![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 23941 | . . . . . 6 β’ (πΊ β TopGrp β π½ β (TopOnβπ΅)) |
5 | 1, 4 | syl 17 | . . . . 5 β’ (π β π½ β (TopOnβπ΅)) |
6 | topontop 22770 | . . . . 5 β’ (π½ β (TopOnβπ΅) β π½ β Top) | |
7 | 5, 6 | syl 17 | . . . 4 β’ (π β π½ β Top) |
8 | 0cld 22897 | . . . 4 β’ (π½ β Top β β β (Clsdβπ½)) | |
9 | 7, 8 | syl 17 | . . 3 β’ (π β β β (Clsdβπ½)) |
10 | eleq1 2815 | . . 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 4341 | . . 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 480 | . . . . . . 7 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β πΊ β CMnd) |
15 | 1 | adantr 480 | . . . . . . 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 480 | . . . . . . 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 480 | . . . . . . 7 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β πΉ:π΄βΆπ΅) |
20 | simpr 484 | . . . . . . 7 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β π₯ β (πΊ tsums πΉ)) | |
21 | 3, 2, 14, 15, 17, 19, 20 | tgptsmscls 24009 | . . . . . 6 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β (πΊ tsums πΉ) = ((clsβπ½)β{π₯})) |
22 | tgptps 23939 | . . . . . . . . . . . 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 23994 | . . . . . . . . . 10 β’ (π β (πΊ tsums πΉ) β π΅) |
25 | toponuni 22771 | . . . . . . . . . . 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 4017 | . . . . . . . . 9 β’ (π β (πΊ tsums πΉ) β βͺ π½) |
28 | 27 | sselda 3977 | . . . . . . . 8 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β π₯ β βͺ π½) |
29 | 28 | snssd 4807 | . . . . . . 7 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β {π₯} β βͺ π½) |
30 | eqid 2726 | . . . . . . . 8 β’ βͺ π½ = βͺ π½ | |
31 | 30 | clscld 22906 | . . . . . . 7 β’ ((π½ β Top β§ {π₯} β βͺ π½) β ((clsβπ½)β{π₯}) β (Clsdβπ½)) |
32 | 7, 29, 31 | syl2an2r 682 | . . . . . 6 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β ((clsβπ½)β{π₯}) β (Clsdβπ½)) |
33 | 21, 32 | eqeltrd 2827 | . . . . 5 β’ ((π β§ π₯ β (πΊ tsums πΉ)) β (πΊ tsums πΉ) β (Clsdβπ½)) |
34 | 33 | ex 412 | . . . 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 3022 | 1 β’ (π β (πΊ tsums πΉ) β (Clsdβπ½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 βwex 1773 β wcel 2098 β wne 2934 β wss 3943 β c0 4317 {csn 4623 βͺ cuni 4902 βΆwf 6533 βcfv 6537 (class class class)co 7405 Basecbs 17153 TopOpenctopn 17376 CMndccmn 19700 Topctop 22750 TopOnctopon 22767 TopSpctps 22789 Clsdccld 22875 clsccl 22877 TopGrpctgp 23930 tsums ctsu 23985 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-ec 8707 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-0g 17396 df-gsum 17397 df-topgen 17398 df-plusf 18572 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19050 df-eqg 19052 df-ghm 19139 df-cntz 19233 df-cmn 19702 df-abl 19703 df-fbas 21237 df-fg 21238 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-cn 23086 df-cnp 23087 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-tmd 23931 df-tgp 23932 df-tsms 23986 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |