Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  0idl Structured version   Unicode version

Theorem 0idl 30398
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 2443 . . . 4  |-  ran  G  =  ran  G
3 0idl.2 . . . 4  |-  Z  =  (GId `  G )
41, 2, 3rngo0cl 25378 . . 3  |-  ( R  e.  RingOps  ->  Z  e.  ran  G )
54snssd 4160 . 2  |-  ( R  e.  RingOps  ->  { Z }  C_ 
ran  G )
6 fvex 5866 . . . . 5  |-  (GId `  G )  e.  _V
73, 6eqeltri 2527 . . . 4  |-  Z  e. 
_V
87snid 4042 . . 3  |-  Z  e. 
{ Z }
98a1i 11 . 2  |-  ( R  e.  RingOps  ->  Z  e.  { Z } )
10 elsn 4028 . . . 4  |-  ( x  e.  { Z }  <->  x  =  Z )
11 elsn 4028 . . . . . . . 8  |-  ( y  e.  { Z }  <->  y  =  Z )
121, 2, 3rngo0rid 25379 . . . . . . . . . . 11  |-  ( ( R  e.  RingOps  /\  Z  e.  ran  G )  -> 
( Z G Z )  =  Z )
134, 12mpdan 668 . . . . . . . . . 10  |-  ( R  e.  RingOps  ->  ( Z G Z )  =  Z )
14 ovex 6309 . . . . . . . . . . 11  |-  ( Z G Z )  e. 
_V
1514elsnc 4038 . . . . . . . . . 10  |-  ( ( Z G Z )  e.  { Z }  <->  ( Z G Z )  =  Z )
1613, 15sylibr 212 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  ( Z G Z )  e.  { Z } )
17 oveq2 6289 . . . . . . . . . 10  |-  ( y  =  Z  ->  ( Z G y )  =  ( Z G Z ) )
1817eleq1d 2512 . . . . . . . . 9  |-  ( y  =  Z  ->  (
( Z G y )  e.  { Z } 
<->  ( Z G Z )  e.  { Z } ) )
1916, 18syl5ibrcom 222 . . . . . . . 8  |-  ( R  e.  RingOps  ->  ( y  =  Z  ->  ( Z G y )  e. 
{ Z } ) )
2011, 19syl5bi 217 . . . . . . 7  |-  ( R  e.  RingOps  ->  ( y  e. 
{ Z }  ->  ( Z G y )  e.  { Z }
) )
2120ralrimiv 2855 . . . . . 6  |-  ( R  e.  RingOps  ->  A. y  e.  { Z }  ( Z G y )  e. 
{ Z } )
22 eqid 2443 . . . . . . . . . 10  |-  ( 2nd `  R )  =  ( 2nd `  R )
233, 2, 1, 22rngorz 25382 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( z ( 2nd `  R ) Z )  =  Z )
24 ovex 6309 . . . . . . . . . 10  |-  ( z ( 2nd `  R
) Z )  e. 
_V
2524elsnc 4038 . . . . . . . . 9  |-  ( ( z ( 2nd `  R
) Z )  e. 
{ Z }  <->  ( z
( 2nd `  R
) Z )  =  Z )
2623, 25sylibr 212 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( z ( 2nd `  R ) Z )  e.  { Z }
)
273, 2, 1, 22rngolz 25381 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( Z ( 2nd `  R ) z )  =  Z )
28 ovex 6309 . . . . . . . . . 10  |-  ( Z ( 2nd `  R
) z )  e. 
_V
2928elsnc 4038 . . . . . . . . 9  |-  ( ( Z ( 2nd `  R
) z )  e. 
{ Z }  <->  ( Z
( 2nd `  R
) z )  =  Z )
3027, 29sylibr 212 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( Z ( 2nd `  R ) z )  e.  { Z }
)
3126, 30jca 532 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( ( z ( 2nd `  R ) Z )  e.  { Z }  /\  ( Z ( 2nd `  R
) z )  e. 
{ Z } ) )
3231ralrimiva 2857 . . . . . 6  |-  ( R  e.  RingOps  ->  A. z  e.  ran  G ( ( z ( 2nd `  R ) Z )  e.  { Z }  /\  ( Z ( 2nd `  R
) z )  e. 
{ Z } ) )
3321, 32jca 532 . . . . 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 6288 . . . . . . . 8  |-  ( x  =  Z  ->  (
x G y )  =  ( Z G y ) )
3534eleq1d 2512 . . . . . . 7  |-  ( x  =  Z  ->  (
( x G y )  e.  { Z } 
<->  ( Z G y )  e.  { Z } ) )
3635ralbidv 2882 . . . . . 6  |-  ( x  =  Z  ->  ( A. y  e.  { Z }  ( x G y )  e.  { Z }  <->  A. y  e.  { Z }  ( Z G y )  e. 
{ Z } ) )
37 oveq2 6289 . . . . . . . . 9  |-  ( x  =  Z  ->  (
z ( 2nd `  R
) x )  =  ( z ( 2nd `  R ) Z ) )
3837eleq1d 2512 . . . . . . . 8  |-  ( x  =  Z  ->  (
( z ( 2nd `  R ) x )  e.  { Z }  <->  ( z ( 2nd `  R
) Z )  e. 
{ Z } ) )
39 oveq1 6288 . . . . . . . . 9  |-  ( x  =  Z  ->  (
x ( 2nd `  R
) z )  =  ( Z ( 2nd `  R ) z ) )
4039eleq1d 2512 . . . . . . . 8  |-  ( x  =  Z  ->  (
( x ( 2nd `  R ) z )  e.  { Z }  <->  ( Z ( 2nd `  R
) z )  e. 
{ Z } ) )
4138, 40anbi12d 710 . . . . . . 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 2882 . . . . . 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 710 . . . . 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 222 . . . 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 217 . . 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 2855 . 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 30387 . 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 1180 1  |-  ( R  e.  RingOps  ->  { Z }  e.  ( Idl `  R
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   _Vcvv 3095    C_ wss 3461   {csn 4014   ran crn 4990   ` cfv 5578  (class class class)co 6281   1stc1st 6783   2ndc2nd 6784  GIdcgi 25167   RingOpscrngo 25355   Idlcidl 30380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-1st 6785  df-2nd 6786  df-grpo 25171  df-gid 25172  df-ginv 25173  df-ablo 25262  df-rngo 25356  df-idl 30383
This theorem is referenced by:  0rngo  30400  divrngidl  30401  smprngopr  30425  isdmn3  30447
  Copyright terms: Public domain W3C validator