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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0tsmsbi | Structured version Visualization version GIF version |
Description: Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 23-Jun-2017.) |
Ref | Expression |
---|---|
xrge0tsmseq.g | ⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞)) |
xrge0tsmseq.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
xrge0tsmseq.f | ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞)) |
Ref | Expression |
---|---|
xrge0tsmsbi | ⊢ (𝜑 → (𝐶 ∈ (𝐺 tsums 𝐹) ↔ 𝐶 = ∪ (𝐺 tsums 𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrge0tsmseq.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | xrge0tsmseq.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞)) | |
3 | xrge0tsmseq.g | . . . . . 6 ⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞)) | |
4 | 3 | xrge0tsms2 23370 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶(0[,]+∞)) → (𝐺 tsums 𝐹) ≈ 1o) |
5 | 1, 2, 4 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐺 tsums 𝐹) ≈ 1o) |
6 | en1b 8565 | . . . 4 ⊢ ((𝐺 tsums 𝐹) ≈ 1o ↔ (𝐺 tsums 𝐹) = {∪ (𝐺 tsums 𝐹)}) | |
7 | 5, 6 | sylib 219 | . . 3 ⊢ (𝜑 → (𝐺 tsums 𝐹) = {∪ (𝐺 tsums 𝐹)}) |
8 | 7 | eleq2d 2895 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐺 tsums 𝐹) ↔ 𝐶 ∈ {∪ (𝐺 tsums 𝐹)})) |
9 | ovex 7178 | . . . . . . 7 ⊢ (𝐺 tsums 𝐹) ∈ V | |
10 | 9 | uniex 7454 | . . . . . 6 ⊢ ∪ (𝐺 tsums 𝐹) ∈ V |
11 | eleq1 2897 | . . . . . 6 ⊢ (𝐶 = ∪ (𝐺 tsums 𝐹) → (𝐶 ∈ V ↔ ∪ (𝐺 tsums 𝐹) ∈ V)) | |
12 | 10, 11 | mpbiri 259 | . . . . 5 ⊢ (𝐶 = ∪ (𝐺 tsums 𝐹) → 𝐶 ∈ V) |
13 | elsng 4571 | . . . . 5 ⊢ (𝐶 ∈ V → (𝐶 ∈ {∪ (𝐺 tsums 𝐹)} ↔ 𝐶 = ∪ (𝐺 tsums 𝐹))) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝐶 = ∪ (𝐺 tsums 𝐹) → (𝐶 ∈ {∪ (𝐺 tsums 𝐹)} ↔ 𝐶 = ∪ (𝐺 tsums 𝐹))) |
15 | 14 | ibir 269 | . . 3 ⊢ (𝐶 = ∪ (𝐺 tsums 𝐹) → 𝐶 ∈ {∪ (𝐺 tsums 𝐹)}) |
16 | elsni 4574 | . . 3 ⊢ (𝐶 ∈ {∪ (𝐺 tsums 𝐹)} → 𝐶 = ∪ (𝐺 tsums 𝐹)) | |
17 | 15, 16 | impbii 210 | . 2 ⊢ (𝐶 = ∪ (𝐺 tsums 𝐹) ↔ 𝐶 ∈ {∪ (𝐺 tsums 𝐹)}) |
18 | 8, 17 | syl6bbr 290 | 1 ⊢ (𝜑 → (𝐶 ∈ (𝐺 tsums 𝐹) ↔ 𝐶 = ∪ (𝐺 tsums 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {csn 4557 ∪ cuni 4830 class class class wbr 5057 ⟶wf 6344 (class class class)co 7145 1oc1o 8084 ≈ cen 8494 0cc0 10525 +∞cpnf 10660 [,]cicc 12729 ↾s cress 16472 ℝ*𝑠cxrs 16761 tsums ctsu 22661 |
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-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-ntr 21556 df-nei 21634 df-cn 21763 df-haus 21851 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-tsms 22662 |
This theorem is referenced by: esumcl 31188 |
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