Theorem tgpgrp 24102

Description: A topological group is a group. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)

Assertion

Ref	Expression
tgpgrp	⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)

Proof of Theorem tgpgrp

Step	Hyp	Ref	Expression
1		eqid 2735	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2735	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 24101	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp1bi 1144	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2106  ‘cfv 6563  (class class class)co 7431  TopOpenctopn 17468  Grpcgrp 18964  inv_gcminusg 18965   Cn ccn 23248  TopMndctmd 24094  TopGrpctgp 24095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-tgp 24097

This theorem is referenced by:  grpinvhmeo  24110  istgp2  24115  oppgtgp  24122  tgplacthmeo  24127  subgtgp  24129  subgntr  24131  opnsubg  24132  clssubg  24133  cldsubg  24135  tgpconncompeqg  24136  tgpconncomp  24137  snclseqg  24140  tgphaus  24141  tgpt1  24142  tgpt0  24143  qustgpopn  24144  qustgplem  24145  qustgphaus  24147  prdstgpd  24149  tsmsinv  24172  tsmssub  24173  tgptsmscls  24174  tsmsxplem1  24177  tsmsxplem2  24178