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Theorem tdrgtrg 20502
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
StepHypRef Expression
1 eqid 2467 . . 3  |-  (mulGrp `  R )  =  (mulGrp `  R )
2 eqid 2467 . . 3  |-  (Unit `  R )  =  (Unit `  R )
31, 2istdrg 20495 . 2  |-  ( R  e. TopDRing 
<->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  (
(mulGrp `  R )s  (Unit `  R ) )  e. 
TopGrp ) )
43simp1bi 1011 1  |-  ( R  e. TopDRing  ->  R  e.  TopRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1767   ` cfv 5588  (class class class)co 6285   ↾s cress 14494  mulGrpcmgp 16955  Unitcui 17101   DivRingcdr 17208   TopGrpctgp 20397   TopRingctrg 20485  TopDRingctdrg 20486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596  df-ov 6288  df-tdrg 20490
This theorem is referenced by:  tdrgrng  20504  tdrgtmd  20505  tdrgtps  20506  dvrcn  20513
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