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 30599 | . . 3 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
6 | xrge00 30600 | . . 3 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
7 | xrge0cmn 20515 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
9 | xrge0tps 31084 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp) |
11 | nfcv 2974 | . . . 4 ⊢ Ⅎ𝑘(0[,]+∞) | |
12 | eqid 2818 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
13 | 1, 2, 11, 4, 12 | fmptdF 30329 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
14 | 4 | ex 413 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞))) |
15 | 1, 14 | ralrimi 3213 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
16 | 2 | fnmptf 6477 | . . . . 5 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞) → (𝑘 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
18 | 0xr 10676 | . . . . 5 ⊢ 0 ∈ ℝ* | |
19 | 18 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ*) |
20 | 17, 3, 19 | fndmfifsupp 8834 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) finSupp 0) |
21 | 5, 6, 8, 10, 3, 13, 20 | tsmsid 22675 | . 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 31202 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 Ⅎwnfc 2958 ∀wral 3135 ↦ cmpt 5137 Fn wfn 6343 (class class class)co 7145 Fincfn 8497 0cc0 10525 +∞cpnf 10660 ℝ*cxr 10662 [,]cicc 12729 ↾s cress 16472 Σg cgsu 16702 ℝ*𝑠cxrs 16761 CMndccmn 18835 TopSpctps 21468 Σ*cesum 31185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-xadd 12496 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-tset 16572 df-ple 16573 df-ds 16575 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-ordt 16762 df-xrs 16763 df-mre 16845 df-mrc 16846 df-acs 16848 df-ps 17798 df-tsr 17799 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-cntz 18385 df-cmn 18837 df-fbas 20470 df-fg 20471 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-cn 21763 df-haus 21851 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-tsms 22662 df-esum 31186 |
This theorem is referenced by: esum2d 31251 |
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