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Mirrors > Home > MPE Home > Th. List > grpinvhmeo | Structured version Visualization version GIF version |
Description: The inverse function in a topological group is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
tgpcn.j | β’ π½ = (TopOpenβπΊ) |
tgpinv.5 | β’ πΌ = (invgβπΊ) |
Ref | Expression |
---|---|
grpinvhmeo | β’ (πΊ β TopGrp β πΌ β (π½Homeoπ½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgpcn.j | . . 3 β’ π½ = (TopOpenβπΊ) | |
2 | tgpinv.5 | . . 3 β’ πΌ = (invgβπΊ) | |
3 | 1, 2 | tgpinv 23588 | . 2 β’ (πΊ β TopGrp β πΌ β (π½ Cn π½)) |
4 | tgpgrp 23581 | . . . 4 β’ (πΊ β TopGrp β πΊ β Grp) | |
5 | eqid 2732 | . . . . 5 β’ (BaseβπΊ) = (BaseβπΊ) | |
6 | 5, 2 | grpinvcnv 18890 | . . . 4 β’ (πΊ β Grp β β‘πΌ = πΌ) |
7 | 4, 6 | syl 17 | . . 3 β’ (πΊ β TopGrp β β‘πΌ = πΌ) |
8 | 7, 3 | eqeltrd 2833 | . 2 β’ (πΊ β TopGrp β β‘πΌ β (π½ Cn π½)) |
9 | ishmeo 23262 | . 2 β’ (πΌ β (π½Homeoπ½) β (πΌ β (π½ Cn π½) β§ β‘πΌ β (π½ Cn π½))) | |
10 | 3, 8, 9 | sylanbrc 583 | 1 β’ (πΊ β TopGrp β πΌ β (π½Homeoπ½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 β‘ccnv 5675 βcfv 6543 (class class class)co 7408 Basecbs 17143 TopOpenctopn 17366 Grpcgrp 18818 invgcminusg 18819 Cn ccn 22727 Homeochmeo 23256 TopGrpctgp 23574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-map 8821 df-0g 17386 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821 df-minusg 18822 df-top 22395 df-topon 22412 df-cn 22730 df-hmeo 23258 df-tgp 23576 |
This theorem is referenced by: tgpconncomp 23616 tsmsxplem1 23656 |
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