MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tdrgdrng Structured version   Unicode version

Theorem tdrgdrng 18203
Description: A topological division ring is a division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
tdrgdrng  |-  ( R  e. TopDRing  ->  R  e.  DivRing )

Proof of Theorem tdrgdrng
StepHypRef Expression
1 eqid 2436 . . 3  |-  (mulGrp `  R )  =  (mulGrp `  R )
2 eqid 2436 . . 3  |-  (Unit `  R )  =  (Unit `  R )
31, 2istdrg 18195 . 2  |-  ( R  e. TopDRing 
<->  ( R  e.  TopRing  /\  R  e.  DivRing  /\  (
(mulGrp `  R )s  (Unit `  R ) )  e. 
TopGrp ) )
43simp2bi 973 1  |-  ( R  e. TopDRing  ->  R  e.  DivRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   ` cfv 5454  (class class class)co 6081   ↾s cress 13470  mulGrpcmgp 15648  Unitcui 15744   DivRingcdr 15835   TopGrpctgp 18101   TopRingctrg 18185  TopDRingctdrg 18186
This theorem is referenced by:  tvclvec  18228
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-tdrg 18190
  Copyright terms: Public domain W3C validator