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Theorem 0idl 26529
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 2408 . . . 4  |-  ran  G  =  ran  G
3 0idl.2 . . . 4  |-  Z  =  (GId `  G )
41, 2, 3rngo0cl 21943 . . 3  |-  ( R  e.  RingOps  ->  Z  e.  ran  G )
54snssd 3907 . 2  |-  ( R  e.  RingOps  ->  { Z }  C_ 
ran  G )
6 fvex 5705 . . . . 5  |-  (GId `  G )  e.  _V
73, 6eqeltri 2478 . . . 4  |-  Z  e. 
_V
87snid 3805 . . 3  |-  Z  e. 
{ Z }
98a1i 11 . 2  |-  ( R  e.  RingOps  ->  Z  e.  { Z } )
10 elsn 3793 . . . 4  |-  ( x  e.  { Z }  <->  x  =  Z )
11 elsn 3793 . . . . . . . 8  |-  ( y  e.  { Z }  <->  y  =  Z )
121, 2, 3rngo0rid 21944 . . . . . . . . . . 11  |-  ( ( R  e.  RingOps  /\  Z  e.  ran  G )  -> 
( Z G Z )  =  Z )
134, 12mpdan 650 . . . . . . . . . 10  |-  ( R  e.  RingOps  ->  ( Z G Z )  =  Z )
14 ovex 6069 . . . . . . . . . . 11  |-  ( Z G Z )  e. 
_V
1514elsnc 3801 . . . . . . . . . 10  |-  ( ( Z G Z )  e.  { Z }  <->  ( Z G Z )  =  Z )
1613, 15sylibr 204 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  ( Z G Z )  e.  { Z } )
17 oveq2 6052 . . . . . . . . . 10  |-  ( y  =  Z  ->  ( Z G y )  =  ( Z G Z ) )
1817eleq1d 2474 . . . . . . . . 9  |-  ( y  =  Z  ->  (
( Z G y )  e.  { Z } 
<->  ( Z G Z )  e.  { Z } ) )
1916, 18syl5ibrcom 214 . . . . . . . 8  |-  ( R  e.  RingOps  ->  ( y  =  Z  ->  ( Z G y )  e. 
{ Z } ) )
2011, 19syl5bi 209 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( y  e. 
{ Z }  ->  ( Z G y )  e.  { Z }
) )
2120ralrimiv 2752 . . . . . 6  |-  ( R  e.  RingOps  ->  A. y  e.  { Z }  ( Z G y )  e. 
{ Z } )
22 eqid 2408 . . . . . . . . . 10  |-  ( 2nd `  R )  =  ( 2nd `  R )
233, 2, 1, 22rngorz 21947 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( z ( 2nd `  R ) Z )  =  Z )
24 ovex 6069 . . . . . . . . . 10  |-  ( z ( 2nd `  R
) Z )  e. 
_V
2524elsnc 3801 . . . . . . . . 9  |-  ( ( z ( 2nd `  R
) Z )  e. 
{ Z }  <->  ( z
( 2nd `  R
) Z )  =  Z )
2623, 25sylibr 204 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( z ( 2nd `  R ) Z )  e.  { Z }
)
273, 2, 1, 22rngolz 21946 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( Z ( 2nd `  R ) z )  =  Z )
28 ovex 6069 . . . . . . . . . 10  |-  ( Z ( 2nd `  R
) z )  e. 
_V
2928elsnc 3801 . . . . . . . . 9  |-  ( ( Z ( 2nd `  R
) z )  e. 
{ Z }  <->  ( Z
( 2nd `  R
) z )  =  Z )
3027, 29sylibr 204 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( Z ( 2nd `  R ) z )  e.  { Z }
)
3126, 30jca 519 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( ( z ( 2nd `  R ) Z )  e.  { Z }  /\  ( Z ( 2nd `  R
) z )  e. 
{ Z } ) )
3231ralrimiva 2753 . . . . . 6  |-  ( R  e.  RingOps  ->  A. z  e.  ran  G ( ( z ( 2nd `  R ) Z )  e.  { Z }  /\  ( Z ( 2nd `  R
) z )  e. 
{ Z } ) )
3321, 32jca 519 . . . . 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 6051 . . . . . . . 8  |-  ( x  =  Z  ->  (
x G y )  =  ( Z G y ) )
3534eleq1d 2474 . . . . . . 7  |-  ( x  =  Z  ->  (
( x G y )  e.  { Z } 
<->  ( Z G y )  e.  { Z } ) )
3635ralbidv 2690 . . . . . 6  |-  ( x  =  Z  ->  ( A. y  e.  { Z }  ( x G y )  e.  { Z }  <->  A. y  e.  { Z }  ( Z G y )  e. 
{ Z } ) )
37 oveq2 6052 . . . . . . . . 9  |-  ( x  =  Z  ->  (
z ( 2nd `  R
) x )  =  ( z ( 2nd `  R ) Z ) )
3837eleq1d 2474 . . . . . . . 8  |-  ( x  =  Z  ->  (
( z ( 2nd `  R ) x )  e.  { Z }  <->  ( z ( 2nd `  R
) Z )  e. 
{ Z } ) )
39 oveq1 6051 . . . . . . . . 9  |-  ( x  =  Z  ->  (
x ( 2nd `  R
) z )  =  ( Z ( 2nd `  R ) z ) )
4039eleq1d 2474 . . . . . . . 8  |-  ( x  =  Z  ->  (
( x ( 2nd `  R ) z )  e.  { Z }  <->  ( Z ( 2nd `  R
) z )  e. 
{ Z } ) )
4138, 40anbi12d 692 . . . . . . 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 2690 . . . . . 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 692 . . . . 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 214 . . . 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 209 . . 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 2752 . 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 26518 . 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 1137 1  |-  ( R  e.  RingOps  ->  { Z }  e.  ( Idl `  R
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2670   _Vcvv 2920    C_ wss 3284   {csn 3778   ran crn 4842   ` cfv 5417  (class class class)co 6044   1stc1st 6310   2ndc2nd 6311  GIdcgi 21732   RingOpscrngo 21920   Idlcidl 26511
This theorem is referenced by:  0rngo  26531  divrngidl  26532  smprngopr  26556  isdmn3  26578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-1st 6312  df-2nd 6313  df-riota 6512  df-grpo 21736  df-gid 21737  df-ginv 21738  df-ablo 21827  df-rngo 21921  df-idl 26514
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