Theorem tgpgrp 24107

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 2740	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2740	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 24106	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp1bi 1145	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2108  ‘cfv 6573  (class class class)co 7448  TopOpenctopn 17481  Grpcgrp 18973  inv_gcminusg 18974   Cn ccn 23253  TopMndctmd 24099  TopGrpctgp 24100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-tgp 24102

This theorem is referenced by:  grpinvhmeo  24115  istgp2  24120  oppgtgp  24127  tgplacthmeo  24132  subgtgp  24134  subgntr  24136  opnsubg  24137  clssubg  24138  cldsubg  24140  tgpconncompeqg  24141  tgpconncomp  24142  snclseqg  24145  tgphaus  24146  tgpt1  24147  tgpt0  24148  qustgpopn  24149  qustgplem  24150  qustgphaus  24152  prdstgpd  24154  tsmsinv  24177  tsmssub  24178  tgptsmscls  24179  tsmsxplem1  24182  tsmsxplem2  24183