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Mirrors > Home > MPE Home > Th. List > haustsms2 | Structured version Visualization version GIF version |
Description: In a Hausdorff topological group, a sum has at most one limit point. (Contributed by Mario Carneiro, 13-Sep-2015.) |
Ref | Expression |
---|---|
tsmscl.b | β’ π΅ = (BaseβπΊ) |
tsmscl.1 | β’ (π β πΊ β CMnd) |
tsmscl.2 | β’ (π β πΊ β TopSp) |
tsmscl.a | β’ (π β π΄ β π) |
tsmscl.f | β’ (π β πΉ:π΄βΆπ΅) |
haustsms.j | β’ π½ = (TopOpenβπΊ) |
haustsms.h | β’ (π β π½ β Haus) |
Ref | Expression |
---|---|
haustsms2 | β’ (π β (π β (πΊ tsums πΉ) β (πΊ tsums πΉ) = {π})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . . . 6 β’ ((π β§ π β (πΊ tsums πΉ)) β π β (πΊ tsums πΉ)) | |
2 | tsmscl.b | . . . . . . . 8 β’ π΅ = (BaseβπΊ) | |
3 | tsmscl.1 | . . . . . . . 8 β’ (π β πΊ β CMnd) | |
4 | tsmscl.2 | . . . . . . . 8 β’ (π β πΊ β TopSp) | |
5 | tsmscl.a | . . . . . . . 8 β’ (π β π΄ β π) | |
6 | tsmscl.f | . . . . . . . 8 β’ (π β πΉ:π΄βΆπ΅) | |
7 | haustsms.j | . . . . . . . 8 β’ π½ = (TopOpenβπΊ) | |
8 | haustsms.h | . . . . . . . 8 β’ (π β π½ β Haus) | |
9 | 2, 3, 4, 5, 6, 7, 8 | haustsms 23995 | . . . . . . 7 β’ (π β β*π₯ π₯ β (πΊ tsums πΉ)) |
10 | 9 | adantr 480 | . . . . . 6 β’ ((π β§ π β (πΊ tsums πΉ)) β β*π₯ π₯ β (πΊ tsums πΉ)) |
11 | eleq1 2815 | . . . . . . . . 9 β’ (π₯ = π β (π₯ β (πΊ tsums πΉ) β π β (πΊ tsums πΉ))) | |
12 | 11 | moi2 3707 | . . . . . . . 8 β’ (((π β (πΊ tsums πΉ) β§ β*π₯ π₯ β (πΊ tsums πΉ)) β§ (π₯ β (πΊ tsums πΉ) β§ π β (πΊ tsums πΉ))) β π₯ = π) |
13 | 12 | ancom2s 647 | . . . . . . 7 β’ (((π β (πΊ tsums πΉ) β§ β*π₯ π₯ β (πΊ tsums πΉ)) β§ (π β (πΊ tsums πΉ) β§ π₯ β (πΊ tsums πΉ))) β π₯ = π) |
14 | 13 | expr 456 | . . . . . 6 β’ (((π β (πΊ tsums πΉ) β§ β*π₯ π₯ β (πΊ tsums πΉ)) β§ π β (πΊ tsums πΉ)) β (π₯ β (πΊ tsums πΉ) β π₯ = π)) |
15 | 1, 10, 1, 14 | syl21anc 835 | . . . . 5 β’ ((π β§ π β (πΊ tsums πΉ)) β (π₯ β (πΊ tsums πΉ) β π₯ = π)) |
16 | velsn 4639 | . . . . 5 β’ (π₯ β {π} β π₯ = π) | |
17 | 15, 16 | imbitrrdi 251 | . . . 4 β’ ((π β§ π β (πΊ tsums πΉ)) β (π₯ β (πΊ tsums πΉ) β π₯ β {π})) |
18 | 17 | ssrdv 3983 | . . 3 β’ ((π β§ π β (πΊ tsums πΉ)) β (πΊ tsums πΉ) β {π}) |
19 | snssi 4806 | . . . 4 β’ (π β (πΊ tsums πΉ) β {π} β (πΊ tsums πΉ)) | |
20 | 19 | adantl 481 | . . 3 β’ ((π β§ π β (πΊ tsums πΉ)) β {π} β (πΊ tsums πΉ)) |
21 | 18, 20 | eqssd 3994 | . 2 β’ ((π β§ π β (πΊ tsums πΉ)) β (πΊ tsums πΉ) = {π}) |
22 | 21 | ex 412 | 1 β’ (π β (π β (πΊ tsums πΉ) β (πΊ tsums πΉ) = {π})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β*wmo 2526 β wss 3943 {csn 4623 βΆwf 6533 βcfv 6537 (class class class)co 7405 Basecbs 17153 TopOpenctopn 17376 CMndccmn 19700 TopSpctps 22789 Hauscha 23167 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-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-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-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-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-seq 13973 df-hash 14296 df-0g 17396 df-gsum 17397 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-cntz 19233 df-cmn 19702 df-fbas 21237 df-fg 21238 df-top 22751 df-topon 22768 df-topsp 22790 df-nei 22957 df-haus 23174 df-fil 23705 df-flim 23798 df-flf 23799 df-tsms 23986 |
This theorem is referenced by: haustsmsid 24000 xrge0tsms 24705 xrge0tsmsd 32715 |
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