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Theorem rngosn 21071
Description: The trivial or zero ring defined on a singleton set  { A } (see http://en.wikipedia.org/wiki/Trivial_ring). (Contributed by Steve Rodriguez, 10-Feb-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
ringsn.1  |-  A  e. 
_V
Assertion
Ref Expression
rngosn  |-  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. } >.  e.  RingOps

Proof of Theorem rngosn
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringsn.1 . . . . 5  |-  A  e. 
_V
21ablosn 21014 . . . 4  |-  { <. <. A ,  A >. ,  A >. }  e.  AbelOp
32a1i 10 . . 3  |-  (  T. 
->  { <. <. A ,  A >. ,  A >. }  e.  AbelOp )
4 opex 4237 . . . . . 6  |-  <. A ,  A >.  e.  _V
54rnsnop 5153 . . . . 5  |-  ran  { <. <. A ,  A >. ,  A >. }  =  { A }
65eqcomi 2287 . . . 4  |-  { A }  =  ran  { <. <. A ,  A >. ,  A >. }
76a1i 10 . . 3  |-  (  T. 
->  { A }  =  ran  { <. <. A ,  A >. ,  A >. } )
8 ablogrpo 20951 . . . . 5  |-  ( {
<. <. A ,  A >. ,  A >. }  e.  AbelOp  ->  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
)
96grpofo 20866 . . . . 5  |-  ( {
<. <. A ,  A >. ,  A >. }  e.  GrpOp  ->  { <. <. A ,  A >. ,  A >. } :
( { A }  X.  { A } )
-onto-> { A } )
10 fof 5451 . . . . 5  |-  ( {
<. <. A ,  A >. ,  A >. } :
( { A }  X.  { A } )
-onto-> { A }  ->  {
<. <. A ,  A >. ,  A >. } :
( { A }  X.  { A } ) --> { A } )
118, 9, 103syl 18 . . . 4  |-  ( {
<. <. A ,  A >. ,  A >. }  e.  AbelOp  ->  { <. <. A ,  A >. ,  A >. } :
( { A }  X.  { A } ) --> { A } )
123, 11syl 15 . . 3  |-  (  T. 
->  { <. <. A ,  A >. ,  A >. } :
( { A }  X.  { A } ) --> { A } )
13 elsni 3664 . . . . . 6  |-  ( x  e.  { A }  ->  x  =  A )
14 elsni 3664 . . . . . 6  |-  ( y  e.  { A }  ->  y  =  A )
15 elsni 3664 . . . . . 6  |-  ( z  e.  { A }  ->  z  =  A )
1613, 14, 153anim123i 1137 . . . . 5  |-  ( ( x  e.  { A }  /\  y  e.  { A }  /\  z  e.  { A } )  ->  ( x  =  A  /\  y  =  A  /\  z  =  A ) )
1716adantl 452 . . . 4  |-  ( (  T.  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  z  e.  { A } ) )  -> 
( x  =  A  /\  y  =  A  /\  z  =  A ) )
18 simp1 955 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  x  =  A )
19 simp2 956 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  y  =  A )
2018, 19oveq12d 5876 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
21 df-ov 5861 . . . . . . . 8  |-  ( A { <. <. A ,  A >. ,  A >. } A
)  =  ( {
<. <. A ,  A >. ,  A >. } `  <. A ,  A >. )
224, 1fvsn 5713 . . . . . . . 8  |-  ( {
<. <. A ,  A >. ,  A >. } `  <. A ,  A >. )  =  A
2321, 22eqtri 2303 . . . . . . 7  |-  ( A { <. <. A ,  A >. ,  A >. } A
)  =  A
2420, 23syl6eq 2331 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  A )
2524, 18eqtr4d 2318 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  x )
26 simp3 957 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  z  =  A )
2719, 26oveq12d 5876 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } z )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
2827, 23syl6eq 2331 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } z )  =  A )
2926, 28eqtr4d 2318 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  z  =  ( y { <. <. A ,  A >. ,  A >. } z ) )
3025, 29oveq12d 5876 . . . 4  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( ( x { <. <. A ,  A >. ,  A >. } y ) { <. <. A ,  A >. ,  A >. } z )  =  ( x { <. <. A ,  A >. ,  A >. }  ( y { <. <. A ,  A >. ,  A >. } z ) ) )
3117, 30syl 15 . . 3  |-  ( (  T.  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  z  e.  { A } ) )  -> 
( ( x { <. <. A ,  A >. ,  A >. } y ) { <. <. A ,  A >. ,  A >. } z )  =  ( x { <. <. A ,  A >. ,  A >. }  ( y { <. <. A ,  A >. ,  A >. } z ) ) )
3218, 24eqtr4d 2318 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  x  =  ( x { <. <. A ,  A >. ,  A >. } y ) )
3319, 18eqtr4d 2318 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  y  =  x )
3433oveq1d 5873 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } z )  =  ( x { <. <. A ,  A >. ,  A >. } z ) )
3532, 34oveq12d 5876 . . . 4  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. }  ( y { <. <. A ,  A >. ,  A >. } z ) )  =  ( ( x { <. <. A ,  A >. ,  A >. } y ) { <. <. A ,  A >. ,  A >. }  (
x { <. <. A ,  A >. ,  A >. } z ) ) )
3617, 35syl 15 . . 3  |-  ( (  T.  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  z  e.  { A } ) )  -> 
( x { <. <. A ,  A >. ,  A >. }  ( y { <. <. A ,  A >. ,  A >. } z ) )  =  ( ( x { <. <. A ,  A >. ,  A >. } y ) { <. <. A ,  A >. ,  A >. }  (
x { <. <. A ,  A >. ,  A >. } z ) ) )
3719, 26eqtr4d 2318 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  y  =  z )
3837oveq2d 5874 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  ( x { <. <. A ,  A >. ,  A >. } z ) )
3938, 29oveq12d 5876 . . . 4  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( ( x { <. <. A ,  A >. ,  A >. } y ) { <. <. A ,  A >. ,  A >. } z )  =  ( ( x { <. <. A ,  A >. ,  A >. } z ) { <. <. A ,  A >. ,  A >. }  (
y { <. <. A ,  A >. ,  A >. } z ) ) )
4017, 39syl 15 . . 3  |-  ( (  T.  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  z  e.  { A } ) )  -> 
( ( x { <. <. A ,  A >. ,  A >. } y ) { <. <. A ,  A >. ,  A >. } z )  =  ( ( x { <. <. A ,  A >. ,  A >. } z ) { <. <. A ,  A >. ,  A >. }  (
y { <. <. A ,  A >. ,  A >. } z ) ) )
411snid 3667 . . . 4  |-  A  e. 
{ A }
4241a1i 10 . . 3  |-  (  T. 
->  A  e.  { A } )
4314oveq2d 5874 . . . . . 6  |-  ( y  e.  { A }  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
4443, 23syl6eq 2331 . . . . 5  |-  ( y  e.  { A }  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  A )
4544, 14eqtr4d 2318 . . . 4  |-  ( y  e.  { A }  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  y )
4645adantl 452 . . 3  |-  ( (  T.  /\  y  e. 
{ A } )  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  y )
4714oveq1d 5873 . . . . . 6  |-  ( y  e.  { A }  ->  ( y { <. <. A ,  A >. ,  A >. } A )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
4847, 23syl6eq 2331 . . . . 5  |-  ( y  e.  { A }  ->  ( y { <. <. A ,  A >. ,  A >. } A )  =  A )
4948, 14eqtr4d 2318 . . . 4  |-  ( y  e.  { A }  ->  ( y { <. <. A ,  A >. ,  A >. } A )  =  y )
5049adantl 452 . . 3  |-  ( (  T.  /\  y  e. 
{ A } )  ->  ( y {
<. <. A ,  A >. ,  A >. } A
)  =  y )
513, 7, 12, 31, 36, 40, 42, 46, 50isrngod 21046 . 2  |-  (  T. 
->  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. }
>.  e.  RingOps )
5251trud 1314 1  |-  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. } >.  e.  RingOps
Colors of variables: wff set class
Syntax hints:    /\ wa 358    /\ w3a 934    T. wtru 1307    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   <.cop 3643    X. cxp 4687   ran crn 4690   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853   AbelOpcablo 20948   RingOpscrngo 21042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-grpo 20858  df-ablo 20949  df-rngo 21043
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