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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 24002 | . . . . . 6 β’ (πΊ β TopGrp β π½ β (TopOnβπ΅)) |
5 | 1, 4 | syl 17 | . . . . 5 β’ (π β π½ β (TopOnβπ΅)) |
6 | topontop 22831 | . . . . 5 β’ (π½ β (TopOnβπ΅) β π½ β Top) | |
7 | 5, 6 | syl 17 | . . . 4 β’ (π β π½ β Top) |
8 | 0cld 22958 | . . . 4 β’ (π½ β Top β β β (Clsdβπ½)) | |
9 | 7, 8 | syl 17 | . . 3 β’ (π β β β (Clsdβπ½)) |
10 | eleq1 2813 | . . 3 β’ ((πΊ tsums πΉ) = β β ((πΊ tsums πΉ) β (Clsdβπ½) β β β (Clsdβπ½))) | |
11 | 9, 10 | syl5ibrcom 246 | . 2 β’ (π β ((πΊ tsums πΉ) = β β (πΊ tsums πΉ) β (Clsdβπ½))) |
12 | n0 4342 | . . 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 479 | . . . . . . 7 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β πΊ β CMnd) |
15 | 1 | adantr 479 | . . . . . . 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 479 | . . . . . . 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 479 | . . . . . . 7 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β πΉ:π΄βΆπ΅) |
20 | simpr 483 | . . . . . . 7 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β π₯ β (πΊ tsums πΉ)) | |
21 | 3, 2, 14, 15, 17, 19, 20 | tgptsmscls 24070 | . . . . . 6 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β (πΊ tsums πΉ) = ((clsβπ½)β{π₯})) |
22 | tgptps 24000 | . . . . . . . . . . . 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 24055 | . . . . . . . . . 10 β’ (π β (πΊ tsums πΉ) β π΅) |
25 | toponuni 22832 | . . . . . . . . . . 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 4013 | . . . . . . . . 9 β’ (π β (πΊ tsums πΉ) β βͺ π½) |
28 | 27 | sselda 3972 | . . . . . . . 8 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β π₯ β βͺ π½) |
29 | 28 | snssd 4808 | . . . . . . 7 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β {π₯} β βͺ π½) |
30 | eqid 2725 | . . . . . . . 8 β’ βͺ π½ = βͺ π½ | |
31 | 30 | clscld 22967 | . . . . . . 7 β’ ((π½ β Top β§ {π₯} β βͺ π½) β ((clsβπ½)β{π₯}) β (Clsdβπ½)) |
32 | 7, 29, 31 | syl2an2r 683 | . . . . . 6 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β ((clsβπ½)β{π₯}) β (Clsdβπ½)) |
33 | 21, 32 | eqeltrd 2825 | . . . . 5 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β (πΊ tsums πΉ) β (Clsdβπ½)) |
34 | 33 | ex 411 | . . . 4 β’ (π β (π₯ β (πΊ tsums πΉ) β (πΊ tsums πΉ) β (Clsdβπ½))) |
35 | 34 | exlimdv 1928 | . . 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 3018 | 1 β’ (π β (πΊ tsums πΉ) β (Clsdβπ½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 βwex 1773 β wcel 2098 β wne 2930 β wss 3940 β c0 4318 {csn 4624 βͺ cuni 4903 βΆwf 6538 βcfv 6542 (class class class)co 7415 Basecbs 17177 TopOpenctopn 17400 CMndccmn 19737 Topctop 22811 TopOnctopon 22828 TopSpctps 22850 Clsdccld 22936 clsccl 22938 TopGrpctgp 23991 tsums ctsu 24046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-ec 8723 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-fzo 13658 df-seq 13997 df-hash 14320 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-0g 17420 df-gsum 17421 df-topgen 17422 df-plusf 18596 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-subg 19080 df-eqg 19082 df-ghm 19170 df-cntz 19270 df-cmn 19739 df-abl 19740 df-fbas 21278 df-fg 21279 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22865 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-cn 23147 df-cnp 23148 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-tmd 23992 df-tgp 23993 df-tsms 24047 |
This theorem is referenced by: (None) |
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