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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 483 | . . . . . 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 24058 | . . . . . . 7 β’ (π β β*π₯ π₯ β (πΊ tsums πΉ)) |
10 | 9 | adantr 479 | . . . . . 6 β’ ((π β§ π β (πΊ tsums πΉ)) β β*π₯ π₯ β (πΊ tsums πΉ)) |
11 | eleq1 2813 | . . . . . . . . 9 β’ (π₯ = π β (π₯ β (πΊ tsums πΉ) β π β (πΊ tsums πΉ))) | |
12 | 11 | moi2 3703 | . . . . . . . 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 648 | . . . . . . 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 455 | . . . . . 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 836 | . . . . 5 β’ ((π β§ π β (πΊ tsums πΉ)) β (π₯ β (πΊ tsums πΉ) β π₯ = π)) |
16 | velsn 4640 | . . . . 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 3978 | . . 3 β’ ((π β§ π β (πΊ tsums πΉ)) β (πΊ tsums πΉ) β {π}) |
19 | snssi 4807 | . . . 4 β’ (π β (πΊ tsums πΉ) β {π} β (πΊ tsums πΉ)) | |
20 | 19 | adantl 480 | . . 3 β’ ((π β§ π β (πΊ tsums πΉ)) β {π} β (πΊ tsums πΉ)) |
21 | 18, 20 | eqssd 3990 | . 2 β’ ((π β§ π β (πΊ tsums πΉ)) β (πΊ tsums πΉ) = {π}) |
22 | 21 | ex 411 | 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 394 = wceq 1533 β wcel 2098 β*wmo 2526 β wss 3939 {csn 4624 βΆwf 6539 βcfv 6543 (class class class)co 7416 Basecbs 17179 TopOpenctopn 17402 CMndccmn 19739 TopSpctps 22852 Hauscha 23230 tsums ctsu 24048 |
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 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 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-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 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-seq 13999 df-hash 14322 df-0g 17422 df-gsum 17423 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-cntz 19272 df-cmn 19741 df-fbas 21280 df-fg 21281 df-top 22814 df-topon 22831 df-topsp 22853 df-nei 23020 df-haus 23237 df-fil 23768 df-flim 23861 df-flf 23862 df-tsms 24049 |
This theorem is referenced by: haustsmsid 24063 xrge0tsms 24768 xrge0tsmsd 32816 |
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