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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 23459 | . 2 β’ (πΊ β TopGrp β πΌ β (π½ Cn π½)) |
4 | tgpgrp 23452 | . . . 4 β’ (πΊ β TopGrp β πΊ β Grp) | |
5 | eqid 2733 | . . . . 5 β’ (BaseβπΊ) = (BaseβπΊ) | |
6 | 5, 2 | grpinvcnv 18823 | . . . 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 2834 | . 2 β’ (πΊ β TopGrp β β‘πΌ β (π½ Cn π½)) |
9 | ishmeo 23133 | . 2 β’ (πΌ β (π½Homeoπ½) β (πΌ β (π½ Cn π½) β§ β‘πΌ β (π½ Cn π½))) | |
10 | 3, 8, 9 | sylanbrc 584 | 1 β’ (πΊ β TopGrp β πΌ β (π½Homeoπ½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β‘ccnv 5636 βcfv 6500 (class class class)co 7361 Basecbs 17091 TopOpenctopn 17311 Grpcgrp 18756 invgcminusg 18757 Cn ccn 22598 Homeochmeo 23127 TopGrpctgp 23445 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-map 8773 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-top 22266 df-topon 22283 df-cn 22601 df-hmeo 23129 df-tgp 23447 |
This theorem is referenced by: tgpconncomp 23487 tsmsxplem1 23527 |
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