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Theorem 0rngo 30586
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 5882 . . . . . . 7  |-  (GId `  G )  e.  _V
31, 2eqeltri 2541 . . . . . 6  |-  Z  e. 
_V
43snid 4060 . . . . 5  |-  Z  e. 
{ Z }
5 eleq1 2529 . . . . 5  |-  ( Z  =  U  ->  ( Z  e.  { Z } 
<->  U  e.  { Z } ) )
64, 5mpbii 211 . . . 4  |-  ( Z  =  U  ->  U  e.  { Z } )
7 0ring.1 . . . . . 6  |-  G  =  ( 1st `  R
)
87, 10idl 30584 . . . . 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 30585 . . . . 5  |-  ( ( R  e.  RingOps  /\  { Z }  e.  ( Idl `  R ) )  ->  ( U  e. 
{ Z }  <->  { Z }  =  X )
)
138, 12mpdan 668 . . . 4  |-  ( R  e.  RingOps  ->  ( U  e. 
{ Z }  <->  { Z }  =  X )
)
146, 13syl5ib 219 . . 3  |-  ( R  e.  RingOps  ->  ( Z  =  U  ->  { Z }  =  X )
)
15 eqcom 2466 . . 3  |-  ( { Z }  =  X  <-> 
X  =  { Z } )
1614, 15syl6ib 226 . 2  |-  ( R  e.  RingOps  ->  ( Z  =  U  ->  X  =  { Z } ) )
177rneqi 5239 . . . . 5  |-  ran  G  =  ran  ( 1st `  R
)
1810, 17eqtri 2486 . . . 4  |-  X  =  ran  ( 1st `  R
)
1918, 9, 11rngo1cl 25557 . . 3  |-  ( R  e.  RingOps  ->  U  e.  X
)
20 eleq2 2530 . . . 4  |-  ( X  =  { Z }  ->  ( U  e.  X  <->  U  e.  { Z }
) )
21 elsni 4057 . . . . 5  |-  ( U  e.  { Z }  ->  U  =  Z )
2221eqcomd 2465 . . . 4  |-  ( U  e.  { Z }  ->  Z  =  U )
2320, 22syl6bi 228 . . 3  |-  ( X  =  { Z }  ->  ( U  e.  X  ->  Z  =  U ) )
2419, 23syl5com 30 . 2  |-  ( R  e.  RingOps  ->  ( X  =  { Z }  ->  Z  =  U ) )
2516, 24impbid 191 1  |-  ( R  e.  RingOps  ->  ( Z  =  U  <->  X  =  { Z } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395    e. wcel 1819   _Vcvv 3109   {csn 4032   ran crn 5009   ` cfv 5594   1stc1st 6797   2ndc2nd 6798  GIdcgi 25315   RingOpscrngo 25503   Idlcidl 30566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-1st 6799  df-2nd 6800  df-grpo 25319  df-gid 25320  df-ginv 25321  df-ablo 25410  df-ass 25441  df-exid 25443  df-mgmOLD 25447  df-sgrOLD 25459  df-mndo 25466  df-rngo 25504  df-idl 30569
This theorem is referenced by:  smprngopr  30611  isfldidl2  30628
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