Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumel | Structured version Visualization version GIF version |
Description: The extended sum is a limit point of the corresponding infinite group sum. (Contributed by Thierry Arnoux, 24-Mar-2017.) |
Ref | Expression |
---|---|
esumel.1 | ⊢ Ⅎ𝑘𝜑 |
esumel.2 | ⊢ Ⅎ𝑘𝐴 |
esumel.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
esumel.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
Ref | Expression |
---|---|
esumel | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumel.3 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | esumel.1 | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
3 | esumel.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
4 | 3 | ex 415 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞))) |
5 | 2, 4 | ralrimi 3216 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
6 | esumel.2 | . . . . 5 ⊢ Ⅎ𝑘𝐴 | |
7 | 6 | esumcl 31284 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
8 | 1, 5, 7 | syl2anc 586 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
9 | snidg 4593 | . . 3 ⊢ (Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞) → Σ*𝑘 ∈ 𝐴𝐵 ∈ {Σ*𝑘 ∈ 𝐴𝐵}) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ {Σ*𝑘 ∈ 𝐴𝐵}) |
11 | eqid 2821 | . . 3 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞)) | |
12 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑘(0[,]+∞) | |
13 | eqid 2821 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
14 | 2, 6, 12, 3, 13 | fmptdF 30395 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
15 | inss1 4205 | . . . . . . . . 9 ⊢ (𝒫 𝐴 ∩ Fin) ⊆ 𝒫 𝐴 | |
16 | simpr 487 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) | |
17 | 15, 16 | sseldi 3965 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ∈ 𝒫 𝐴) |
18 | 17 | elpwid 4553 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑥 ⊆ 𝐴) |
19 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑘𝑥 | |
20 | 6, 19 | resmptf 5902 | . . . . . . 7 ⊢ (𝑥 ⊆ 𝐴 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥) = (𝑘 ∈ 𝑥 ↦ 𝐵)) |
21 | 18, 20 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥) = (𝑘 ∈ 𝑥 ↦ 𝐵)) |
22 | 21 | eqcomd 2827 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → (𝑘 ∈ 𝑥 ↦ 𝐵) = ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥)) |
23 | 22 | oveq2d 7166 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝐴 ∩ Fin)) → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝑥 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥))) |
24 | 2, 6, 1, 3, 23 | esumval 31300 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = sup(ran (𝑥 ∈ (𝒫 𝐴 ∩ Fin) ↦ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg ((𝑘 ∈ 𝐴 ↦ 𝐵) ↾ 𝑥))), ℝ*, < )) |
25 | 11, 1, 14, 24 | xrge0tsmsd 30687 | . 2 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = {Σ*𝑘 ∈ 𝐴𝐵}) |
26 | 10, 25 | eleqtrrd 2916 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 Ⅎwnfc 2961 ∀wral 3138 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 {csn 4561 ↦ cmpt 5139 ↾ cres 5552 (class class class)co 7150 Fincfn 8503 0cc0 10531 +∞cpnf 10666 [,]cicc 12735 ↾s cress 16478 Σg cgsu 16708 ℝ*𝑠cxrs 16767 tsums ctsu 22728 Σ*cesum 31281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-xadd 12502 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-seq 13364 df-hash 13685 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-tset 16578 df-ple 16579 df-ds 16581 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-ordt 16768 df-xrs 16769 df-mre 16851 df-mrc 16852 df-acs 16854 df-ps 17804 df-tsr 17805 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-cntz 18441 df-cmn 18902 df-fbas 20536 df-fg 20537 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-ntr 21622 df-nei 21700 df-cn 21829 df-haus 21917 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-tsms 22729 df-esum 31282 |
This theorem is referenced by: esumsplit 31307 esumadd 31311 esumaddf 31315 esumcocn 31334 |
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