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Theorem 0rngo 32324
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 5889 . . . . . . 7  |-  (GId `  G )  e.  _V
31, 2eqeltri 2545 . . . . . 6  |-  Z  e. 
_V
43snid 3988 . . . . 5  |-  Z  e. 
{ Z }
5 eleq1 2537 . . . . 5  |-  ( Z  =  U  ->  ( Z  e.  { Z } 
<->  U  e.  { Z } ) )
64, 5mpbii 216 . . . 4  |-  ( Z  =  U  ->  U  e.  { Z } )
7 0ring.1 . . . . . 6  |-  G  =  ( 1st `  R
)
87, 10idl 32322 . . . . 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 32323 . . . . 5  |-  ( ( R  e.  RingOps  /\  { Z }  e.  ( Idl `  R ) )  ->  ( U  e. 
{ Z }  <->  { Z }  =  X )
)
138, 12mpdan 681 . . . 4  |-  ( R  e.  RingOps  ->  ( U  e. 
{ Z }  <->  { Z }  =  X )
)
146, 13syl5ib 227 . . 3  |-  ( R  e.  RingOps  ->  ( Z  =  U  ->  { Z }  =  X )
)
15 eqcom 2478 . . 3  |-  ( { Z }  =  X  <-> 
X  =  { Z } )
1614, 15syl6ib 234 . 2  |-  ( R  e.  RingOps  ->  ( Z  =  U  ->  X  =  { Z } ) )
177rneqi 5067 . . . . 5  |-  ran  G  =  ran  ( 1st `  R
)
1810, 17eqtri 2493 . . . 4  |-  X  =  ran  ( 1st `  R
)
1918, 9, 11rngo1cl 26238 . . 3  |-  ( R  e.  RingOps  ->  U  e.  X
)
20 eleq2 2538 . . . 4  |-  ( X  =  { Z }  ->  ( U  e.  X  <->  U  e.  { Z }
) )
21 elsni 3985 . . . . 5  |-  ( U  e.  { Z }  ->  U  =  Z )
2221eqcomd 2477 . . . 4  |-  ( U  e.  { Z }  ->  Z  =  U )
2320, 22syl6bi 236 . . 3  |-  ( X  =  { Z }  ->  ( U  e.  X  ->  Z  =  U ) )
2419, 23syl5com 30 . 2  |-  ( R  e.  RingOps  ->  ( X  =  { Z }  ->  Z  =  U ) )
2516, 24impbid 195 1  |-  ( R  e.  RingOps  ->  ( Z  =  U  <->  X  =  { Z } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1452    e. wcel 1904   _Vcvv 3031   {csn 3959   ran crn 4840   ` cfv 5589   1stc1st 6810   2ndc2nd 6811  GIdcgi 25996   RingOpscrngo 26184   Idlcidl 32304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-1st 6812  df-2nd 6813  df-grpo 26000  df-gid 26001  df-ginv 26002  df-ablo 26091  df-ass 26122  df-exid 26124  df-mgmOLD 26128  df-sgrOLD 26140  df-mndo 26147  df-rngo 26185  df-idl 32307
This theorem is referenced by:  smprngopr  32349  isfldidl2  32366
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