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Theorem tgpinv 22108
Description: In a topological group, the inverse function is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by FL, 27-Jun-2014.)
Hypotheses
Ref Expression
tgpcn.j 𝐽 = (TopOpen‘𝐺)
tgpinv.5 𝐼 = (invg𝐺)
Assertion
Ref Expression
tgpinv (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))

Proof of Theorem tgpinv
StepHypRef Expression
1 tgpcn.j . . 3 𝐽 = (TopOpen‘𝐺)
2 tgpinv.5 . . 3 𝐼 = (invg𝐺)
31, 2istgp 22100 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ 𝐼 ∈ (𝐽 Cn 𝐽)))
43simp3bi 1141 1 (𝐺 ∈ TopGrp → 𝐼 ∈ (𝐽 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cfv 6031  (class class class)co 6792  TopOpenctopn 16289  Grpcgrp 17629  invgcminusg 17630   Cn ccn 21248  TopMndctmd 22093  TopGrpctgp 22094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6795  df-tgp 22096
This theorem is referenced by:  grpinvhmeo  22109  tgpsubcn  22113  tgpmulg  22116  oppgtgp  22121  subgtgp  22128  prdstgpd  22147  tsmsinv  22170  invrcn2  22202
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