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Theorem 0idl 26649
Description: The set containing only  0 is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
Hypotheses
Ref Expression
0idl.1  |-  G  =  ( 1st `  R
)
0idl.2  |-  Z  =  (GId `  G )
Assertion
Ref Expression
0idl  |-  ( R  e.  RingOps  ->  { Z }  e.  ( Idl `  R
) )

Proof of Theorem 0idl
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0idl.1 . . . 4  |-  G  =  ( 1st `  R
)
2 eqid 2438 . . . 4  |-  ran  G  =  ran  G
3 0idl.2 . . . 4  |-  Z  =  (GId `  G )
41, 2, 3rngo0cl 21991 . . 3  |-  ( R  e.  RingOps  ->  Z  e.  ran  G )
54snssd 3945 . 2  |-  ( R  e.  RingOps  ->  { Z }  C_ 
ran  G )
6 fvex 5745 . . . . 5  |-  (GId `  G )  e.  _V
73, 6eqeltri 2508 . . . 4  |-  Z  e. 
_V
87snid 3843 . . 3  |-  Z  e. 
{ Z }
98a1i 11 . 2  |-  ( R  e.  RingOps  ->  Z  e.  { Z } )
10 elsn 3831 . . . 4  |-  ( x  e.  { Z }  <->  x  =  Z )
11 elsn 3831 . . . . . . . 8  |-  ( y  e.  { Z }  <->  y  =  Z )
121, 2, 3rngo0rid 21992 . . . . . . . . . . 11  |-  ( ( R  e.  RingOps  /\  Z  e.  ran  G )  -> 
( Z G Z )  =  Z )
134, 12mpdan 651 . . . . . . . . . 10  |-  ( R  e.  RingOps  ->  ( Z G Z )  =  Z )
14 ovex 6109 . . . . . . . . . . 11  |-  ( Z G Z )  e. 
_V
1514elsnc 3839 . . . . . . . . . 10  |-  ( ( Z G Z )  e.  { Z }  <->  ( Z G Z )  =  Z )
1613, 15sylibr 205 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  ( Z G Z )  e.  { Z } )
17 oveq2 6092 . . . . . . . . . 10  |-  ( y  =  Z  ->  ( Z G y )  =  ( Z G Z ) )
1817eleq1d 2504 . . . . . . . . 9  |-  ( y  =  Z  ->  (
( Z G y )  e.  { Z } 
<->  ( Z G Z )  e.  { Z } ) )
1916, 18syl5ibrcom 215 . . . . . . . 8  |-  ( R  e.  RingOps  ->  ( y  =  Z  ->  ( Z G y )  e. 
{ Z } ) )
2011, 19syl5bi 210 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( y  e. 
{ Z }  ->  ( Z G y )  e.  { Z }
) )
2120ralrimiv 2790 . . . . . 6  |-  ( R  e.  RingOps  ->  A. y  e.  { Z }  ( Z G y )  e. 
{ Z } )
22 eqid 2438 . . . . . . . . . 10  |-  ( 2nd `  R )  =  ( 2nd `  R )
233, 2, 1, 22rngorz 21995 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( z ( 2nd `  R ) Z )  =  Z )
24 ovex 6109 . . . . . . . . . 10  |-  ( z ( 2nd `  R
) Z )  e. 
_V
2524elsnc 3839 . . . . . . . . 9  |-  ( ( z ( 2nd `  R
) Z )  e. 
{ Z }  <->  ( z
( 2nd `  R
) Z )  =  Z )
2623, 25sylibr 205 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( z ( 2nd `  R ) Z )  e.  { Z }
)
273, 2, 1, 22rngolz 21994 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( Z ( 2nd `  R ) z )  =  Z )
28 ovex 6109 . . . . . . . . . 10  |-  ( Z ( 2nd `  R
) z )  e. 
_V
2928elsnc 3839 . . . . . . . . 9  |-  ( ( Z ( 2nd `  R
) z )  e. 
{ Z }  <->  ( Z
( 2nd `  R
) z )  =  Z )
3027, 29sylibr 205 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( Z ( 2nd `  R ) z )  e.  { Z }
)
3126, 30jca 520 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( ( z ( 2nd `  R ) Z )  e.  { Z }  /\  ( Z ( 2nd `  R
) z )  e. 
{ Z } ) )
3231ralrimiva 2791 . . . . . 6  |-  ( R  e.  RingOps  ->  A. z  e.  ran  G ( ( z ( 2nd `  R ) Z )  e.  { Z }  /\  ( Z ( 2nd `  R
) z )  e. 
{ Z } ) )
3321, 32jca 520 . . . . 5  |-  ( R  e.  RingOps  ->  ( A. y  e.  { Z }  ( Z G y )  e. 
{ Z }  /\  A. z  e.  ran  G
( ( z ( 2nd `  R ) Z )  e.  { Z }  /\  ( Z ( 2nd `  R
) z )  e. 
{ Z } ) ) )
34 oveq1 6091 . . . . . . . 8  |-  ( x  =  Z  ->  (
x G y )  =  ( Z G y ) )
3534eleq1d 2504 . . . . . . 7  |-  ( x  =  Z  ->  (
( x G y )  e.  { Z } 
<->  ( Z G y )  e.  { Z } ) )
3635ralbidv 2727 . . . . . 6  |-  ( x  =  Z  ->  ( A. y  e.  { Z }  ( x G y )  e.  { Z }  <->  A. y  e.  { Z }  ( Z G y )  e. 
{ Z } ) )
37 oveq2 6092 . . . . . . . . 9  |-  ( x  =  Z  ->  (
z ( 2nd `  R
) x )  =  ( z ( 2nd `  R ) Z ) )
3837eleq1d 2504 . . . . . . . 8  |-  ( x  =  Z  ->  (
( z ( 2nd `  R ) x )  e.  { Z }  <->  ( z ( 2nd `  R
) Z )  e. 
{ Z } ) )
39 oveq1 6091 . . . . . . . . 9  |-  ( x  =  Z  ->  (
x ( 2nd `  R
) z )  =  ( Z ( 2nd `  R ) z ) )
4039eleq1d 2504 . . . . . . . 8  |-  ( x  =  Z  ->  (
( x ( 2nd `  R ) z )  e.  { Z }  <->  ( Z ( 2nd `  R
) z )  e. 
{ Z } ) )
4138, 40anbi12d 693 . . . . . . 7  |-  ( x  =  Z  ->  (
( ( z ( 2nd `  R ) x )  e.  { Z }  /\  (
x ( 2nd `  R
) z )  e. 
{ Z } )  <-> 
( ( z ( 2nd `  R ) Z )  e.  { Z }  /\  ( Z ( 2nd `  R
) z )  e. 
{ Z } ) ) )
4241ralbidv 2727 . . . . . 6  |-  ( x  =  Z  ->  ( A. z  e.  ran  G ( ( z ( 2nd `  R ) x )  e.  { Z }  /\  (
x ( 2nd `  R
) z )  e. 
{ Z } )  <->  A. z  e.  ran  G ( ( z ( 2nd `  R ) Z )  e.  { Z }  /\  ( Z ( 2nd `  R
) z )  e. 
{ Z } ) ) )
4336, 42anbi12d 693 . . . . 5  |-  ( x  =  Z  ->  (
( A. y  e. 
{ Z }  (
x G y )  e.  { Z }  /\  A. z  e.  ran  G ( ( z ( 2nd `  R ) x )  e.  { Z }  /\  (
x ( 2nd `  R
) z )  e. 
{ Z } ) )  <->  ( A. y  e.  { Z }  ( Z G y )  e. 
{ Z }  /\  A. z  e.  ran  G
( ( z ( 2nd `  R ) Z )  e.  { Z }  /\  ( Z ( 2nd `  R
) z )  e. 
{ Z } ) ) ) )
4433, 43syl5ibrcom 215 . . . 4  |-  ( R  e.  RingOps  ->  ( x  =  Z  ->  ( A. y  e.  { Z }  ( x G y )  e.  { Z }  /\  A. z  e.  ran  G ( ( z ( 2nd `  R
) x )  e. 
{ Z }  /\  ( x ( 2nd `  R ) z )  e.  { Z }
) ) ) )
4510, 44syl5bi 210 . . 3  |-  ( R  e.  RingOps  ->  ( x  e. 
{ Z }  ->  ( A. y  e.  { Z }  ( x G y )  e. 
{ Z }  /\  A. z  e.  ran  G
( ( z ( 2nd `  R ) x )  e.  { Z }  /\  (
x ( 2nd `  R
) z )  e. 
{ Z } ) ) ) )
4645ralrimiv 2790 . 2  |-  ( R  e.  RingOps  ->  A. x  e.  { Z }  ( A. y  e.  { Z }  ( x G y )  e.  { Z }  /\  A. z  e.  ran  G ( ( z ( 2nd `  R
) x )  e. 
{ Z }  /\  ( x ( 2nd `  R ) z )  e.  { Z }
) ) )
471, 22, 2, 3isidl 26638 . 2  |-  ( R  e.  RingOps  ->  ( { Z }  e.  ( Idl `  R )  <->  ( { Z }  C_  ran  G  /\  Z  e.  { Z }  /\  A. x  e. 
{ Z }  ( A. y  e.  { Z }  ( x G y )  e.  { Z }  /\  A. z  e.  ran  G ( ( z ( 2nd `  R
) x )  e. 
{ Z }  /\  ( x ( 2nd `  R ) z )  e.  { Z }
) ) ) ) )
485, 9, 46, 47mpbir3and 1138 1  |-  ( R  e.  RingOps  ->  { Z }  e.  ( Idl `  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    C_ wss 3322   {csn 3816   ran crn 4882   ` cfv 5457  (class class class)co 6084   1stc1st 6350   2ndc2nd 6351  GIdcgi 21780   RingOpscrngo 21968   Idlcidl 26631
This theorem is referenced by:  0rngo  26651  divrngidl  26652  smprngopr  26676  isdmn3  26698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-1st 6352  df-2nd 6353  df-riota 6552  df-grpo 21784  df-gid 21785  df-ginv 21786  df-ablo 21875  df-rngo 21969  df-idl 26634
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