Theorem tdrgtrg 21976

Description: A topological division ring is a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015.)

Assertion

Ref	Expression
tdrgtrg	|- ( R e. TopDRing -> R e. TopRing )

Proof of Theorem tdrgtrg

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- (mulGrp ` R ) = (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 /\ ( (mulGrp ` R ) ↾s (Unit ` R ) ) e. TopGrp ) )
4	3	simp1bi 1076	1 |- ( R e. TopDRing -> R e. TopRing )

Syntax hints:   -> wi 4   e. wcel 1990   ` 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:  tdrgring  21978  tdrgtmd  21979  tdrgtps  21980  dvrcn  21987