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Mirrors > Home > MPE Home > Th. List > tgptsmscld | Structured version Visualization version GIF version |
Description: The set of limit points to an infinite sum in a topological group is closed. (Contributed by Mario Carneiro, 22-Sep-2015.) |
Ref | Expression |
---|---|
tgptsmscls.b | ⊢ 𝐵 = (Base‘𝐺) |
tgptsmscls.j | ⊢ 𝐽 = (TopOpen‘𝐺) |
tgptsmscls.1 | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
tgptsmscls.2 | ⊢ (𝜑 → 𝐺 ∈ TopGrp) |
tgptsmscls.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
tgptsmscls.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
Ref | Expression |
---|---|
tgptsmscld | ⊢ (𝜑 → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgptsmscls.2 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TopGrp) | |
2 | tgptsmscls.j | . . . . . . 7 ⊢ 𝐽 = (TopOpen‘𝐺) | |
3 | tgptsmscls.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 2, 3 | tgptopon 23417 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝐵)) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐵)) |
6 | topontop 22246 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) |
8 | 0cld 22373 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽)) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → ∅ ∈ (Clsd‘𝐽)) |
10 | eleq1 2825 | . . 3 ⊢ ((𝐺 tsums 𝐹) = ∅ → ((𝐺 tsums 𝐹) ∈ (Clsd‘𝐽) ↔ ∅ ∈ (Clsd‘𝐽))) | |
11 | 9, 10 | syl5ibrcom 246 | . 2 ⊢ (𝜑 → ((𝐺 tsums 𝐹) = ∅ → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) |
12 | n0 4304 | . . 3 ⊢ ((𝐺 tsums 𝐹) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐺 tsums 𝐹)) | |
13 | tgptsmscls.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ CMnd) | |
14 | 13 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ CMnd) |
15 | 1 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐺 ∈ TopGrp) |
16 | tgptsmscls.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
17 | 16 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐴 ∈ 𝑉) |
18 | tgptsmscls.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) | |
19 | 18 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝐹:𝐴⟶𝐵) |
20 | simpr 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ (𝐺 tsums 𝐹)) | |
21 | 3, 2, 14, 15, 17, 19, 20 | tgptsmscls 23485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑥})) |
22 | tgptps 23415 | . . . . . . . . . . . 12 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp) | |
23 | 1, 22 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ TopSp) |
24 | 3, 13, 23, 16, 18 | tsmscl 23470 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵) |
25 | toponuni 22247 | . . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
26 | 5, 25 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = ∪ 𝐽) |
27 | 24, 26 | sseqtrd 3982 | . . . . . . . . 9 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ ∪ 𝐽) |
28 | 27 | sselda 3942 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ ∪ 𝐽) |
29 | 28 | snssd 4767 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → {𝑥} ⊆ ∪ 𝐽) |
30 | eqid 2736 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
31 | 30 | clscld 22382 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ {𝑥} ⊆ ∪ 𝐽) → ((cls‘𝐽)‘{𝑥}) ∈ (Clsd‘𝐽)) |
32 | 7, 29, 31 | syl2an2r 683 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{𝑥}) ∈ (Clsd‘𝐽)) |
33 | 21, 32 | eqeltrd 2838 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽)) |
34 | 33 | ex 413 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐺 tsums 𝐹) → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) |
35 | 34 | exlimdv 1936 | . . 3 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ (𝐺 tsums 𝐹) → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) |
36 | 12, 35 | biimtrid 241 | . 2 ⊢ (𝜑 → ((𝐺 tsums 𝐹) ≠ ∅ → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) |
37 | 11, 36 | pm2.61dne 3029 | 1 ⊢ (𝜑 → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2941 ⊆ wss 3908 ∅c0 4280 {csn 4584 ∪ cuni 4863 ⟶wf 6489 ‘cfv 6493 (class class class)co 7353 Basecbs 17075 TopOpenctopn 17295 CMndccmn 19553 Topctop 22226 TopOnctopon 22243 TopSpctps 22265 Clsdccld 22351 clsccl 22353 TopGrpctgp 23406 tsums ctsu 23461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7613 df-om 7799 df-1st 7917 df-2nd 7918 df-supp 8089 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-ec 8646 df-map 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9302 df-oi 9442 df-card 9871 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-n0 12410 df-z 12496 df-uz 12760 df-fz 13417 df-fzo 13560 df-seq 13899 df-hash 14223 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-ress 17105 df-plusg 17138 df-0g 17315 df-gsum 17316 df-topgen 17317 df-plusf 18488 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-mhm 18593 df-submnd 18594 df-grp 18743 df-minusg 18744 df-sbg 18745 df-subg 18916 df-eqg 18918 df-ghm 18997 df-cntz 19088 df-cmn 19555 df-abl 19556 df-fbas 20778 df-fg 20779 df-top 22227 df-topon 22244 df-topsp 22266 df-bases 22280 df-cld 22354 df-ntr 22355 df-cls 22356 df-nei 22433 df-cn 22562 df-cnp 22563 df-tx 22897 df-hmeo 23090 df-fil 23181 df-fm 23273 df-flim 23274 df-flf 23275 df-tmd 23407 df-tgp 23408 df-tsms 23462 |
This theorem is referenced by: (None) |
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