Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 29685 | . . 3 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
6 | xrge00 29686 | . . 3 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
7 | xrge0cmn 19788 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
9 | xrge0tps 29988 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp) |
11 | nfcv 2764 | . . . 4 ⊢ Ⅎ𝑘(0[,]+∞) | |
12 | eqid 2622 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
13 | 1, 2, 11, 4, 12 | fmptdF 29456 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
14 | 4 | ex 450 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐵 ∈ (0[,]+∞))) |
15 | 1, 14 | ralrimi 2957 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
16 | 2 | fnmptf 6016 | . . . . 5 ⊢ (∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞) → (𝑘 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
18 | 0xr 10086 | . . . . 5 ⊢ 0 ∈ ℝ* | |
19 | 18 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ*) |
20 | 17, 3, 19 | fndmfifsupp 8288 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) finSupp 0) |
21 | 5, 6, 8, 10, 3, 13, 20 | tsmsid 21943 | . 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 30106 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 ∀wral 2912 ↦ cmpt 4729 Fn wfn 5883 (class class class)co 6650 Fincfn 7955 0cc0 9936 +∞cpnf 10071 ℝ*cxr 10073 [,]cicc 12178 ↾s cress 15858 Σg cgsu 16101 ℝ*𝑠cxrs 16160 CMndccmn 18193 TopSpctps 20736 Σ*cesum 30089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-xadd 11947 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-tset 15960 df-ple 15961 df-ds 15964 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-ordt 16161 df-xrs 16162 df-mre 16246 df-mrc 16247 df-acs 16249 df-ps 17200 df-tsr 17201 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-cntz 17750 df-cmn 18195 df-fbas 19743 df-fg 19744 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-cn 21031 df-haus 21119 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-tsms 21930 df-esum 30090 |
This theorem is referenced by: esum2d 30155 |
Copyright terms: Public domain | W3C validator |