Theorem tgpgrp 24397

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 2761	. . 3 ⊢ (TopOpen‘𝐺) = (TopOpen‘𝐺)
2		eqid 2761	. . 3 ⊢ (invg‘𝐺) = (invg‘𝐺)
3	1, 2	istgp 24396	. 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 6538  (class class class)co 7420  TopOpenctopn 17592  Grpcgrp 19144  inv_gcminusg 19145   Cn ccn 23542  TopMndctmd 24389  TopGrpctgp 24390

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 2733  ax-nul 5260

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 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3453  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 6494  df-fv 6546  df-ov 7423  df-tgp 24392

This theorem is used by:  grpinvhmeo  24405  istgp2  24410  oppgtgp  24417  tgplacthmeo  24422  subgtgp  24424  subgntr  24426  opnsubg  24427  clssubg  24428  cldsubg  24430  tgpconncompeqg  24431  tgpconncomp  24432  snclseqg  24435  tgphaus  24436  tgpt1  24437  tgpt0  24438  qustgpopn  24439  qustgplem  24440  qustgphaus  24442  prdstgpd  24444  tsmsinv  24467  tsmssub  24468  tgptsmscls  24469  tsmsxplem1  24472  tsmsxplem2  24473