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Theorem 0rngo 32225
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 5892 . . . . . . 7  |-  (GId `  G )  e.  _V
31, 2eqeltri 2503 . . . . . 6  |-  Z  e. 
_V
43snid 4026 . . . . 5  |-  Z  e. 
{ Z }
5 eleq1 2495 . . . . 5  |-  ( Z  =  U  ->  ( Z  e.  { Z } 
<->  U  e.  { Z } ) )
64, 5mpbii 214 . . . 4  |-  ( Z  =  U  ->  U  e.  { Z } )
7 0ring.1 . . . . . 6  |-  G  =  ( 1st `  R
)
87, 10idl 32223 . . . . 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 32224 . . . . 5  |-  ( ( R  e.  RingOps  /\  { Z }  e.  ( Idl `  R ) )  ->  ( U  e. 
{ Z }  <->  { Z }  =  X )
)
138, 12mpdan 672 . . . 4  |-  ( R  e.  RingOps  ->  ( U  e. 
{ Z }  <->  { Z }  =  X )
)
146, 13syl5ib 222 . . 3  |-  ( R  e.  RingOps  ->  ( Z  =  U  ->  { Z }  =  X )
)
15 eqcom 2431 . . 3  |-  ( { Z }  =  X  <-> 
X  =  { Z } )
1614, 15syl6ib 229 . 2  |-  ( R  e.  RingOps  ->  ( Z  =  U  ->  X  =  { Z } ) )
177rneqi 5080 . . . . 5  |-  ran  G  =  ran  ( 1st `  R
)
1810, 17eqtri 2451 . . . 4  |-  X  =  ran  ( 1st `  R
)
1918, 9, 11rngo1cl 26156 . . 3  |-  ( R  e.  RingOps  ->  U  e.  X
)
20 eleq2 2496 . . . 4  |-  ( X  =  { Z }  ->  ( U  e.  X  <->  U  e.  { Z }
) )
21 elsni 4023 . . . . 5  |-  ( U  e.  { Z }  ->  U  =  Z )
2221eqcomd 2430 . . . 4  |-  ( U  e.  { Z }  ->  Z  =  U )
2320, 22syl6bi 231 . . 3  |-  ( X  =  { Z }  ->  ( U  e.  X  ->  Z  =  U ) )
2419, 23syl5com 31 . 2  |-  ( R  e.  RingOps  ->  ( X  =  { Z }  ->  Z  =  U ) )
2516, 24impbid 193 1  |-  ( R  e.  RingOps  ->  ( Z  =  U  <->  X  =  { Z } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1872   _Vcvv 3080   {csn 3998   ran crn 4854   ` cfv 5601   1stc1st 6806   2ndc2nd 6807  GIdcgi 25914   RingOpscrngo 26102   Idlcidl 32205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-1st 6808  df-2nd 6809  df-grpo 25918  df-gid 25919  df-ginv 25920  df-ablo 26009  df-ass 26040  df-exid 26042  df-mgmOLD 26046  df-sgrOLD 26058  df-mndo 26065  df-rngo 26103  df-idl 32208
This theorem is referenced by:  smprngopr  32250  isfldidl2  32267
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