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Theorem rngosn3 21109
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 21073 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 on1el3.2 . . . . . . . . . 10  |-  X  =  ran  G
43grpofo 20882 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )
5 fof 5467 . . . . . . . . 9  |-  ( G : ( X  X.  X ) -onto-> X  ->  G : ( X  X.  X ) --> X )
62, 4, 53syl 18 . . . . . . . 8  |-  ( R  e.  RingOps  ->  G : ( X  X.  X ) --> X )
76adantr 451 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  G : ( X  X.  X ) --> X )
8 id 19 . . . . . . . . 9  |-  ( X  =  { A }  ->  X  =  { A } )
98, 8xpeq12d 4730 . . . . . . . 8  |-  ( X  =  { A }  ->  ( X  X.  X
)  =  ( { A }  X.  { A } ) )
109, 8feq23d 5402 . . . . . . 7  |-  ( X  =  { A }  ->  ( G : ( X  X.  X ) --> X  <->  G : ( { A }  X.  { A } ) --> { A } ) )
117, 10syl5ibcom 211 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A }  ->  G : ( { A }  X.  { A } ) --> { A } ) )
12 fdm 5409 . . . . . . . . . 10  |-  ( G : ( X  X.  X ) --> X  ->  dom  G  =  ( X  X.  X ) )
137, 12syl 15 . . . . . . . . 9  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  dom  G  =  ( X  X.  X ) )
1413eqcomd 2301 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  X.  X )  =  dom  G )
15 fdm 5409 . . . . . . . . 9  |-  ( G : ( { A }  X.  { A }
) --> { A }  ->  dom  G  =  ( { A }  X.  { A } ) )
1615eqeq2d 2307 . . . . . . . 8  |-  ( G : ( { A }  X.  { A }
) --> { A }  ->  ( ( X  X.  X )  =  dom  G  <-> 
( X  X.  X
)  =  ( { A }  X.  { A } ) ) )
1714, 16syl5ibcom 211 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( G : ( { A }  X.  { A }
) --> { A }  ->  ( X  X.  X
)  =  ( { A }  X.  { A } ) ) )
18 xpid11 4916 . . . . . . 7  |-  ( ( X  X.  X )  =  ( { A }  X.  { A }
)  <->  X  =  { A } )
1917, 18syl6ib 217 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( G : ( { A }  X.  { A }
) --> { A }  ->  X  =  { A } ) )
2011, 19impbid 183 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A } 
<->  G : ( { A }  X.  { A } ) --> { A } ) )
21 simpr 447 . . . . . . 7  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  A  e.  B )
22 xpsng 5715 . . . . . . 7  |-  ( ( A  e.  B  /\  A  e.  B )  ->  ( { A }  X.  { A } )  =  { <. A ,  A >. } )
2321, 22sylancom 648 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( { A }  X.  { A } )  =  { <. A ,  A >. } )
2423feq2d 5396 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( G : ( { A }  X.  { A }
) --> { A }  <->  G : { <. A ,  A >. } --> { A } ) )
25 opex 4253 . . . . . 6  |-  <. A ,  A >.  e.  _V
26 fsng 5713 . . . . . 6  |-  ( (
<. A ,  A >.  e. 
_V  /\  A  e.  B )  ->  ( G : { <. A ,  A >. } --> { A } 
<->  G  =  { <. <. A ,  A >. ,  A >. } ) )
2725, 21, 26sylancr 644 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( G : { <. A ,  A >. } --> { A } 
<->  G  =  { <. <. A ,  A >. ,  A >. } ) )
2820, 24, 273bitrd 270 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A } 
<->  G  =  { <. <. A ,  A >. ,  A >. } ) )
291eqeq1i 2303 . . . 4  |-  ( G  =  { <. <. A ,  A >. ,  A >. }  <-> 
( 1st `  R
)  =  { <. <. A ,  A >. ,  A >. } )
3028, 29syl6bb 252 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A } 
<->  ( 1st `  R
)  =  { <. <. A ,  A >. ,  A >. } ) )
3130anbi1d 685 . 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 2296 . . . . . . 7  |-  ( 2nd `  R )  =  ( 2nd `  R )
331, 32, 3rngosm 21064 . . . . . 6  |-  ( R  e.  RingOps  ->  ( 2nd `  R
) : ( X  X.  X ) --> X )
3433adantr 451 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( 2nd `  R ) : ( X  X.  X
) --> X )
359, 8feq23d 5402 . . . . 5  |-  ( X  =  { A }  ->  ( ( 2nd `  R
) : ( X  X.  X ) --> X  <-> 
( 2nd `  R
) : ( { A }  X.  { A } ) --> { A } ) )
3634, 35syl5ibcom 211 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A }  ->  ( 2nd `  R
) : ( { A }  X.  { A } ) --> { A } ) )
3723feq2d 5396 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  (
( 2nd `  R
) : ( { A }  X.  { A } ) --> { A } 
<->  ( 2nd `  R
) : { <. A ,  A >. } --> { A } ) )
38 fsng 5713 . . . . . 6  |-  ( (
<. A ,  A >.  e. 
_V  /\  A  e.  B )  ->  (
( 2nd `  R
) : { <. A ,  A >. } --> { A } 
<->  ( 2nd `  R
)  =  { <. <. A ,  A >. ,  A >. } ) )
3925, 21, 38sylancr 644 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  (
( 2nd `  R
) : { <. A ,  A >. } --> { A } 
<->  ( 2nd `  R
)  =  { <. <. A ,  A >. ,  A >. } ) )
4037, 39bitrd 244 . . . 4  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  (
( 2nd `  R
) : ( { A }  X.  { A } ) --> { A } 
<->  ( 2nd `  R
)  =  { <. <. A ,  A >. ,  A >. } ) )
4136, 40sylibd 205 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A }  ->  ( 2nd `  R
)  =  { <. <. A ,  A >. ,  A >. } ) )
4241pm4.71d 615 . 2  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  =  { A } 
<->  ( X  =  { A }  /\  ( 2nd `  R )  =  { <. <. A ,  A >. ,  A >. } ) ) )
43 relrngo 21060 . . . . . 6  |-  Rel  RingOps
44 df-rel 4712 . . . . . 6  |-  ( Rel  RingOps  <->  RingOps  C_  ( _V  X.  _V ) )
4543, 44mpbi 199 . . . . 5  |-  RingOps  C_  ( _V  X.  _V )
4645sseli 3189 . . . 4  |-  ( R  e.  RingOps  ->  R  e.  ( _V  X.  _V )
)
4746adantr 451 . . 3  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  R  e.  ( _V  X.  _V ) )
48 eqop 6178 . . 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 15 . 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 276 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   {csn 3653   <.cop 3656    X. cxp 4703   dom cdm 4705   ran crn 4706   Rel wrel 4710   -->wf 5267   -onto->wfo 5269   ` cfv 5271   1stc1st 6136   2ndc2nd 6137   GrpOpcgr 20869   RingOpscrngo 21058
This theorem is referenced by:  rngosn4  21110
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-1st 6138  df-2nd 6139  df-grpo 20874  df-ablo 20965  df-rngo 21059
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