Theorem isdrng 16834

Description: The predicate "is a division ring". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 2-Dec-2014.)

Hypotheses

Ref	Expression
isdrng.b	|- B = ( Base ` R )
isdrng.u	|- U = (Unit ` R )
isdrng.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
isdrng	|- ( R e. DivRing <-> ( R e. Ring /\ U = ( B \ { .0. } ) ) )

Proof of Theorem isdrng

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5689	. . . 4 |- ( r = R -> (Unit ` r ) = (Unit ` R ) )
2		isdrng.u	. . . 4 |- U = (Unit ` R )
3	1, 2	syl6eqr 2491	. . 3 |- ( r = R -> (Unit ` r ) = U )
4		fveq2 5689	. . . . 5 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
5		isdrng.b	. . . . 5 |- B = ( Base ` R )
6	4, 5	syl6eqr 2491	. . . 4 |- ( r = R -> ( Base ` r ) = B )
7		fveq2 5689	. . . . . 6 |- ( r = R -> ( 0g ` r ) = ( 0g ` R ) )
8		isdrng.z	. . . . . 6 |- .0. = ( 0g ` R )
9	7, 8	syl6eqr 2491	. . . . 5 |- ( r = R -> ( 0g ` r ) = .0. )
10	9	sneqd 3887	. . . 4 |- ( r = R -> { ( 0g ` r ) } = { .0. } )
11	6, 10	difeq12d 3473	. . 3 |- ( r = R -> ( ( Base ` r ) \ { ( 0g ` r ) } ) = ( B \ { .0. } ) )
12	3, 11	eqeq12d 2455	. 2 |- ( r = R -> ( (Unit ` r ) = ( ( Base ` r ) \ { ( 0g ` r ) } ) <-> U = ( B \ { .0. } ) ) )
13		df-drng 16832	. 2 |- DivRing = { r e. Ring | (Unit ` r ) = ( ( Base ` r ) \ { ( 0g ` r ) } ) }
14	12, 13	elrab2 3117	1 |- ( R e. DivRing <-> ( R e. Ring /\ U = ( B \ { .0. } ) ) )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   \ cdif 3323   {csn 3875   ` cfv 5416   Basecbs 14172   0gc0g 14376   Ringcrg 16643  Unitcui 16729   DivRingcdr 16830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-iota 5379  df-fv 5424  df-drng 16832

This theorem is referenced by:  drngunit  16835  drngui  16836  drngrng  16837  isdrng2  16840  drngprop  16841  drngid  16844  opprdrng  16854  drngpropd  16857  issubdrg  16888  drngdomn  17373  fidomndrng  17377  istdrg2  19750  cvsunit  20678  cphreccllem  20695  zrhunitpreima  26405  cntzsdrg  29556