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Theorem 0rngo 26637
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 5742 . . . . . . 7  |-  (GId `  G )  e.  _V
31, 2eqeltri 2506 . . . . . 6  |-  Z  e. 
_V
43snid 3841 . . . . 5  |-  Z  e. 
{ Z }
5 eleq1 2496 . . . . 5  |-  ( Z  =  U  ->  ( Z  e.  { Z } 
<->  U  e.  { Z } ) )
64, 5mpbii 203 . . . 4  |-  ( Z  =  U  ->  U  e.  { Z } )
7 0ring.1 . . . . . 6  |-  G  =  ( 1st `  R
)
87, 10idl 26635 . . . . 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 26636 . . . . 5  |-  ( ( R  e.  RingOps  /\  { Z }  e.  ( Idl `  R ) )  ->  ( U  e. 
{ Z }  <->  { Z }  =  X )
)
138, 12mpdan 650 . . . 4  |-  ( R  e.  RingOps  ->  ( U  e. 
{ Z }  <->  { Z }  =  X )
)
146, 13syl5ib 211 . . 3  |-  ( R  e.  RingOps  ->  ( Z  =  U  ->  { Z }  =  X )
)
15 eqcom 2438 . . 3  |-  ( { Z }  =  X  <-> 
X  =  { Z } )
1614, 15syl6ib 218 . 2  |-  ( R  e.  RingOps  ->  ( Z  =  U  ->  X  =  { Z } ) )
177rneqi 5096 . . . . 5  |-  ran  G  =  ran  ( 1st `  R
)
1810, 17eqtri 2456 . . . 4  |-  X  =  ran  ( 1st `  R
)
1918, 9, 11rngo1cl 22017 . . 3  |-  ( R  e.  RingOps  ->  U  e.  X
)
20 eleq2 2497 . . . 4  |-  ( X  =  { Z }  ->  ( U  e.  X  <->  U  e.  { Z }
) )
21 elsni 3838 . . . . 5  |-  ( U  e.  { Z }  ->  U  =  Z )
2221eqcomd 2441 . . . 4  |-  ( U  e.  { Z }  ->  Z  =  U )
2320, 22syl6bi 220 . . 3  |-  ( X  =  { Z }  ->  ( U  e.  X  ->  Z  =  U ) )
2419, 23syl5com 28 . 2  |-  ( R  e.  RingOps  ->  ( X  =  { Z }  ->  Z  =  U ) )
2516, 24impbid 184 1  |-  ( R  e.  RingOps  ->  ( Z  =  U  <->  X  =  { Z } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   _Vcvv 2956   {csn 3814   ran crn 4879   ` cfv 5454   1stc1st 6347   2ndc2nd 6348  GIdcgi 21775   RingOpscrngo 21963   Idlcidl 26617
This theorem is referenced by:  smprngopr  26662  isfldidl2  26679
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-1st 6349  df-2nd 6350  df-riota 6549  df-grpo 21779  df-gid 21780  df-ginv 21781  df-ablo 21870  df-ass 21901  df-exid 21903  df-mgm 21907  df-sgr 21919  df-mndo 21926  df-rngo 21964  df-idl 26620
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