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

Theorem 0idl 26753
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 2296 . . . 4  |-  ran  G  =  ran  G
3 0idl.2 . . . 4  |-  Z  =  (GId `  G )
41, 2, 3rngo0cl 21081 . . 3  |-  ( R  e.  RingOps  ->  Z  e.  ran  G )
54snssd 3776 . 2  |-  ( R  e.  RingOps  ->  { Z }  C_ 
ran  G )
6 fvex 5555 . . . . 5  |-  (GId `  G )  e.  _V
73, 6eqeltri 2366 . . . 4  |-  Z  e. 
_V
87snid 3680 . . 3  |-  Z  e. 
{ Z }
98a1i 10 . 2  |-  ( R  e.  RingOps  ->  Z  e.  { Z } )
10 elsn 3668 . . . 4  |-  ( x  e.  { Z }  <->  x  =  Z )
11 elsn 3668 . . . . . . . 8  |-  ( y  e.  { Z }  <->  y  =  Z )
121, 2, 3rngo0rid 21082 . . . . . . . . . . 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 5899 . . . . . . . . . . 11  |-  ( Z G Z )  e. 
_V
1514elsnc 3676 . . . . . . . . . 10  |-  ( ( Z G Z )  e.  { Z }  <->  ( Z G Z )  =  Z )
1613, 15sylibr 203 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  ( Z G Z )  e.  { Z } )
17 oveq2 5882 . . . . . . . . . 10  |-  ( y  =  Z  ->  ( Z G y )  =  ( Z G Z ) )
1817eleq1d 2362 . . . . . . . . 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 2638 . . . . . 6  |-  ( R  e.  RingOps  ->  A. y  e.  { Z }  ( Z G y )  e. 
{ Z } )
22 eqid 2296 . . . . . . . . . 10  |-  ( 2nd `  R )  =  ( 2nd `  R )
233, 2, 1, 22rngorz 21085 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( z ( 2nd `  R ) Z )  =  Z )
24 ovex 5899 . . . . . . . . . 10  |-  ( z ( 2nd `  R
) Z )  e. 
_V
2524elsnc 3676 . . . . . . . . 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 21084 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  z  e.  ran  G )  -> 
( Z ( 2nd `  R ) z )  =  Z )
28 ovex 5899 . . . . . . . . . 10  |-  ( Z ( 2nd `  R
) z )  e. 
_V
2928elsnc 3676 . . . . . . . . 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 2639 . . . . . 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 5881 . . . . . . . 8  |-  ( x  =  Z  ->  (
x G y )  =  ( Z G y ) )
3534eleq1d 2362 . . . . . . 7  |-  ( x  =  Z  ->  (
( x G y )  e.  { Z } 
<->  ( Z G y )  e.  { Z } ) )
3635ralbidv 2576 . . . . . 6  |-  ( x  =  Z  ->  ( A. y  e.  { Z }  ( x G y )  e.  { Z }  <->  A. y  e.  { Z }  ( Z G y )  e. 
{ Z } ) )
37 oveq2 5882 . . . . . . . . 9  |-  ( x  =  Z  ->  (
z ( 2nd `  R
) x )  =  ( z ( 2nd `  R ) Z ) )
3837eleq1d 2362 . . . . . . . 8  |-  ( x  =  Z  ->  (
( z ( 2nd `  R ) x )  e.  { Z }  <->  ( z ( 2nd `  R
) Z )  e. 
{ Z } ) )
39 oveq1 5881 . . . . . . . . 9  |-  ( x  =  Z  ->  (
x ( 2nd `  R
) z )  =  ( Z ( 2nd `  R ) z ) )
4039eleq1d 2362 . . . . . . . 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 2576 . . . . . 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 2638 . 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 26742 . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   {csn 3653   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137  GIdcgi 20870   RingOpscrngo 21058   Idlcidl 26735
This theorem is referenced by:  0rngo  26755  divrngidl  26756  smprngopr  26780  isdmn3  26802
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876  df-ablo 20965  df-rngo 21059  df-idl 26738
  Copyright terms: Public domain W3C validator