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Theorem tgptmd 22103
Description: A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
Assertion
Ref Expression
tgptmd (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)

Proof of Theorem tgptmd
StepHypRef Expression
1 eqid 2771 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
2 eqid 2771 . . 3 (invg𝐺) = (invg𝐺)
31, 2istgp 22101 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
43simp2bi 1140 1 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  cfv 6031  (class class class)co 6793  TopOpenctopn 16290  Grpcgrp 17630  invgcminusg 17631   Cn ccn 21249  TopMndctmd 22094  TopGrpctgp 22095
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 837  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 6796  df-tgp 22097
This theorem is referenced by:  tgptps  22104  tgpcn  22108  tgpsubcn  22114  tgpmulg  22117  oppgtgp  22122  tgplacthmeo  22127  subgtgp  22129  clsnsg  22133  tgpt0  22142  prdstgpd  22148  tsmssub  22172  tsmsxp  22178  trgtmd2  22192  nlmtlm  22718  qqhcn  30375
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