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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 24847 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶(0[,]+∞)) → (𝐺 tsums 𝐹) ≈ 1o) |
6 | 2, 3, 5 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐺 tsums 𝐹) ≈ 1o) |
7 | en1eqsn 9304 | . . . 4 ⊢ ((𝐶 ∈ (𝐺 tsums 𝐹) ∧ (𝐺 tsums 𝐹) ≈ 1o) → (𝐺 tsums 𝐹) = {𝐶}) | |
8 | 1, 6, 7 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐺 tsums 𝐹) = {𝐶}) |
9 | 8 | unieqd 4918 | . 2 ⊢ (𝜑 → ∪ (𝐺 tsums 𝐹) = ∪ {𝐶}) |
10 | unisng 4923 | . . 3 ⊢ (𝐶 ∈ (𝐺 tsums 𝐹) → ∪ {𝐶} = 𝐶) | |
11 | 1, 10 | syl 17 | . 2 ⊢ (𝜑 → ∪ {𝐶} = 𝐶) |
12 | 9, 11 | eqtr2d 2777 | 1 ⊢ (𝜑 → 𝐶 = ∪ (𝐺 tsums 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {csn 4624 ∪ cuni 4905 class class class wbr 5141 ⟶wf 6555 (class class class)co 7429 1oc1o 8495 ≈ cen 8978 0cc0 11151 +∞cpnf 11288 [,]cicc 13386 ↾s cress 17270 ℝ*𝑠cxrs 17541 tsums ctsu 24124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 ax-pre-sup 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-iin 4992 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-se 5636 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-isom 6568 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-of 7694 df-om 7884 df-1st 8010 df-2nd 8011 df-supp 8182 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-2o 8503 df-er 8741 df-map 8864 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9398 df-fi 9447 df-sup 9478 df-inf 9479 df-oi 9546 df-card 9975 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-div 11917 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12875 df-q 12987 df-xadd 13151 df-ioo 13387 df-ioc 13388 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-seq 14039 df-hash 14366 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17244 df-ress 17271 df-plusg 17306 df-mulr 17307 df-tset 17312 df-ple 17313 df-ds 17315 df-rest 17463 df-topn 17464 df-0g 17482 df-gsum 17483 df-topgen 17484 df-ordt 17542 df-xrs 17543 df-mre 17625 df-mrc 17626 df-acs 17628 df-ps 18607 df-tsr 18608 df-mgm 18649 df-sgrp 18728 df-mnd 18744 df-submnd 18793 df-cntz 19331 df-cmn 19796 df-fbas 21353 df-fg 21354 df-top 22890 df-topon 22907 df-topsp 22929 df-bases 22943 df-ntr 23018 df-nei 23096 df-cn 23225 df-haus 23313 df-fil 23844 df-fm 23936 df-flim 23937 df-flf 23938 df-tsms 24125 |
This theorem is referenced by: esumid 34023 |
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