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Theorem rngosn3 21862
Description: The only unital ring with a base set consisting in one element is the zero ring. (Contributed by FL, 13-Feb-2010.) (Proof shortened by Mario Carneiro, 30-Apr-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
on1el3.1  |-  G  =  ( 1st `  R
)
on1el3.2  |-  X  =  ran  G
Assertion
Ref Expression
rngosn3  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A } 
<->  R  =  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. } >. ) )

Proof of Theorem rngosn3
StepHypRef Expression
1 on1el3.1 . . . . . . . . . 10  |-  G  =  ( 1st `  R
)
21rngogrpo 21826 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 on1el3.2 . . . . . . . . . 10  |-  X  =  ran  G
43grpofo 21635 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )
5 fof 5593 . . . . . . . . 9  |-  ( G : ( X  X.  X ) -onto-> X  ->  G : ( X  X.  X ) --> X )
62, 4, 53syl 19 . . . . . . . 8  |-  ( R  e.  RingOps  ->  G : ( X  X.  X ) --> X )
76adantr 452 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  G : ( X  X.  X ) --> X )
8 id 20 . . . . . . . . 9  |-  ( X  =  { A }  ->  X  =  { A } )
98, 8xpeq12d 4843 . . . . . . . 8  |-  ( X  =  { A }  ->  ( X  X.  X
)  =  ( { A }  X.  { A } ) )
109, 8feq23d 5528 . . . . . . 7  |-  ( X  =  { A }  ->  ( G : ( X  X.  X ) --> X  <->  G : ( { A }  X.  { A } ) --> { A } ) )
117, 10syl5ibcom 212 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A }  ->  G : ( { A }  X.  { A } ) --> { A } ) )
12 fdm 5535 . . . . . . . . . 10  |-  ( G : ( X  X.  X ) --> X  ->  dom  G  =  ( X  X.  X ) )
137, 12syl 16 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  dom  G  =  ( X  X.  X ) )
1413eqcomd 2392 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  X.  X )  =  dom  G )
15 fdm 5535 . . . . . . . . 9  |-  ( G : ( { A }  X.  { A }
) --> { A }  ->  dom  G  =  ( { A }  X.  { A } ) )
1615eqeq2d 2398 . . . . . . . 8  |-  ( G : ( { A }  X.  { A }
) --> { A }  ->  ( ( X  X.  X )  =  dom  G  <-> 
( X  X.  X
)  =  ( { A }  X.  { A } ) ) )
1714, 16syl5ibcom 212 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( G : ( { A }  X.  { A }
) --> { A }  ->  ( X  X.  X
)  =  ( { A }  X.  { A } ) ) )
18 xpid11 5031 . . . . . . 7  |-  ( ( X  X.  X )  =  ( { A }  X.  { A }
)  <->  X  =  { A } )
1917, 18syl6ib 218 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( G : ( { A }  X.  { A }
) --> { A }  ->  X  =  { A } ) )
2011, 19impbid 184 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A } 
<->  G : ( { A }  X.  { A } ) --> { A } ) )
21 simpr 448 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  A  e.  B )
22 xpsng 5848 . . . . . . 7  |-  ( ( A  e.  B  /\  A  e.  B )  ->  ( { A }  X.  { A } )  =  { <. A ,  A >. } )
2321, 22sylancom 649 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( { A }  X.  { A } )  =  { <. A ,  A >. } )
2423feq2d 5521 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( G : ( { A }  X.  { A }
) --> { A }  <->  G : { <. A ,  A >. } --> { A } ) )
25 opex 4368 . . . . . 6  |-  <. A ,  A >.  e.  _V
26 fsng 5846 . . . . . 6  |-  ( (
<. A ,  A >.  e. 
_V  /\  A  e.  B )  ->  ( G : { <. A ,  A >. } --> { A } 
<->  G  =  { <. <. A ,  A >. ,  A >. } ) )
2725, 21, 26sylancr 645 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( G : { <. A ,  A >. } --> { A } 
<->  G  =  { <. <. A ,  A >. ,  A >. } ) )
2820, 24, 273bitrd 271 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A } 
<->  G  =  { <. <. A ,  A >. ,  A >. } ) )
291eqeq1i 2394 . . . 4  |-  ( G  =  { <. <. A ,  A >. ,  A >. }  <-> 
( 1st `  R
)  =  { <. <. A ,  A >. ,  A >. } )
3028, 29syl6bb 253 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A } 
<->  ( 1st `  R
)  =  { <. <. A ,  A >. ,  A >. } ) )
3130anbi1d 686 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  (
( X  =  { A }  /\  ( 2nd `  R )  =  { <. <. A ,  A >. ,  A >. } )  <-> 
( ( 1st `  R
)  =  { <. <. A ,  A >. ,  A >. }  /\  ( 2nd `  R )  =  { <. <. A ,  A >. ,  A >. } ) ) )
32 eqid 2387 . . . . . . 7  |-  ( 2nd `  R )  =  ( 2nd `  R )
331, 32, 3rngosm 21817 . . . . . 6  |-  ( R  e.  RingOps  ->  ( 2nd `  R
) : ( X  X.  X ) --> X )
3433adantr 452 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( 2nd `  R ) : ( X  X.  X
) --> X )
359, 8feq23d 5528 . . . . 5  |-  ( X  =  { A }  ->  ( ( 2nd `  R
) : ( X  X.  X ) --> X  <-> 
( 2nd `  R
) : ( { A }  X.  { A } ) --> { A } ) )
3634, 35syl5ibcom 212 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A }  ->  ( 2nd `  R
) : ( { A }  X.  { A } ) --> { A } ) )
3723feq2d 5521 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  (
( 2nd `  R
) : ( { A }  X.  { A } ) --> { A } 
<->  ( 2nd `  R
) : { <. A ,  A >. } --> { A } ) )
38 fsng 5846 . . . . . 6  |-  ( (
<. A ,  A >.  e. 
_V  /\  A  e.  B )  ->  (
( 2nd `  R
) : { <. A ,  A >. } --> { A } 
<->  ( 2nd `  R
)  =  { <. <. A ,  A >. ,  A >. } ) )
3925, 21, 38sylancr 645 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  (
( 2nd `  R
) : { <. A ,  A >. } --> { A } 
<->  ( 2nd `  R
)  =  { <. <. A ,  A >. ,  A >. } ) )
4037, 39bitrd 245 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  (
( 2nd `  R
) : ( { A }  X.  { A } ) --> { A } 
<->  ( 2nd `  R
)  =  { <. <. A ,  A >. ,  A >. } ) )
4136, 40sylibd 206 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A }  ->  ( 2nd `  R
)  =  { <. <. A ,  A >. ,  A >. } ) )
4241pm4.71d 616 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A } 
<->  ( X  =  { A }  /\  ( 2nd `  R )  =  { <. <. A ,  A >. ,  A >. } ) ) )
43 relrngo 21813 . . . . . 6  |-  Rel  RingOps
44 df-rel 4825 . . . . . 6  |-  ( Rel  RingOps  <->  RingOps  C_  ( _V  X.  _V ) )
4543, 44mpbi 200 . . . . 5  |-  RingOps  C_  ( _V  X.  _V )
4645sseli 3287 . . . 4  |-  ( R  e.  RingOps  ->  R  e.  ( _V  X.  _V )
)
4746adantr 452 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  R  e.  ( _V  X.  _V ) )
48 eqop 6328 . . 3  |-  ( R  e.  ( _V  X.  _V )  ->  ( R  =  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. }
>. 
<->  ( ( 1st `  R
)  =  { <. <. A ,  A >. ,  A >. }  /\  ( 2nd `  R )  =  { <. <. A ,  A >. ,  A >. } ) ) )
4947, 48syl 16 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( R  =  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. } >.  <->  (
( 1st `  R
)  =  { <. <. A ,  A >. ,  A >. }  /\  ( 2nd `  R )  =  { <. <. A ,  A >. ,  A >. } ) ) )
5031, 42, 493bitr4d 277 1  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A } 
<->  R  =  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. } >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899    C_ wss 3263   {csn 3757   <.cop 3760    X. cxp 4816   dom cdm 4818   ran crn 4819   Rel wrel 4823   -->wf 5390   -onto->wfo 5392   ` cfv 5394   1stc1st 6286   2ndc2nd 6287   GrpOpcgr 21622   RingOpscrngo 21811
This theorem is referenced by:  rngosn4  21863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-1st 6288  df-2nd 6289  df-grpo 21627  df-ablo 21718  df-rngo 21812
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