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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 13405 | . . 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 412 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞))) |
6 | 3, 5 | ralrimi 3246 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
7 | esumpinfsum.a | . . . . 5 ⊢ Ⅎ𝑘𝐴 | |
8 | 7 | esumcl 33520 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
9 | 2, 6, 8 | syl2anc 583 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
10 | 1, 9 | sselid 3973 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ*) |
11 | esumpinfsum.5 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ*) | |
12 | esumpinfsum.6 | . . . . . . 7 ⊢ (𝜑 → 0 < 𝑀) | |
13 | 0xr 11259 | . . . . . . . 8 ⊢ 0 ∈ ℝ* | |
14 | xrltle 13126 | . . . . . . . 8 ⊢ ((0 ∈ ℝ* ∧ 𝑀 ∈ ℝ*) → (0 < 𝑀 → 0 ≤ 𝑀)) | |
15 | 13, 11, 14 | sylancr 586 | . . . . . . 7 ⊢ (𝜑 → (0 < 𝑀 → 0 ≤ 𝑀)) |
16 | 12, 15 | mpd 15 | . . . . . 6 ⊢ (𝜑 → 0 ≤ 𝑀) |
17 | pnfge 13108 | . . . . . . 7 ⊢ (𝑀 ∈ ℝ* → 𝑀 ≤ +∞) | |
18 | 11, 17 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑀 ≤ +∞) |
19 | pnfxr 11266 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
20 | elicc1 13366 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (𝑀 ∈ (0[,]+∞) ↔ (𝑀 ∈ ℝ* ∧ 0 ≤ 𝑀 ∧ 𝑀 ≤ +∞))) | |
21 | 13, 19, 20 | mp2an 689 | . . . . . 6 ⊢ (𝑀 ∈ (0[,]+∞) ↔ (𝑀 ∈ ℝ* ∧ 0 ≤ 𝑀 ∧ 𝑀 ≤ +∞)) |
22 | 11, 16, 18, 21 | syl3anbrc 1340 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (0[,]+∞)) |
23 | nfcv 2895 | . . . . . 6 ⊢ Ⅎ𝑘𝑀 | |
24 | 7, 23 | esumcst 33553 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑀 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝑀 = ((♯‘𝐴) ·e 𝑀)) |
25 | 2, 22, 24 | syl2anc 583 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝑀 = ((♯‘𝐴) ·e 𝑀)) |
26 | esumpinfsum.2 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐴 ∈ Fin) | |
27 | hashinf 14293 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘𝐴) = +∞) | |
28 | 2, 26, 27 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (♯‘𝐴) = +∞) |
29 | 28 | oveq1d 7417 | . . . 4 ⊢ (𝜑 → ((♯‘𝐴) ·e 𝑀) = (+∞ ·e 𝑀)) |
30 | xmulpnf2 13252 | . . . . 5 ⊢ ((𝑀 ∈ ℝ* ∧ 0 < 𝑀) → (+∞ ·e 𝑀) = +∞) | |
31 | 11, 12, 30 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (+∞ ·e 𝑀) = +∞) |
32 | 25, 29, 31 | 3eqtrd 2768 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝑀 = +∞) |
33 | 22 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ∈ (0[,]+∞)) |
34 | esumpinfsum.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑀 ≤ 𝐵) | |
35 | 3, 7, 2, 33, 4, 34 | esumlef 33552 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝑀 ≤ Σ*𝑘 ∈ 𝐴𝐵) |
36 | 32, 35 | eqbrtrrd 5163 | . 2 ⊢ (𝜑 → +∞ ≤ Σ*𝑘 ∈ 𝐴𝐵) |
37 | xgepnf 13142 | . . 3 ⊢ (Σ*𝑘 ∈ 𝐴𝐵 ∈ ℝ* → (+∞ ≤ Σ*𝑘 ∈ 𝐴𝐵 ↔ Σ*𝑘 ∈ 𝐴𝐵 = +∞)) | |
38 | 37 | biimpd 228 | . 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 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2875 ∀wral 3053 class class class wbr 5139 ‘cfv 6534 (class class class)co 7402 Fincfn 8936 0cc0 11107 +∞cpnf 11243 ℝ*cxr 11245 < clt 11246 ≤ cle 11247 ·e cxmu 13089 [,]cicc 13325 ♯chash 14288 Σ*cesum 33517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-inf2 9633 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-xnn0 12543 df-z 12557 df-dec 12676 df-uz 12821 df-q 12931 df-rp 12973 df-xneg 13090 df-xadd 13091 df-xmul 13092 df-ioo 13326 df-ioc 13327 df-ico 13328 df-icc 13329 df-fz 13483 df-fzo 13626 df-fl 13755 df-mod 13833 df-seq 13965 df-exp 14026 df-fac 14232 df-bc 14261 df-hash 14289 df-shft 15012 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 df-abs 15181 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15631 df-ef 16009 df-sin 16011 df-cos 16012 df-pi 16014 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-starv 17213 df-sca 17214 df-vsca 17215 df-ip 17216 df-tset 17217 df-ple 17218 df-ds 17220 df-unif 17221 df-hom 17222 df-cco 17223 df-rest 17369 df-topn 17370 df-0g 17388 df-gsum 17389 df-topgen 17390 df-pt 17391 df-prds 17394 df-ordt 17448 df-xrs 17449 df-qtop 17454 df-imas 17455 df-xps 17457 df-mre 17531 df-mrc 17532 df-acs 17534 df-ps 18523 df-tsr 18524 df-plusf 18564 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18988 df-subg 19042 df-cntz 19225 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-ring 20132 df-cring 20133 df-subrng 20438 df-subrg 20463 df-abv 20652 df-lmod 20700 df-scaf 20701 df-sra 21013 df-rgmod 21014 df-psmet 21222 df-xmet 21223 df-met 21224 df-bl 21225 df-mopn 21226 df-fbas 21227 df-fg 21228 df-cnfld 21231 df-top 22720 df-topon 22737 df-topsp 22759 df-bases 22773 df-cld 22847 df-ntr 22848 df-cls 22849 df-nei 22926 df-lp 22964 df-perf 22965 df-cn 23055 df-cnp 23056 df-haus 23143 df-tx 23390 df-hmeo 23583 df-fil 23674 df-fm 23766 df-flim 23767 df-flf 23768 df-tmd 23900 df-tgp 23901 df-tsms 23955 df-trg 23988 df-xms 24150 df-ms 24151 df-tms 24152 df-nm 24415 df-ngp 24416 df-nrg 24418 df-nlm 24419 df-ii 24721 df-cncf 24722 df-limc 25719 df-dv 25720 df-log 26409 df-esum 33518 |
This theorem is referenced by: hasheuni 33575 |
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