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Mirrors > Home > MPE Home > Th. List > tgpinv | Structured version Visualization version GIF version |
Description: In a topological group, the inverse function is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by FL, 27-Jun-2014.) |
Ref | Expression |
---|---|
tgpcn.j | β’ π½ = (TopOpenβπΊ) |
tgpinv.5 | β’ πΌ = (invgβπΊ) |
Ref | Expression |
---|---|
tgpinv | β’ (πΊ β TopGrp β πΌ β (π½ Cn π½)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgpcn.j | . . 3 β’ π½ = (TopOpenβπΊ) | |
2 | tgpinv.5 | . . 3 β’ πΌ = (invgβπΊ) | |
3 | 1, 2 | istgp 23451 | . 2 β’ (πΊ β TopGrp β (πΊ β Grp β§ πΊ β TopMnd β§ πΌ β (π½ Cn π½))) |
4 | 3 | simp3bi 1148 | 1 β’ (πΊ β TopGrp β πΌ β (π½ Cn π½)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βcfv 6500 (class class class)co 7361 TopOpenctopn 17311 Grpcgrp 18756 invgcminusg 18757 Cn ccn 22598 TopMndctmd 23444 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-ext 2704 ax-nul 5267 |
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-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 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-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-iota 6452 df-fv 6508 df-ov 7364 df-tgp 23447 |
This theorem is referenced by: grpinvhmeo 23460 tgpsubcn 23464 tgpmulg 23467 oppgtgp 23472 subgtgp 23479 prdstgpd 23499 tsmsinv 23522 invrcn2 23554 |
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