Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumcl | Structured version Visualization version GIF version |
Description: Closure for extended sum in the extended positive reals. (Contributed by Thierry Arnoux, 2-Jan-2017.) |
Ref | Expression |
---|---|
esumcl.1 | ⊢ Ⅎ𝑘𝐴 |
Ref | Expression |
---|---|
esumcl | ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrge0base 30967 | . . 3 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
2 | xrge0cmn 20359 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
4 | xrge0tps 31560 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp) |
6 | simpl 486 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → 𝐴 ∈ 𝑉) | |
7 | esumcl.1 | . . . . . 6 ⊢ Ⅎ𝑘𝐴 | |
8 | 7 | nfel1 2913 | . . . . 5 ⊢ Ⅎ𝑘 𝐴 ∈ 𝑉 |
9 | nfra1 3130 | . . . . 5 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞) | |
10 | 8, 9 | nfan 1907 | . . . 4 ⊢ Ⅎ𝑘(𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) |
11 | nfcv 2897 | . . . 4 ⊢ Ⅎ𝑘(0[,]+∞) | |
12 | simpr 488 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) | |
13 | 12 | r19.21bi 3120 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
14 | eqid 2736 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
15 | 10, 7, 11, 13, 14 | fmptdF 30667 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
16 | 1, 3, 5, 6, 15 | tsmscl 22986 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ⊆ (0[,]+∞)) |
17 | df-esum 31662 | . . 3 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
18 | eqid 2736 | . . . 4 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞)) | |
19 | 18, 6, 15 | xrge0tsmsbi 30991 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → (Σ*𝑘 ∈ 𝐴𝐵 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) ↔ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)))) |
20 | 17, 19 | mpbiri 261 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))) |
21 | 16, 20 | sseldd 3888 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐴 𝐵 ∈ (0[,]+∞)) → Σ*𝑘 ∈ 𝐴𝐵 ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Ⅎwnfc 2877 ∀wral 3051 ∪ cuni 4805 ↦ cmpt 5120 (class class class)co 7191 0cc0 10694 +∞cpnf 10829 [,]cicc 12903 ↾s cress 16667 ℝ*𝑠cxrs 16959 CMndccmn 19124 TopSpctps 21783 tsums ctsu 22977 Σ*cesum 31661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-fi 9005 df-sup 9036 df-inf 9037 df-oi 9104 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-xadd 12670 df-ioo 12904 df-ioc 12905 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-seq 13540 df-hash 13862 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-tset 16768 df-ple 16769 df-ds 16771 df-rest 16881 df-topn 16882 df-0g 16900 df-gsum 16901 df-topgen 16902 df-ordt 16960 df-xrs 16961 df-mre 17043 df-mrc 17044 df-acs 17046 df-ps 18026 df-tsr 18027 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-cntz 18665 df-cmn 19126 df-fbas 20314 df-fg 20315 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-ntr 21871 df-nei 21949 df-cn 22078 df-haus 22166 df-fil 22697 df-fm 22789 df-flim 22790 df-flf 22791 df-tsms 22978 df-esum 31662 |
This theorem is referenced by: esumel 31681 esummono 31688 esumpad 31689 esumpad2 31690 esumle 31692 esumlef 31696 esumrnmpt2 31702 esumfsup 31704 esumpinfval 31707 esumpinfsum 31711 esumpmono 31713 esummulc1 31715 esummulc2 31716 esumdivc 31717 hasheuni 31719 esumcvg 31720 esumgect 31724 esum2dlem 31726 esum2d 31727 measiun 31852 omscl 31928 oms0 31930 omsmon 31931 omssubadd 31933 carsggect 31951 carsgclctunlem2 31952 omsmeas 31956 |
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