Theorem tgpgrp 21882

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 e. TopGrp -> G e. Grp )

Proof of Theorem tgpgrp

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( TopOpen ` G ) = ( TopOpen ` G )
2		eqid 2622	. . 3 |- ( invg ` G ) = ( invg ` G )
3	1, 2	istgp 21881	. 2 |- ( G e. TopGrp <-> ( G e. Grp /\ G e. TopMnd /\ ( invg ` G ) e. ( ( TopOpen ` G ) Cn ( TopOpen ` G ) ) ) )
4	3	simp1bi 1076	1 |- ( G e. TopGrp -> G e. Grp )

Syntax hints:   -> wi 4   e. wcel 1990   ` cfv 5888  (class class class)co 6650   TopOpenctopn 16082   Grpcgrp 17422   invgcminusg 17423   Cn ccn 21028  TopMndctmd 21874   TopGrpctgp 21875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-tgp 21877

This theorem is referenced by:  grpinvhmeo  21890  istgp2  21895  oppgtgp  21902  tgplacthmeo  21907  subgtgp  21909  subgntr  21910  opnsubg  21911  clssubg  21912  cldsubg  21914  tgpconncompeqg  21915  tgpconncomp  21916  snclseqg  21919  tgphaus  21920  tgpt1  21921  tgpt0  21922  qustgpopn  21923  qustgplem  21924  qustgphaus  21926  prdstgpd  21928  tsmsinv  21951  tsmssub  21952  tgptsmscls  21953  tsmsxplem1  21956  tsmsxplem2  21957