Theorem indistgp 21904

Description: Any group equipped with the indiscrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)

Hypotheses

Ref	Expression
distgp.1	|- B = ( Base ` G )
distgp.2	|- J = ( TopOpen ` G )

Assertion

Ref	Expression
indistgp	|- ( ( G e. Grp /\ J = { (/) , B } ) -> G e. TopGrp )

Proof of Theorem indistgp

Step	Hyp	Ref	Expression
1		simpl 473	. 2 |- ( ( G e. Grp /\ J = { (/) , B } ) -> G e. Grp )
2		simpr 477	. . . 4 |- ( ( G e. Grp /\ J = { (/) , B } ) -> J = { (/) , B } )
3		distgp.1	. . . . . 6 |- B = ( Base ` G )
4		fvex 6201	. . . . . 6 |- ( Base ` G ) e. _V
5	3, 4	eqeltri 2697	. . . . 5 |- B e. _V
6		indistopon 20805	. . . . 5 |- ( B e. _V -> { (/) , B } e. (TopOn ` B ) )
7	5, 6	ax-mp 5	. . . 4 |- { (/) , B } e. (TopOn ` B )
8	2, 7	syl6eqel 2709	. . 3 |- ( ( G e. Grp /\ J = { (/) , B } ) -> J e. (TopOn ` B ) )
9		distgp.2	. . . 4 |- J = ( TopOpen ` G )
10	3, 9	istps 20738	. . 3 |- ( G e. TopSp <-> J e. (TopOn ` B ) )
11	8, 10	sylibr 224	. 2 |- ( ( G e. Grp /\ J = { (/) , B } ) -> G e. TopSp )
12		eqid 2622	. . . . . 6 |- ( -g ` G ) = ( -g ` G )
13	3, 12	grpsubf 17494	. . . . 5 |- ( G e. Grp -> ( -g ` G ) : ( B X. B ) --> B )
14	13	adantr 481	. . . 4 |- ( ( G e. Grp /\ J = { (/) , B } ) -> ( -g ` G ) : ( B X. B ) --> B )
15	5, 5	xpex 6962	. . . . 5 |- ( B X. B ) e. _V
16	5, 15	elmap 7886	. . . 4 |- ( ( -g ` G ) e. ( B ^m ( B X. B ) ) <-> ( -g ` G ) : ( B X. B ) --> B )
17	14, 16	sylibr 224	. . 3 |- ( ( G e. Grp /\ J = { (/) , B } ) -> ( -g ` G ) e. ( B ^m ( B X. B ) ) )
18	2	oveq2d 6666	. . . 4 |- ( ( G e. Grp /\ J = { (/) , B } ) -> ( ( J tX J ) Cn J ) = ( ( J tX J ) Cn { (/) , B } ) )
19		txtopon 21394	. . . . . 6 |- ( ( J e. (TopOn ` B ) /\ J e. (TopOn ` B ) ) -> ( J tX J ) e. (TopOn ` ( B X. B ) ) )
20	8, 8, 19	syl2anc 693	. . . . 5 |- ( ( G e. Grp /\ J = { (/) , B } ) -> ( J tX J ) e. (TopOn ` ( B X. B ) ) )
21		cnindis 21096	. . . . 5 |- ( ( ( J tX J ) e. (TopOn ` ( B X. B ) ) /\ B e. _V ) -> ( ( J tX J ) Cn { (/) , B } ) = ( B ^m ( B X. B ) ) )
22	20, 5, 21	sylancl 694	. . . 4 |- ( ( G e. Grp /\ J = { (/) , B } ) -> ( ( J tX J ) Cn { (/) , B } ) = ( B ^m ( B X. B ) ) )
23	18, 22	eqtrd 2656	. . 3 |- ( ( G e. Grp /\ J = { (/) , B } ) -> ( ( J tX J ) Cn J ) = ( B ^m ( B X. B ) ) )
24	17, 23	eleqtrrd 2704	. 2 |- ( ( G e. Grp /\ J = { (/) , B } ) -> ( -g ` G ) e. ( ( J tX J ) Cn J ) )
25	9, 12	istgp2 21895	. 2 |- ( G e. TopGrp <-> ( G e. Grp /\ G e. TopSp /\ ( -g ` G ) e. ( ( J tX J ) Cn J ) ) )
26	1, 11, 24, 25	syl3anbrc 1246	1 |- ( ( G e. Grp /\ J = { (/) , B } ) -> G e. TopGrp )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   {cpr 4179   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Basecbs 15857   TopOpenctopn 16082   Grpcgrp 17422   -gcsg 17424  TopOnctopon 20715   TopSpctps 20736   Cn ccn 21028   tX ctx 21363   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-0g 16102  df-topgen 16104  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-tx 21365  df-tmd 21876  df-tgp 21877

This theorem is referenced by: (None)