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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0tsmseq | 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, 24-Mar-2017.) |
Ref | Expression |
---|---|
xrge0tsmseq.g | ⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞)) |
xrge0tsmseq.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
xrge0tsmseq.f | ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞)) |
xrge0tsmseq.h | ⊢ (𝜑 → 𝐶 ∈ (𝐺 tsums 𝐹)) |
Ref | Expression |
---|---|
xrge0tsmseq | ⊢ (𝜑 → 𝐶 = ∪ (𝐺 tsums 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrge0tsmseq.h | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝐺 tsums 𝐹)) | |
2 | xrge0tsmseq.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | xrge0tsmseq.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞)) | |
4 | xrge0tsmseq.g | . . . . . 6 ⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞)) | |
5 | 4 | xrge0tsms2 23057 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶(0[,]+∞)) → (𝐺 tsums 𝐹) ≈ 1o) |
6 | 2, 3, 5 | syl2anc 579 | . . . 4 ⊢ (𝜑 → (𝐺 tsums 𝐹) ≈ 1o) |
7 | en1eqsn 8480 | . . . 4 ⊢ ((𝐶 ∈ (𝐺 tsums 𝐹) ∧ (𝐺 tsums 𝐹) ≈ 1o) → (𝐺 tsums 𝐹) = {𝐶}) | |
8 | 1, 6, 7 | syl2anc 579 | . . 3 ⊢ (𝜑 → (𝐺 tsums 𝐹) = {𝐶}) |
9 | 8 | unieqd 4683 | . 2 ⊢ (𝜑 → ∪ (𝐺 tsums 𝐹) = ∪ {𝐶}) |
10 | unisng 4688 | . . 3 ⊢ (𝐶 ∈ (𝐺 tsums 𝐹) → ∪ {𝐶} = 𝐶) | |
11 | 1, 10 | syl 17 | . 2 ⊢ (𝜑 → ∪ {𝐶} = 𝐶) |
12 | 9, 11 | eqtr2d 2815 | 1 ⊢ (𝜑 → 𝐶 = ∪ (𝐺 tsums 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 {csn 4398 ∪ cuni 4673 class class class wbr 4888 ⟶wf 6133 (class class class)co 6924 1oc1o 7838 ≈ cen 8240 0cc0 10274 +∞cpnf 10410 [,]cicc 12495 ↾s cress 16267 ℝ*𝑠cxrs 16557 tsums ctsu 22348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-4 11445 df-5 11446 df-6 11447 df-7 11448 df-8 11449 df-9 11450 df-n0 11648 df-z 11734 df-dec 11851 df-uz 11998 df-q 12101 df-xadd 12263 df-ioo 12496 df-ioc 12497 df-ico 12498 df-icc 12499 df-fz 12649 df-fzo 12790 df-seq 13125 df-hash 13442 df-struct 16268 df-ndx 16269 df-slot 16270 df-base 16272 df-sets 16273 df-ress 16274 df-plusg 16362 df-mulr 16363 df-tset 16368 df-ple 16369 df-ds 16371 df-rest 16480 df-topn 16481 df-0g 16499 df-gsum 16500 df-topgen 16501 df-ordt 16558 df-xrs 16559 df-mre 16643 df-mrc 16644 df-acs 16646 df-ps 17597 df-tsr 17598 df-mgm 17639 df-sgrp 17681 df-mnd 17692 df-submnd 17733 df-cntz 18144 df-cmn 18592 df-fbas 20150 df-fg 20151 df-top 21117 df-topon 21134 df-topsp 21156 df-bases 21169 df-ntr 21243 df-nei 21321 df-cn 21450 df-haus 21538 df-fil 22069 df-fm 22161 df-flim 22162 df-flf 22163 df-tsms 22349 |
This theorem is referenced by: esumid 30712 |
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