Theorem istdrg 21969

Description: Express the predicate " R is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
istrg.1	|- M = (mulGrp ` R )
istdrg.1	|- U = (Unit ` R )

Assertion

Ref	Expression
istdrg	|- ( R e. TopDRing <-> ( R e. TopRing /\ R e. DivRing /\ ( M ↾s U ) e. TopGrp ) )

Proof of Theorem istdrg

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3796	. . 3 |- ( R e. ( TopRing i^i DivRing ) <-> ( R e. TopRing /\ R e. DivRing ) )
2	1	anbi1i 731	. 2 |- ( ( R e. ( TopRing i^i DivRing ) /\ ( M ↾s U ) e. TopGrp ) <-> ( ( R e. TopRing /\ R e. DivRing ) /\ ( M ↾s U ) e. TopGrp ) )
3		fveq2 6191	. . . . . 6 |- ( r = R -> (mulGrp ` r ) = (mulGrp ` R ) )
4		istrg.1	. . . . . 6 |- M = (mulGrp ` R )
5	3, 4	syl6eqr 2674	. . . . 5 |- ( r = R -> (mulGrp ` r ) = M )
6		fveq2 6191	. . . . . 6 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
7		istdrg.1	. . . . . 6 |- U = (Unit ` R )
8	6, 7	syl6eqr 2674	. . . . 5 |- ( r = R -> (Unit ` r ) = U )
9	5, 8	oveq12d 6668	. . . 4 |- ( r = R -> ( (mulGrp ` r ) ↾s (Unit ` r ) ) = ( M ↾s U ) )
10	9	eleq1d 2686	. . 3 |- ( r = R -> ( ( (mulGrp ` r ) ↾s (Unit ` r ) ) e. TopGrp <-> ( M ↾s U ) e. TopGrp ) )
11		df-tdrg 21964	. . 3 |- TopDRing = { r e. ( TopRing i^i DivRing ) | ( (mulGrp ` r ) ↾s (Unit ` r ) ) e. TopGrp }
12	10, 11	elrab2 3366	. 2 |- ( R e. TopDRing <-> ( R e. ( TopRing i^i DivRing ) /\ ( M ↾s U ) e. TopGrp ) )
13		df-3an 1039	. 2 |- ( ( R e. TopRing /\ R e. DivRing /\ ( M ↾s U ) e. TopGrp ) <-> ( ( R e. TopRing /\ R e. DivRing ) /\ ( M ↾s U ) e. TopGrp ) )
14	2, 12, 13	3bitr4i 292	1 |- ( R e. TopDRing <-> ( R e. TopRing /\ R e. DivRing /\ ( M ↾s U ) e. TopGrp ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   i^i cin 3573   ` cfv 5888  (class class class)co 6650   ↾_s cress 15858  mulGrpcmgp 18489  Unitcui 18639   DivRingcdr 18747   TopGrpctgp 21875   TopRingctrg 21959  TopDRingctdrg 21960

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  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-tdrg 21964

This theorem is referenced by:  tdrgunit  21970  tdrgtrg  21976  tdrgdrng  21977  istdrg2  21981  nrgtdrg  22497