Theorem istgp 21881

Description: The predicate "is a topological group". Definition of [BourbakiTop1] p. III.1. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)

Hypotheses

Ref	Expression
istgp.1	|- J = ( TopOpen ` G )
istgp.2	|- I = ( invg ` G )

Assertion

Ref	Expression
istgp	|- ( G e. TopGrp <-> ( G e. Grp /\ G e. TopMnd /\ I e. ( J Cn J ) ) )

Proof of Theorem istgp

Dummy variables f j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3796	. . 3 |- ( G e. ( Grp i^i TopMnd ) <-> ( G e. Grp /\ G e. TopMnd ) )
2	1	anbi1i 731	. 2 |- ( ( G e. ( Grp i^i TopMnd ) /\ I e. ( J Cn J ) ) <-> ( ( G e. Grp /\ G e. TopMnd ) /\ I e. ( J Cn J ) ) )
3		fvexd 6203	. . . 4 |- ( f = G -> ( TopOpen ` f ) e. _V )
4		simpl 473	. . . . . . 7 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> f = G )
5	4	fveq2d 6195	. . . . . 6 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> ( invg ` f ) = ( invg ` G ) )
6		istgp.2	. . . . . 6 |- I = ( invg ` G )
7	5, 6	syl6eqr 2674	. . . . 5 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> ( invg ` f ) = I )
8		id 22	. . . . . . 7 |- ( j = ( TopOpen ` f ) -> j = ( TopOpen ` f ) )
9		fveq2 6191	. . . . . . . 8 |- ( f = G -> ( TopOpen ` f ) = ( TopOpen ` G ) )
10		istgp.1	. . . . . . . 8 |- J = ( TopOpen ` G )
11	9, 10	syl6eqr 2674	. . . . . . 7 |- ( f = G -> ( TopOpen ` f ) = J )
12	8, 11	sylan9eqr 2678	. . . . . 6 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> j = J )
13	12, 12	oveq12d 6668	. . . . 5 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> ( j Cn j ) = ( J Cn J ) )
14	7, 13	eleq12d 2695	. . . 4 |- ( ( f = G /\ j = ( TopOpen ` f ) ) -> ( ( invg ` f ) e. ( j Cn j ) <-> I e. ( J Cn J ) ) )
15	3, 14	sbcied 3472	. . 3 |- ( f = G -> ( [. ( TopOpen ` f ) / j ]. ( invg ` f ) e. ( j Cn j ) <-> I e. ( J Cn J ) ) )
16		df-tgp 21877	. . 3 |- TopGrp = { f e. ( Grp i^i TopMnd ) | [. ( TopOpen ` f ) / j ]. ( invg ` f ) e. ( j Cn j ) }
17	15, 16	elrab2 3366	. 2 |- ( G e. TopGrp <-> ( G e. ( Grp i^i TopMnd ) /\ I e. ( J Cn J ) ) )
18		df-3an 1039	. 2 |- ( ( G e. Grp /\ G e. TopMnd /\ I e. ( J Cn J ) ) <-> ( ( G e. Grp /\ G e. TopMnd ) /\ I e. ( J Cn J ) ) )
19	2, 17, 18	3bitr4i 292	1 |- ( G e. TopGrp <-> ( G e. Grp /\ G e. TopMnd /\ I e. ( J Cn J ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   i^i cin 3573   ` 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:  tgpgrp  21882  tgptmd  21883  tgpinv  21889  istgp2  21895  oppgtgp  21902  symgtgp  21905  subgtgp  21909  prdstgpd  21928  tlmtgp  21999  nrgtdrg  22497