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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 24834 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶(0[,]+∞)) → (𝐺 tsums 𝐹) ≈ 1o) |
6 | 2, 3, 5 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (𝐺 tsums 𝐹) ≈ 1o) |
7 | en1eqsn 9311 | . . . 4 ⊢ ((𝐶 ∈ (𝐺 tsums 𝐹) ∧ (𝐺 tsums 𝐹) ≈ 1o) → (𝐺 tsums 𝐹) = {𝐶}) | |
8 | 1, 6, 7 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝐺 tsums 𝐹) = {𝐶}) |
9 | 8 | unieqd 4925 | . 2 ⊢ (𝜑 → ∪ (𝐺 tsums 𝐹) = ∪ {𝐶}) |
10 | unisng 4932 | . . 3 ⊢ (𝐶 ∈ (𝐺 tsums 𝐹) → ∪ {𝐶} = 𝐶) | |
11 | 1, 10 | syl 17 | . 2 ⊢ (𝜑 → ∪ {𝐶} = 𝐶) |
12 | 9, 11 | eqtr2d 2766 | 1 ⊢ (𝜑 → 𝐶 = ∪ (𝐺 tsums 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {csn 4632 ∪ cuni 4912 class class class wbr 5152 ⟶wf 6549 (class class class)co 7423 1oc1o 8488 ≈ cen 8970 0cc0 11154 +∞cpnf 11291 [,]cicc 13376 ↾s cress 17237 ℝ*𝑠cxrs 17510 tsums ctsu 24113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 ax-pre-sup 11232 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5579 df-eprel 5585 df-po 5593 df-so 5594 df-fr 5636 df-se 5637 df-we 5638 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-pred 6311 df-ord 6378 df-on 6379 df-lim 6380 df-suc 6381 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7689 df-om 7876 df-1st 8002 df-2nd 8003 df-supp 8174 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-map 8856 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9402 df-fi 9450 df-sup 9481 df-inf 9482 df-oi 9549 df-card 9978 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 df-nn 12260 df-2 12322 df-3 12323 df-4 12324 df-5 12325 df-6 12326 df-7 12327 df-8 12328 df-9 12329 df-n0 12520 df-z 12606 df-dec 12725 df-uz 12870 df-q 12980 df-xadd 13142 df-ioo 13377 df-ioc 13378 df-ico 13379 df-icc 13380 df-fz 13534 df-fzo 13677 df-seq 14017 df-hash 14343 df-struct 17144 df-sets 17161 df-slot 17179 df-ndx 17191 df-base 17209 df-ress 17238 df-plusg 17274 df-mulr 17275 df-tset 17280 df-ple 17281 df-ds 17283 df-rest 17432 df-topn 17433 df-0g 17451 df-gsum 17452 df-topgen 17453 df-ordt 17511 df-xrs 17512 df-mre 17594 df-mrc 17595 df-acs 17597 df-ps 18586 df-tsr 18587 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-submnd 18769 df-cntz 19306 df-cmn 19775 df-fbas 21332 df-fg 21333 df-top 22879 df-topon 22896 df-topsp 22918 df-bases 22932 df-ntr 23007 df-nei 23085 df-cn 23214 df-haus 23302 df-fil 23833 df-fm 23925 df-flim 23926 df-flf 23927 df-tsms 24114 |
This theorem is referenced by: esumid 33833 |
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