Theorem drngui 16962

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 16960	. . . 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 462	. 2 |- (Unit ` R ) = ( B \ { .0. } )
8	7	eqcomi 2467	1 |- ( B \ { .0. } ) = (Unit ` R )

Syntax hints:   /\ wa 369   = wceq 1370   e. wcel 1758   \ cdif 3434   {csn 3986   ` cfv 5527   Basecbs 14293   0gc0g 14498   Ringcrg 16769  Unitcui 16855   DivRingcdr 16956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-iota 5490  df-fv 5535  df-drng 16958

This theorem is referenced by:  cnflddiv  17972  cnfldinv  17973  cnsubdrglem  17990  cnmgpabl  18000  cnmsubglem  18001  gzrngunit  18004  zringunit  18040  zrngunit  18041  expghm  18049  expghmOLD  18050  psgninv  18138  zrhpsgnmhm  18140  amgmlem  22517  dchrghm  22729  dchrabs  22733  sum2dchr  22747  lgseisenlem4  22825  qrngdiv  23007  proot1ex  29718