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Theorem rngosn3 25554
Description: Obsolete as of 25-Jan-2020. Use ring1zr 18049 or srg1zr 17306 instead. 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 25518 . . . . . . . . 9  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 on1el3.2 . . . . . . . . . 10  |-  X  =  ran  G
43grpofo 25327 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  G :
( X  X.  X
) -onto-> X )
5 fof 5801 . . . . . . . . 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 } )
98sqxpeqd 5034 . . . . . . . 8  |-  ( X  =  { A }  ->  ( X  X.  X
)  =  ( { A }  X.  { A } ) )
109, 8feq23d 5732 . . . . . . 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 5741 . . . . . . . . . 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 2465 . . . . . . . 8  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( X  X.  X )  =  dom  G )
15 fdm 5741 . . . . . . . . 9  |-  ( G : ( { A }  X.  { A }
) --> { A }  ->  dom  G  =  ( { A }  X.  { A } ) )
1615eqeq2d 2471 . . . . . . . 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 5234 . . . . . . 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 6073 . . . . . . 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 5724 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  ( G : ( { A }  X.  { A }
) --> { A }  <->  G : { <. A ,  A >. } --> { A } ) )
25 opex 4720 . . . . . 6  |-  <. A ,  A >.  e.  _V
26 fsng 6071 . . . . . 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 2464 . . . 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 2457 . . . . . . 7  |-  ( 2nd `  R )  =  ( 2nd `  R )
331, 32, 3rngosm 25509 . . . . . 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 5732 . . . . 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 5724 . . . . 5  |-  ( ( R  e.  RingOps  /\  A  e.  B )  ->  (
( 2nd `  R
) : ( { A }  X.  { A } ) --> { A } 
<->  ( 2nd `  R
) : { <. A ,  A >. } --> { A } ) )
38 fsng 6071 . . . . . 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 25505 . . . . . 6  |-  Rel  RingOps
44 df-rel 5015 . . . . . 6  |-  ( Rel  RingOps  <->  RingOps  C_  ( _V  X.  _V ) )
4543, 44mpbi 208 . . . . 5  |-  RingOps  C_  ( _V  X.  _V )
4645sseli 3495 . . . 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 6839 . . 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 1395    e. wcel 1819   _Vcvv 3109    C_ wss 3471   {csn 4032   <.cop 4038    X. cxp 5006   dom cdm 5008   ran crn 5009   Rel wrel 5013   -->wf 5590   -onto->wfo 5592   ` cfv 5594   1stc1st 6797   2ndc2nd 6798   GrpOpcgr 25314   RingOpscrngo 25503
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-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-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-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-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-1st 6799  df-2nd 6800  df-grpo 25319  df-ablo 25410  df-rngo 25504
This theorem is referenced by:  rngosn4  25555
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