Theorem tgpgrp 24305

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 2760	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2760	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 24304	. 2 ⊢ (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg‘𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
4	3	simp1bi 1163	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  ‘cfv 6533  (class class class)co 7414  TopOpenctopn 17507  Grpcgrp 19058  inv_gcminusg 19059   Cn ccn 23450  TopMndctmd 24297  TopGrpctgp 24298

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7417  df-tgp 24300

This theorem is used by:  grpinvhmeo  24313  istgp2  24318  oppgtgp  24325  tgplacthmeo  24330  subgtgp  24332  subgntr  24334  opnsubg  24335  clssubg  24336  cldsubg  24338  tgpconncompeqg  24339  tgpconncomp  24340  snclseqg  24343  tgphaus  24344  tgpt1  24345  tgpt0  24346  qustgpopn  24347  qustgplem  24348  qustgphaus  24350  prdstgpd  24352  tsmsinv  24375  tsmssub  24376  tgptsmscls  24377  tsmsxplem1  24380  tsmsxplem2  24381