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Theorem 0rngo 26755
Description: In a ring,  0  =  1 iff the ring contains only 
0. (Contributed by Jeff Madsen, 6-Jan-2011.)
Hypotheses
Ref Expression
0ring.1  |-  G  =  ( 1st `  R
)
0ring.2  |-  H  =  ( 2nd `  R
)
0ring.3  |-  X  =  ran  G
0ring.4  |-  Z  =  (GId `  G )
0ring.5  |-  U  =  (GId `  H )
Assertion
Ref Expression
0rngo  |-  ( R  e.  RingOps  ->  ( Z  =  U  <->  X  =  { Z } ) )

Proof of Theorem 0rngo
StepHypRef Expression
1 0ring.4 . . . . . . 7  |-  Z  =  (GId `  G )
2 fvex 5555 . . . . . . 7  |-  (GId `  G )  e.  _V
31, 2eqeltri 2366 . . . . . 6  |-  Z  e. 
_V
43snid 3680 . . . . 5  |-  Z  e. 
{ Z }
5 eleq1 2356 . . . . 5  |-  ( Z  =  U  ->  ( Z  e.  { Z } 
<->  U  e.  { Z } ) )
64, 5mpbii 202 . . . 4  |-  ( Z  =  U  ->  U  e.  { Z } )
7 0ring.1 . . . . . 6  |-  G  =  ( 1st `  R
)
87, 10idl 26753 . . . . 5  |-  ( R  e.  RingOps  ->  { Z }  e.  ( Idl `  R
) )
9 0ring.2 . . . . . 6  |-  H  =  ( 2nd `  R
)
10 0ring.3 . . . . . 6  |-  X  =  ran  G
11 0ring.5 . . . . . 6  |-  U  =  (GId `  H )
127, 9, 10, 111idl 26754 . . . . 5  |-  ( ( R  e.  RingOps  /\  { Z }  e.  ( Idl `  R ) )  ->  ( U  e. 
{ Z }  <->  { Z }  =  X )
)
138, 12mpdan 649 . . . 4  |-  ( R  e.  RingOps  ->  ( U  e. 
{ Z }  <->  { Z }  =  X )
)
146, 13syl5ib 210 . . 3  |-  ( R  e.  RingOps  ->  ( Z  =  U  ->  { Z }  =  X )
)
15 eqcom 2298 . . 3  |-  ( { Z }  =  X  <-> 
X  =  { Z } )
1614, 15syl6ib 217 . 2  |-  ( R  e.  RingOps  ->  ( Z  =  U  ->  X  =  { Z } ) )
177rneqi 4921 . . . . 5  |-  ran  G  =  ran  ( 1st `  R
)
1810, 17eqtri 2316 . . . 4  |-  X  =  ran  ( 1st `  R
)
1918, 9, 11rngo1cl 21112 . . 3  |-  ( R  e.  RingOps  ->  U  e.  X
)
20 eleq2 2357 . . . 4  |-  ( X  =  { Z }  ->  ( U  e.  X  <->  U  e.  { Z }
) )
21 elsni 3677 . . . . 5  |-  ( U  e.  { Z }  ->  U  =  Z )
2221eqcomd 2301 . . . 4  |-  ( U  e.  { Z }  ->  Z  =  U )
2320, 22syl6bi 219 . . 3  |-  ( X  =  { Z }  ->  ( U  e.  X  ->  Z  =  U ) )
2419, 23syl5com 26 . 2  |-  ( R  e.  RingOps  ->  ( X  =  { Z }  ->  Z  =  U ) )
2516, 24impbid 183 1  |-  ( R  e.  RingOps  ->  ( Z  =  U  <->  X  =  { Z } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   ran crn 4706   ` cfv 5271   1stc1st 6136   2ndc2nd 6137  GIdcgi 20870   RingOpscrngo 21058   Idlcidl 26735
This theorem is referenced by:  smprngopr  26780  isfldidl2  26797
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-rmo 2564  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-ass 20996  df-exid 20998  df-mgm 21002  df-sgr 21014  df-mndo 21021  df-rngo 21059  df-idl 26738
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