Theorem istdrg2 21981

Description: A topological-ring division ring is a topological division ring iff the group of nonzero elements is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
istdrg2.m	|- M = (mulGrp ` R )
istdrg2.b	|- B = ( Base ` R )
istdrg2.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
istdrg2	|- ( R e. TopDRing <-> ( R e. TopRing /\ R e. DivRing /\ ( M ↾s ( B \ { .0. } ) ) e. TopGrp ) )

Proof of Theorem istdrg2

Step	Hyp	Ref	Expression
1		istdrg2.m	. . 3 |- M = (mulGrp ` R )
2		eqid 2622	. . 3 |- (Unit ` R ) = (Unit ` R )
3	1, 2	istdrg 21969	. 2 |- ( R e. TopDRing <-> ( R e. TopRing /\ R e. DivRing /\ ( M ↾s (Unit ` R ) ) e. TopGrp ) )
4		istdrg2.b	. . . . . . . . 9 |- B = ( Base ` R )
5		istdrg2.z	. . . . . . . . 9 |- .0. = ( 0g ` R )
6	4, 2, 5	isdrng 18751	. . . . . . . 8 |- ( R e. DivRing <-> ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) )
7	6	simprbi 480	. . . . . . 7 |- ( R e. DivRing -> (Unit ` R ) = ( B \ { .0. } ) )
8	7	adantl 482	. . . . . 6 |- ( ( R e. TopRing /\ R e. DivRing ) -> (Unit ` R ) = ( B \ { .0. } ) )
9	8	oveq2d 6666	. . . . 5 |- ( ( R e. TopRing /\ R e. DivRing ) -> ( M ↾s (Unit ` R ) ) = ( M ↾s ( B \ { .0. } ) ) )
10	9	eleq1d 2686	. . . 4 |- ( ( R e. TopRing /\ R e. DivRing ) -> ( ( M ↾s (Unit ` R ) ) e. TopGrp <-> ( M ↾s ( B \ { .0. } ) ) e. TopGrp ) )
11	10	pm5.32i 669	. . 3 |- ( ( ( R e. TopRing /\ R e. DivRing ) /\ ( M ↾s (Unit ` R ) ) e. TopGrp ) <-> ( ( R e. TopRing /\ R e. DivRing ) /\ ( M ↾s ( B \ { .0. } ) ) e. TopGrp ) )
12		df-3an 1039	. . 3 |- ( ( R e. TopRing /\ R e. DivRing /\ ( M ↾s (Unit ` R ) ) e. TopGrp ) <-> ( ( R e. TopRing /\ R e. DivRing ) /\ ( M ↾s (Unit ` R ) ) e. TopGrp ) )
13		df-3an 1039	. . 3 |- ( ( R e. TopRing /\ R e. DivRing /\ ( M ↾s ( B \ { .0. } ) ) e. TopGrp ) <-> ( ( R e. TopRing /\ R e. DivRing ) /\ ( M ↾s ( B \ { .0. } ) ) e. TopGrp ) )
14	11, 12, 13	3bitr4i 292	. 2 |- ( ( R e. TopRing /\ R e. DivRing /\ ( M ↾s (Unit ` R ) ) e. TopGrp ) <-> ( R e. TopRing /\ R e. DivRing /\ ( M ↾s ( B \ { .0. } ) ) e. TopGrp ) )
15	3, 14	bitri 264	1 |- ( R e. TopDRing <-> ( R e. TopRing /\ R e. DivRing /\ ( M ↾s ( B \ { .0. } ) ) e. TopGrp ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   \ cdif 3571   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857   ↾_s cress 15858   0gc0g 16100  mulGrpcmgp 18489   Ringcrg 18547  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-ral 2917  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-drng 18749  df-tdrg 21964

This theorem is referenced by: (None)