Fin) | |
27 | hashinf 13510 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) | |
28 | 2, 26, 27 | syl2anc 576 | . . . . 5 ⊢ (𝜑 → (♯‘𝐴) = +∞) |
29 | 28 | oveq1d 6991 | . . . 4 ⊢ (𝜑 → ((♯‘𝐴) ·e 𝑀) = (+∞ ·e 𝑀)) |
30 | xmulpnf2 12484 | . . . . 5 ⊢ ((𝑀 ∈ ℝ* ∧ 0 < 𝑀) → (+∞ ·e 𝑀) = +∞) | |
31 | 11, 12, 30 | syl2anc 576 | . . . 4 ⊢ (𝜑 → (+∞ ·e 𝑀) = +∞) |
32 | 25, 29, 31 | 3eqtrd 2818 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝑀 = +∞) |
33 | 22 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ∈ (0[,]+∞)) |
34 | esumpinfsum.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ≤ 𝐵) | |
35 | 3, 7, 2, 33, 4, 34 | esumlef 30971 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝑀 ≤ Σ*𝑘 ∈ 𝐴𝐵) |
36 | 32, 35 | eqbrtrrd 4953 | . 2 ⊢ (𝜑 → +∞ ≤ Σ*𝑘 ∈ 𝐴𝐵) |
37 | xgepnf 12375 | . . 3 ⊢ (Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ* → (+∞ ≤ Σ*𝑘 ∈ 𝐴𝐵 ↔ Σ*𝑘 ∈ 𝐴𝐵 = +∞)) | |
38 | 37 | biimpd 221 | . 2 ⊢ (Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ* → (+∞ ≤ Σ*𝑘 ∈ 𝐴𝐵 → Σ*𝑘 ∈ 𝐴𝐵 = +∞)) |
39 | 10, 36, 38 | sylc 65 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = +∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 Ⅎwnf 1746 ∈ wcel 2050 Ⅎwnfc 2916 ∀wral 3088 class class class wbr 4929 ‘cfv 6188 (class class class)co 6976 Fincfn 8306 0cc0 10335 +∞cpnf 10471 ℝ*cxr 10473 < clt 10474 ≤ cle 10475 ·e cxmu 12323 [,]cicc 12557 ♯chash 13505 Σ*cesum 30936 |
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 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-inf2 8898 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-pre-sup 10413 ax-addf 10414 ax-mulf 10415 |
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 2584 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 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-of 7227 df-om 7397 df-1st 7501 df-2nd 7502 df-supp 7634 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-2o 7906 df-oadd 7909 df-er 8089 df-map 8208 df-pm 8209 df-ixp 8260 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-fsupp 8629 df-fi 8670 df-sup 8701 df-inf 8702 df-oi 8769 df-card 9162 df-cda 9388 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-div 11099 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-xnn0 11780 df-z 11794 df-dec 11912 df-uz 12059 df-q 12163 df-rp 12205 df-xneg 12324 df-xadd 12325 df-xmul 12326 df-ioo 12558 df-ioc 12559 df-ico 12560 df-icc 12561 df-fz 12709 df-fzo 12850 df-fl 12977 df-mod 13053 df-seq 13185 df-exp 13245 df-fac 13449 df-bc 13478 df-hash 13506 df-shft 14287 df-cj 14319 df-re 14320 df-im 14321 df-sqrt 14455 df-abs 14456 df-limsup 14689 df-clim 14706 df-rlim 14707 df-sum 14904 df-ef 15281 df-sin 15283 df-cos 15284 df-pi 15286 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-starv 16436 df-sca 16437 df-vsca 16438 df-ip 16439 df-tset 16440 df-ple 16441 df-ds 16443 df-unif 16444 df-hom 16445 df-cco 16446 df-rest 16552 df-topn 16553 df-0g 16571 df-gsum 16572 df-topgen 16573 df-pt 16574 df-prds 16577 df-ordt 16630 df-xrs 16631 df-qtop 16636 df-imas 16637 df-xps 16639 df-mre 16715 df-mrc 16716 df-acs 16718 df-ps 17668 df-tsr 17669 df-plusf 17709 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-mhm 17803 df-submnd 17804 df-grp 17894 df-minusg 17895 df-sbg 17896 df-mulg 18012 df-subg 18060 df-cntz 18218 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-ring 19022 df-cring 19023 df-subrg 19256 df-abv 19310 df-lmod 19358 df-scaf 19359 df-sra 19666 df-rgmod 19667 df-psmet 20239 df-xmet 20240 df-met 20241 df-bl 20242 df-mopn 20243 df-fbas 20244 df-fg 20245 df-cnfld 20248 df-top 21206 df-topon 21223 df-topsp 21245 df-bases 21258 df-cld 21331 df-ntr 21332 df-cls 21333 df-nei 21410 df-lp 21448 df-perf 21449 df-cn 21539 df-cnp 21540 df-haus 21627 df-tx 21874 df-hmeo 22067 df-fil 22158 df-fm 22250 df-flim 22251 df-flf 22252 df-tmd 22384 df-tgp 22385 df-tsms 22438 df-trg 22471 df-xms 22633 df-ms 22634 df-tms 22635 df-nm 22895 df-ngp 22896 df-nrg 22898 df-nlm 22899 df-ii 23188 df-cncf 23189 df-limc 24167 df-dv 24168 df-log 24841 df-esum 30937 |
This theorem is referenced by: hasheuni 30994 |
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