Theorem isprrngo 33849

Description: The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.)

Hypotheses

Ref	Expression
isprrng.1	|- G = ( 1st ` R )
isprrng.2	|- Z = (GId ` G )

Assertion

Ref	Expression
isprrngo	|- ( R e. PrRing <-> ( R e. RingOps /\ { Z } e. ( PrIdl ` R ) ) )

Proof of Theorem isprrngo

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . 7 |- ( r = R -> ( 1st ` r ) = ( 1st ` R ) )
2		isprrng.1	. . . . . . 7 |- G = ( 1st ` R )
3	1, 2	syl6eqr 2674	. . . . . 6 |- ( r = R -> ( 1st ` r ) = G )
4	3	fveq2d 6195	. . . . 5 |- ( r = R -> (GId ` ( 1st ` r ) ) = (GId ` G ) )
5		isprrng.2	. . . . 5 |- Z = (GId ` G )
6	4, 5	syl6eqr 2674	. . . 4 |- ( r = R -> (GId ` ( 1st ` r ) ) = Z )
7	6	sneqd 4189	. . 3 |- ( r = R -> { (GId ` ( 1st ` r ) ) } = { Z } )
8		fveq2 6191	. . 3 |- ( r = R -> ( PrIdl ` r ) = ( PrIdl ` R ) )
9	7, 8	eleq12d 2695	. 2 |- ( r = R -> ( { (GId ` ( 1st ` r ) ) } e. ( PrIdl ` r ) <-> { Z } e. ( PrIdl ` R ) ) )
10		df-prrngo 33847	. 2 |- PrRing = { r e. RingOps | { (GId ` ( 1st ` r ) ) } e. ( PrIdl ` r ) }
11	9, 10	elrab2 3366	1 |- ( R e. PrRing <-> ( R e. RingOps /\ { Z } e. ( PrIdl ` R ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   ` cfv 5888   1stc1st 7166  GIdcgi 27344   RingOpscrngo 33693   PrIdlcpridl 33807   PrRingcprrng 33845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-prrngo 33847

This theorem is referenced by:  prrngorngo  33850  smprngopr  33851  isdmn3  33873