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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 22618 | . . . . . 6 ⊢ (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝐵)) |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐵)) |
6 | topontop 21449 | . . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐽 ∈ Top) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) |
8 | 0cld 21574 | . . . 4 ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽)) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → ∅ ∈ (Clsd‘𝐽)) |
10 | eleq1 2897 | . . 3 ⊢ ((𝐺 tsums 𝐹) = ∅ → ((𝐺 tsums 𝐹) ∈ (Clsd‘𝐽) ↔ ∅ ∈ (Clsd‘𝐽))) | |
11 | 9, 10 | syl5ibrcom 248 | . 2 ⊢ (𝜑 → ((𝐺 tsums 𝐹) = ∅ → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) |
12 | n0 4307 | . . 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 22685 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → (𝐺 tsums 𝐹) = ((cls‘𝐽)‘{𝑥})) |
22 | tgptps 22616 | . . . . . . . . . . . 12 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp) | |
23 | 1, 22 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ TopSp) |
24 | 3, 13, 23, 16, 18 | tsmscl 22670 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ 𝐵) |
25 | toponuni 21450 | . . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝐵) → 𝐵 = ∪ 𝐽) | |
26 | 5, 25 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = ∪ 𝐽) |
27 | 24, 26 | sseqtrd 4004 | . . . . . . . . 9 ⊢ (𝜑 → (𝐺 tsums 𝐹) ⊆ ∪ 𝐽) |
28 | 27 | sselda 3964 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → 𝑥 ∈ ∪ 𝐽) |
29 | 28 | snssd 4734 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → {𝑥} ⊆ ∪ 𝐽) |
30 | eqid 2818 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
31 | 30 | clscld 21583 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ {𝑥} ⊆ ∪ 𝐽) → ((cls‘𝐽)‘{𝑥}) ∈ (Clsd‘𝐽)) |
32 | 7, 29, 31 | syl2an2r 681 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐺 tsums 𝐹)) → ((cls‘𝐽)‘{𝑥}) ∈ (Clsd‘𝐽)) |
33 | 21, 32 | eqeltrd 2910 | . . . . 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 1925 | . . 3 ⊢ (𝜑 → (∃𝑥 𝑥 ∈ (𝐺 tsums 𝐹) → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) |
36 | 12, 35 | syl5bi 243 | . 2 ⊢ (𝜑 → ((𝐺 tsums 𝐹) ≠ ∅ → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽))) |
37 | 11, 36 | pm2.61dne 3100 | 1 ⊢ (𝜑 → (𝐺 tsums 𝐹) ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ≠ wne 3013 ⊆ wss 3933 ∅c0 4288 {csn 4557 ∪ cuni 4830 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 TopOpenctopn 16683 CMndccmn 18835 Topctop 21429 TopOnctopon 21446 TopSpctps 21468 Clsdccld 21552 clsccl 21554 TopGrpctgp 22607 tsums ctsu 22661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-ec 8280 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-0g 16703 df-gsum 16704 df-topgen 16705 df-plusf 17839 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-eqg 18216 df-ghm 18294 df-cntz 18385 df-cmn 18837 df-abl 18838 df-fbas 20470 df-fg 20471 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-cn 21763 df-cnp 21764 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-tmd 22608 df-tgp 22609 df-tsms 22662 |
This theorem is referenced by: (None) |
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