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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 24214 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶(0[,]+∞)) → (𝐺 tsums 𝐹) ≈ 1o) |
5 | 1, 2, 4 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (𝐺 tsums 𝐹) ≈ 1o) |
6 | en1b 8970 | . . . 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 2820 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐺 tsums 𝐹) ↔ 𝐶 ∈ {∪ (𝐺 tsums 𝐹)})) |
9 | ovex 7391 | . . . . . . 7 ⊢ (𝐺 tsums 𝐹) ∈ V | |
10 | 9 | uniex 7679 | . . . . . 6 ⊢ ∪ (𝐺 tsums 𝐹) ∈ V |
11 | eleq1 2822 | . . . . . 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 4601 | . . . . 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 4604 | . . 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 1542 ∈ wcel 2107 Vcvv 3444 {csn 4587 ∪ cuni 4866 class class class wbr 5106 ⟶wf 6493 (class class class)co 7358 1oc1o 8406 ≈ cen 8883 0cc0 11056 +∞cpnf 11191 [,]cicc 13273 ↾s cress 17117 ℝ*𝑠cxrs 17387 tsums ctsu 23493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-om 7804 df-1st 7922 df-2nd 7923 df-supp 8094 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 df-fi 9352 df-sup 9383 df-inf 9384 df-oi 9451 df-card 9880 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-q 12879 df-xadd 13039 df-ioo 13274 df-ioc 13275 df-ico 13276 df-icc 13277 df-fz 13431 df-fzo 13574 df-seq 13913 df-hash 14237 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-tset 17157 df-ple 17158 df-ds 17160 df-rest 17309 df-topn 17310 df-0g 17328 df-gsum 17329 df-topgen 17330 df-ordt 17388 df-xrs 17389 df-mre 17471 df-mrc 17472 df-acs 17474 df-ps 18460 df-tsr 18461 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-submnd 18607 df-cntz 19102 df-cmn 19569 df-fbas 20809 df-fg 20810 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-ntr 22387 df-nei 22465 df-cn 22594 df-haus 22682 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-tsms 23494 |
This theorem is referenced by: esumcl 32686 |
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