Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumgsum | Structured version Visualization version GIF version |
Description: A finite extended sum is the group sum over the extended nonnegative real numbers. (Contributed by Thierry Arnoux, 24-Apr-2020.) |
Ref | Expression |
---|---|
esumgsum.1 | ⊢ Ⅎ𝑘𝜑 |
esumgsum.2 | ⊢ Ⅎ𝑘𝐴 |
esumgsum.3 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
esumgsum.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
Ref | Expression |
---|---|
esumgsum | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumgsum.1 | . 2 ⊢ Ⅎ𝑘𝜑 | |
2 | esumgsum.2 | . 2 ⊢ Ⅎ𝑘𝐴 | |
3 | esumgsum.3 | . 2 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
4 | esumgsum.4 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
5 | xrge0base 30667 | . . 3 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
6 | xrge00 30668 | . . 3 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
7 | xrge0cmn 20581 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
9 | xrge0tps 31180 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp) |
11 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑘(0[,]+∞) | |
12 | eqid 2821 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
13 | 1, 2, 11, 4, 12 | fmptdF 30395 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
14 | 4 | ex 415 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞))) |
15 | 1, 14 | ralrimi 3216 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
16 | 2 | fnmptf 6479 | . . . . 5 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞) → (𝑘 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
18 | 0xr 10682 | . . . . 5 ⊢ 0 ∈ ℝ* | |
19 | 18 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ*) |
20 | 17, 3, 19 | fndmfifsupp 8840 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) finSupp 0) |
21 | 5, 6, 8, 10, 3, 13, 20 | tsmsid 22742 | . 2 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))) |
22 | 1, 2, 3, 4, 21 | esumid 31298 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (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 ↦ cmpt 5139 Fn wfn 6345 (class class class)co 7150 Fincfn 8503 0cc0 10531 +∞cpnf 10666 ℝ*cxr 10668 [,]cicc 12735 ↾s cress 16478 Σg cgsu 16708 ℝ*𝑠cxrs 16767 CMndccmn 18900 TopSpctps 21534 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-cld 21621 df-ntr 21622 df-cls 21623 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: esum2d 31347 |
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