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Theorem 0idl 26650
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 2283 . . . 4  |-  ran  G  =  ran  G
3 0idl.2 . . . 4  |-  Z  =  (GId `  G )
41, 2, 3rngo0cl 21065 . . 3  |-  ( R  e.  RingOps  ->  Z  e.  ran  G )
54snssd 3760 . 2  |-  ( R  e.  RingOps  ->  { Z }  C_ 
ran  G )
6 fvex 5539 . . . . 5  |-  (GId `  G )  e.  _V
73, 6eqeltri 2353 . . . 4  |-  Z  e. 
_V
87snid 3667 . . 3  |-  Z  e. 
{ Z }
98a1i 10 . 2  |-  ( R  e.  RingOps  ->  Z  e.  { Z } )
10 elsn 3655 . . . 4  |-  ( x  e.  { Z }  <->  x  =  Z )
11 elsn 3655 . . . . . . . 8  |-  ( y  e.  { Z }  <->  y  =  Z )
121, 2, 3rngo0rid 21066 . . . . . . . . . . 11  |-  ( ( R  e.  RingOps  /\  Z  e.  ran  G )  -> 
( Z G Z )  =  Z )
134, 12mpdan 649 . . . . . . . . . 10  |-  ( R  e.  RingOps  ->  ( Z G Z )  =  Z )
14 ovex 5883 . . . . . . . . . . 11  |-  ( Z G Z )  e. 
_V
1514elsnc 3663 . . . . . . . . . 10  |-  ( ( Z G Z )  e.  { Z }  <->  ( Z G Z )  =  Z )
1613, 15sylibr 203 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  ( Z G Z )  e.  { Z } )
17 oveq2 5866 . . . . . . . . . 10  |-  ( y  =  Z  ->  ( Z G y )  =  ( Z G Z ) )
1817eleq1d 2349 . . . . . . . . 9  |-  ( y  =  Z  ->  (
( Z G y )  e.  { Z } 
<->  ( Z G Z )  e.  { Z } ) )
1916, 18syl5ibrcom 213 . . . . . . . 8  |-  ( R  e.  RingOps  ->  ( y  =  Z  ->  ( Z G y )  e. 
{ Z } ) )
2011, 19syl5bi 208 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( y  e. 
{ Z }  ->  ( Z G y )  e.  { Z }
) )
2120ralrimiv 2625 . . . . . 6  |-  ( R  e.  RingOps  ->  A. y  e.  { Z }  ( Z G y )  e. 
{ Z } )
22 eqid 2283 . . . . . . . . . 10  |-  ( 2nd `  R )  =  ( 2nd `  R )
233, 2, 1, 22rngorz 21069 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( z ( 2nd `  R ) Z )  =  Z )
24 ovex 5883 . . . . . . . . . 10  |-  ( z ( 2nd `  R
) Z )  e. 
_V
2524elsnc 3663 . . . . . . . . 9  |-  ( ( z ( 2nd `  R
) Z )  e. 
{ Z }  <->  ( z
( 2nd `  R
) Z )  =  Z )
2623, 25sylibr 203 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( z ( 2nd `  R ) Z )  e.  { Z }
)
273, 2, 1, 22rngolz 21068 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( Z ( 2nd `  R ) z )  =  Z )
28 ovex 5883 . . . . . . . . . 10  |-  ( Z ( 2nd `  R
) z )  e. 
_V
2928elsnc 3663 . . . . . . . . 9  |-  ( ( Z ( 2nd `  R
) z )  e. 
{ Z }  <->  ( Z
( 2nd `  R
) z )  =  Z )
3027, 29sylibr 203 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( Z ( 2nd `  R ) z )  e.  { Z }
)
3126, 30jca 518 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( ( z ( 2nd `  R ) Z )  e.  { Z }  /\  ( Z ( 2nd `  R
) z )  e. 
{ Z } ) )
3231ralrimiva 2626 . . . . . 6  |-  ( R  e.  RingOps  ->  A. z  e.  ran  G ( ( z ( 2nd `  R ) Z )  e.  { Z }  /\  ( Z ( 2nd `  R
) z )  e. 
{ Z } ) )
3321, 32jca 518 . . . . 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 5865 . . . . . . . 8  |-  ( x  =  Z  ->  (
x G y )  =  ( Z G y ) )
3534eleq1d 2349 . . . . . . 7  |-  ( x  =  Z  ->  (
( x G y )  e.  { Z } 
<->  ( Z G y )  e.  { Z } ) )
3635ralbidv 2563 . . . . . 6  |-  ( x  =  Z  ->  ( A. y  e.  { Z }  ( x G y )  e.  { Z }  <->  A. y  e.  { Z }  ( Z G y )  e. 
{ Z } ) )
37 oveq2 5866 . . . . . . . . 9  |-  ( x  =  Z  ->  (
z ( 2nd `  R
) x )  =  ( z ( 2nd `  R ) Z ) )
3837eleq1d 2349 . . . . . . . 8  |-  ( x  =  Z  ->  (
( z ( 2nd `  R ) x )  e.  { Z }  <->  ( z ( 2nd `  R
) Z )  e. 
{ Z } ) )
39 oveq1 5865 . . . . . . . . 9  |-  ( x  =  Z  ->  (
x ( 2nd `  R
) z )  =  ( Z ( 2nd `  R ) z ) )
4039eleq1d 2349 . . . . . . . 8  |-  ( x  =  Z  ->  (
( x ( 2nd `  R ) z )  e.  { Z }  <->  ( Z ( 2nd `  R
) z )  e. 
{ Z } ) )
4138, 40anbi12d 691 . . . . . . 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 2563 . . . . . 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 691 . . . . 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 213 . . . 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 208 . . 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 2625 . 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 26639 . 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 1135 1  |-  ( R  e.  RingOps  ->  { Z }  e.  ( Idl `  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   {csn 3640   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121  GIdcgi 20854   RingOpscrngo 21042   Idlcidl 26632
This theorem is referenced by:  0rngo  26652  divrngidl  26653  smprngopr  26677  isdmn3  26699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-rngo 21043  df-idl 26635
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