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Mathbox for Thierry Arnoux |
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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 24725 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶(0[,]+∞)) → (𝐺 tsums 𝐹) ≈ 1o) |
5 | 1, 2, 4 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝐺 tsums 𝐹) ≈ 1o) |
6 | en1b 9037 | . . . 4 ⊢ ((𝐺 tsums 𝐹) ≈ 1o ↔ (𝐺 tsums 𝐹) = {∪ (𝐺 tsums 𝐹)}) | |
7 | 5, 6 | sylib 217 | . . 3 ⊢ (𝜑 → (𝐺 tsums 𝐹) = {∪ (𝐺 tsums 𝐹)}) |
8 | 7 | eleq2d 2814 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐺 tsums 𝐹) ↔ 𝐶 ∈ {∪ (𝐺 tsums 𝐹)})) |
9 | ovex 7447 | . . . . . . 7 ⊢ (𝐺 tsums 𝐹) ∈ V | |
10 | 9 | uniex 7738 | . . . . . 6 ⊢ ∪ (𝐺 tsums 𝐹) ∈ V |
11 | eleq1 2816 | . . . . . 6 ⊢ (𝐶 = ∪ (𝐺 tsums 𝐹) → (𝐶 ∈ V ↔ ∪ (𝐺 tsums 𝐹) ∈ V)) | |
12 | 10, 11 | mpbiri 258 | . . . . 5 ⊢ (𝐶 = ∪ (𝐺 tsums 𝐹) → 𝐶 ∈ V) |
13 | elsng 4638 | . . . . 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 268 | . . 3 ⊢ (𝐶 = ∪ (𝐺 tsums 𝐹) → 𝐶 ∈ {∪ (𝐺 tsums 𝐹)}) |
16 | elsni 4641 | . . 3 ⊢ (𝐶 ∈ {∪ (𝐺 tsums 𝐹)} → 𝐶 = ∪ (𝐺 tsums 𝐹)) | |
17 | 15, 16 | impbii 208 | . 2 ⊢ (𝐶 = ∪ (𝐺 tsums 𝐹) ↔ 𝐶 ∈ {∪ (𝐺 tsums 𝐹)}) |
18 | 8, 17 | bitr4di 289 | 1 ⊢ (𝜑 → (𝐶 ∈ (𝐺 tsums 𝐹) ↔ 𝐶 = ∪ (𝐺 tsums 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 Vcvv 3469 {csn 4624 ∪ cuni 4903 class class class wbr 5142 ⟶wf 6538 (class class class)co 7414 1oc1o 8471 ≈ cen 8950 0cc0 11124 +∞cpnf 11261 [,]cicc 13345 ↾s cress 17194 ℝ*𝑠cxrs 17467 tsums ctsu 24004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-fi 9420 df-sup 9451 df-inf 9452 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-q 12949 df-xadd 13111 df-ioo 13346 df-ioc 13347 df-ico 13348 df-icc 13349 df-fz 13503 df-fzo 13646 df-seq 13985 df-hash 14308 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-tset 17237 df-ple 17238 df-ds 17240 df-rest 17389 df-topn 17390 df-0g 17408 df-gsum 17409 df-topgen 17410 df-ordt 17468 df-xrs 17469 df-mre 17551 df-mrc 17552 df-acs 17554 df-ps 18543 df-tsr 18544 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-submnd 18726 df-cntz 19252 df-cmn 19721 df-fbas 21256 df-fg 21257 df-top 22770 df-topon 22787 df-topsp 22809 df-bases 22823 df-ntr 22898 df-nei 22976 df-cn 23105 df-haus 23193 df-fil 23724 df-fm 23816 df-flim 23817 df-flf 23818 df-tsms 24005 |
This theorem is referenced by: esumcl 33572 |
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