Theorem drngui 17597

Description: The set of units of a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)

Hypotheses

Ref	Expression
drngui.b	|- B = ( Base ` R )
drngui.z	|- .0. = ( 0g ` R )
drngui.r	|- R e. DivRing

Assertion

Ref	Expression
drngui	|- ( B \ { .0. } ) = (Unit ` R )

Proof of Theorem drngui

Step	Hyp	Ref	Expression
1		drngui.r	. . . 4 |- R e. DivRing
2		drngui.b	. . . . 5 |- B = ( Base ` R )
3		eqid 2454	. . . . 5 |- (Unit ` R ) = (Unit ` R )
4		drngui.z	. . . . 5 |- .0. = ( 0g ` R )
5	2, 3, 4	isdrng 17595	. . . 4 |- ( R e. DivRing <-> ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) ) )
6	1, 5	mpbi 208	. . 3 |- ( R e. Ring /\ (Unit ` R ) = ( B \ { .0. } ) )
7	6	simpri 460	. 2 |- (Unit ` R ) = ( B \ { .0. } )
8	7	eqcomi 2467	1 |- ( B \ { .0. } ) = (Unit ` R )

Syntax hints:   /\ wa 367   = wceq 1398   e. wcel 1823   \ cdif 3458   {csn 4016   ` cfv 5570   Basecbs 14716   0gc0g 14929   Ringcrg 17393  Unitcui 17483   DivRingcdr 17591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-drng 17593

This theorem is referenced by:  cnflddiv  18643  cnfldinv  18644  cnsubdrglem  18664  cnmgpabl  18674  cnmsubglem  18675  gzrngunit  18678  zringunit  18703  expghm  18708  psgninv  18791  zrhpsgnmhm  18793  amgmlem  23517  dchrghm  23729  dchrabs  23733  sum2dchr  23747  lgseisenlem4  23825  qrngdiv  24007  proot1ex  31402