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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 | β’ π΅ = (BaseβπΊ) |
tgptsmscls.j | β’ π½ = (TopOpenβπΊ) |
tgptsmscls.1 | β’ (π β πΊ β CMnd) |
tgptsmscls.2 | β’ (π β πΊ β TopGrp) |
tgptsmscls.a | β’ (π β π΄ β π) |
tgptsmscls.f | β’ (π β πΉ:π΄βΆπ΅) |
Ref | Expression |
---|---|
tgptsmscld | β’ (π β (πΊ tsums πΉ) β (Clsdβπ½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgptsmscls.2 | . . . . . 6 β’ (π β πΊ β TopGrp) | |
2 | tgptsmscls.j | . . . . . . 7 β’ π½ = (TopOpenβπΊ) | |
3 | tgptsmscls.b | . . . . . . 7 β’ π΅ = (BaseβπΊ) | |
4 | 2, 3 | tgptopon 23449 | . . . . . 6 β’ (πΊ β TopGrp β π½ β (TopOnβπ΅)) |
5 | 1, 4 | syl 17 | . . . . 5 β’ (π β π½ β (TopOnβπ΅)) |
6 | topontop 22278 | . . . . 5 β’ (π½ β (TopOnβπ΅) β π½ β Top) | |
7 | 5, 6 | syl 17 | . . . 4 β’ (π β π½ β Top) |
8 | 0cld 22405 | . . . 4 β’ (π½ β Top β β β (Clsdβπ½)) | |
9 | 7, 8 | syl 17 | . . 3 β’ (π β β β (Clsdβπ½)) |
10 | eleq1 2822 | . . 3 β’ ((πΊ tsums πΉ) = β β ((πΊ tsums πΉ) β (Clsdβπ½) β β β (Clsdβπ½))) | |
11 | 9, 10 | syl5ibrcom 247 | . 2 β’ (π β ((πΊ tsums πΉ) = β β (πΊ tsums πΉ) β (Clsdβπ½))) |
12 | n0 4307 | . . 3 β’ ((πΊ tsums πΉ) β β β βπ₯ π₯ β (πΊ tsums πΉ)) | |
13 | tgptsmscls.1 | . . . . . . . 8 β’ (π β πΊ β CMnd) | |
14 | 13 | adantr 482 | . . . . . . 7 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β πΊ β CMnd) |
15 | 1 | adantr 482 | . . . . . . 7 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β πΊ β TopGrp) |
16 | tgptsmscls.a | . . . . . . . 8 β’ (π β π΄ β π) | |
17 | 16 | adantr 482 | . . . . . . 7 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β π΄ β π) |
18 | tgptsmscls.f | . . . . . . . 8 β’ (π β πΉ:π΄βΆπ΅) | |
19 | 18 | adantr 482 | . . . . . . 7 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β πΉ:π΄βΆπ΅) |
20 | simpr 486 | . . . . . . 7 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β π₯ β (πΊ tsums πΉ)) | |
21 | 3, 2, 14, 15, 17, 19, 20 | tgptsmscls 23517 | . . . . . 6 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β (πΊ tsums πΉ) = ((clsβπ½)β{π₯})) |
22 | tgptps 23447 | . . . . . . . . . . . 12 β’ (πΊ β TopGrp β πΊ β TopSp) | |
23 | 1, 22 | syl 17 | . . . . . . . . . . 11 β’ (π β πΊ β TopSp) |
24 | 3, 13, 23, 16, 18 | tsmscl 23502 | . . . . . . . . . 10 β’ (π β (πΊ tsums πΉ) β π΅) |
25 | toponuni 22279 | . . . . . . . . . . 11 β’ (π½ β (TopOnβπ΅) β π΅ = βͺ π½) | |
26 | 5, 25 | syl 17 | . . . . . . . . . 10 β’ (π β π΅ = βͺ π½) |
27 | 24, 26 | sseqtrd 3985 | . . . . . . . . 9 β’ (π β (πΊ tsums πΉ) β βͺ π½) |
28 | 27 | sselda 3945 | . . . . . . . 8 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β π₯ β βͺ π½) |
29 | 28 | snssd 4770 | . . . . . . 7 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β {π₯} β βͺ π½) |
30 | eqid 2733 | . . . . . . . 8 β’ βͺ π½ = βͺ π½ | |
31 | 30 | clscld 22414 | . . . . . . 7 β’ ((π½ β Top β§ {π₯} β βͺ π½) β ((clsβπ½)β{π₯}) β (Clsdβπ½)) |
32 | 7, 29, 31 | syl2an2r 684 | . . . . . 6 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β ((clsβπ½)β{π₯}) β (Clsdβπ½)) |
33 | 21, 32 | eqeltrd 2834 | . . . . 5 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β (πΊ tsums πΉ) β (Clsdβπ½)) |
34 | 33 | ex 414 | . . . 4 β’ (π β (π₯ β (πΊ tsums πΉ) β (πΊ tsums πΉ) β (Clsdβπ½))) |
35 | 34 | exlimdv 1937 | . . 3 β’ (π β (βπ₯ π₯ β (πΊ tsums πΉ) β (πΊ tsums πΉ) β (Clsdβπ½))) |
36 | 12, 35 | biimtrid 241 | . 2 β’ (π β ((πΊ tsums πΉ) β β β (πΊ tsums πΉ) β (Clsdβπ½))) |
37 | 11, 36 | pm2.61dne 3028 | 1 β’ (π β (πΊ tsums πΉ) β (Clsdβπ½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 βwex 1782 β wcel 2107 β wne 2940 β wss 3911 β c0 4283 {csn 4587 βͺ cuni 4866 βΆwf 6493 βcfv 6497 (class class class)co 7358 Basecbs 17088 TopOpenctopn 17308 CMndccmn 19567 Topctop 22258 TopOnctopon 22275 TopSpctps 22297 Clsdccld 22383 clsccl 22385 TopGrpctgp 23438 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 |
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-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-ec 8653 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9309 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-nn 12159 df-2 12221 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 df-seq 13913 df-hash 14237 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-0g 17328 df-gsum 17329 df-topgen 17330 df-plusf 18501 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-mhm 18606 df-submnd 18607 df-grp 18756 df-minusg 18757 df-sbg 18758 df-subg 18930 df-eqg 18932 df-ghm 19011 df-cntz 19102 df-cmn 19569 df-abl 19570 df-fbas 20809 df-fg 20810 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-cn 22594 df-cnp 22595 df-tx 22929 df-hmeo 23122 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-tmd 23439 df-tgp 23440 df-tsms 23494 |
This theorem is referenced by: (None) |
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