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Theorem rngosn3 23918
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 23882 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 on1el3.2 . . . . . . . . . 10  |-  X  =  ran  G
43grpofo 23691 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )
5 fof 5625 . . . . . . . . 9  |-  ( G : ( X  X.  X ) -onto-> X  ->  G : ( X  X.  X ) --> X )
62, 4, 53syl 20 . . . . . . . 8  |-  ( R  e.  RingOps  ->  G : ( X  X.  X ) --> X )
76adantr 465 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  G : ( X  X.  X ) --> X )
8 id 22 . . . . . . . . 9  |-  ( X  =  { A }  ->  X  =  { A } )
98, 8xpeq12d 4870 . . . . . . . 8  |-  ( X  =  { A }  ->  ( X  X.  X
)  =  ( { A }  X.  { A } ) )
109, 8feq23d 5559 . . . . . . 7  |-  ( X  =  { A }  ->  ( G : ( X  X.  X ) --> X  <->  G : ( { A }  X.  { A } ) --> { A } ) )
117, 10syl5ibcom 220 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A }  ->  G : ( { A }  X.  { A } ) --> { A } ) )
12 fdm 5568 . . . . . . . . . 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 2448 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  X.  X )  =  dom  G )
15 fdm 5568 . . . . . . . . 9  |-  ( G : ( { A }  X.  { A }
) --> { A }  ->  dom  G  =  ( { A }  X.  { A } ) )
1615eqeq2d 2454 . . . . . . . 8  |-  ( G : ( { A }  X.  { A }
) --> { A }  ->  ( ( X  X.  X )  =  dom  G  <-> 
( X  X.  X
)  =  ( { A }  X.  { A } ) ) )
1714, 16syl5ibcom 220 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( G : ( { A }  X.  { A }
) --> { A }  ->  ( X  X.  X
)  =  ( { A }  X.  { A } ) ) )
18 xpid11 5066 . . . . . . 7  |-  ( ( X  X.  X )  =  ( { A }  X.  { A }
)  <->  X  =  { A } )
1917, 18syl6ib 226 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( G : ( { A }  X.  { A }
) --> { A }  ->  X  =  { A } ) )
2011, 19impbid 191 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A } 
<->  G : ( { A }  X.  { A } ) --> { A } ) )
21 simpr 461 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  A  e.  B )
22 xpsng 5889 . . . . . . 7  |-  ( ( A  e.  B  /\  A  e.  B )  ->  ( { A }  X.  { A } )  =  { <. A ,  A >. } )
2321, 22sylancom 667 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( { A }  X.  { A } )  =  { <. A ,  A >. } )
2423feq2d 5552 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( G : ( { A }  X.  { A }
) --> { A }  <->  G : { <. A ,  A >. } --> { A } ) )
25 opex 4561 . . . . . 6  |-  <. A ,  A >.  e.  _V
26 fsng 5887 . . . . . 6  |-  ( (
<. A ,  A >.  e. 
_V  /\  A  e.  B )  ->  ( G : { <. A ,  A >. } --> { A } 
<->  G  =  { <. <. A ,  A >. ,  A >. } ) )
2725, 21, 26sylancr 663 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( G : { <. A ,  A >. } --> { A } 
<->  G  =  { <. <. A ,  A >. ,  A >. } ) )
2820, 24, 273bitrd 279 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A } 
<->  G  =  { <. <. A ,  A >. ,  A >. } ) )
291eqeq1i 2450 . . . 4  |-  ( G  =  { <. <. A ,  A >. ,  A >. }  <-> 
( 1st `  R
)  =  { <. <. A ,  A >. ,  A >. } )
3028, 29syl6bb 261 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A } 
<->  ( 1st `  R
)  =  { <. <. A ,  A >. ,  A >. } ) )
3130anbi1d 704 . 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 2443 . . . . . . 7  |-  ( 2nd `  R )  =  ( 2nd `  R )
331, 32, 3rngosm 23873 . . . . . 6  |-  ( R  e.  RingOps  ->  ( 2nd `  R
) : ( X  X.  X ) --> X )
3433adantr 465 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( 2nd `  R ) : ( X  X.  X
) --> X )
359, 8feq23d 5559 . . . . 5  |-  ( X  =  { A }  ->  ( ( 2nd `  R
) : ( X  X.  X ) --> X  <-> 
( 2nd `  R
) : ( { A }  X.  { A } ) --> { A } ) )
3634, 35syl5ibcom 220 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A }  ->  ( 2nd `  R
) : ( { A }  X.  { A } ) --> { A } ) )
3723feq2d 5552 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  (
( 2nd `  R
) : ( { A }  X.  { A } ) --> { A } 
<->  ( 2nd `  R
) : { <. A ,  A >. } --> { A } ) )
38 fsng 5887 . . . . . 6  |-  ( (
<. A ,  A >.  e. 
_V  /\  A  e.  B )  ->  (
( 2nd `  R
) : { <. A ,  A >. } --> { A } 
<->  ( 2nd `  R
)  =  { <. <. A ,  A >. ,  A >. } ) )
3925, 21, 38sylancr 663 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  (
( 2nd `  R
) : { <. A ,  A >. } --> { A } 
<->  ( 2nd `  R
)  =  { <. <. A ,  A >. ,  A >. } ) )
4037, 39bitrd 253 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  (
( 2nd `  R
) : ( { A }  X.  { A } ) --> { A } 
<->  ( 2nd `  R
)  =  { <. <. A ,  A >. ,  A >. } ) )
4136, 40sylibd 214 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A }  ->  ( 2nd `  R
)  =  { <. <. A ,  A >. ,  A >. } ) )
4241pm4.71d 634 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A } 
<->  ( X  =  { A }  /\  ( 2nd `  R )  =  { <. <. A ,  A >. ,  A >. } ) ) )
43 relrngo 23869 . . . . . 6  |-  Rel  RingOps
44 df-rel 4852 . . . . . 6  |-  ( Rel  RingOps  <->  RingOps  C_  ( _V  X.  _V ) )
4543, 44mpbi 208 . . . . 5  |-  RingOps  C_  ( _V  X.  _V )
4645sseli 3357 . . . 4  |-  ( R  e.  RingOps  ->  R  e.  ( _V  X.  _V )
)
4746adantr 465 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  R  e.  ( _V  X.  _V ) )
48 eqop 6621 . . 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 285 1  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A } 
<->  R  =  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. } >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2977    C_ wss 3333   {csn 3882   <.cop 3888    X. cxp 4843   dom cdm 4845   ran crn 4846   Rel wrel 4850   -->wf 5419   -onto->wfo 5421   ` cfv 5423   1stc1st 6580   2ndc2nd 6581   GrpOpcgr 23678   RingOpscrngo 23867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-1st 6582  df-2nd 6583  df-grpo 23683  df-ablo 23774  df-rngo 23868
This theorem is referenced by:  rngosn4  23919
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