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Theorem rngosn 21087
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 21030 . . . 4  |-  { <. <. A ,  A >. ,  A >. }  e.  AbelOp
32a1i 10 . . 3  |-  (  T. 
->  { <. <. A ,  A >. ,  A >. }  e.  AbelOp )
4 opex 4253 . . . . . 6  |-  <. A ,  A >.  e.  _V
54rnsnop 5169 . . . . 5  |-  ran  { <. <. A ,  A >. ,  A >. }  =  { A }
65eqcomi 2300 . . . 4  |-  { A }  =  ran  { <. <. A ,  A >. ,  A >. }
76a1i 10 . . 3  |-  (  T. 
->  { A }  =  ran  { <. <. A ,  A >. ,  A >. } )
8 ablogrpo 20967 . . . . 5  |-  ( {
<. <. A ,  A >. ,  A >. }  e.  AbelOp  ->  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
)
96grpofo 20882 . . . . 5  |-  ( {
<. <. A ,  A >. ,  A >. }  e.  GrpOp  ->  { <. <. A ,  A >. ,  A >. } :
( { A }  X.  { A } )
-onto-> { A } )
10 fof 5467 . . . . 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 3677 . . . . . 6  |-  ( x  e.  { A }  ->  x  =  A )
14 elsni 3677 . . . . . 6  |-  ( y  e.  { A }  ->  y  =  A )
15 elsni 3677 . . . . . 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 5892 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
21 df-ov 5877 . . . . . . . 8  |-  ( A { <. <. A ,  A >. ,  A >. } A
)  =  ( {
<. <. A ,  A >. ,  A >. } `  <. A ,  A >. )
224, 1fvsn 5729 . . . . . . . 8  |-  ( {
<. <. A ,  A >. ,  A >. } `  <. A ,  A >. )  =  A
2321, 22eqtri 2316 . . . . . . 7  |-  ( A { <. <. A ,  A >. ,  A >. } A
)  =  A
2420, 23syl6eq 2344 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  A )
2524, 18eqtr4d 2331 . . . . 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 5892 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } z )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
2827, 23syl6eq 2344 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } z )  =  A )
2926, 28eqtr4d 2331 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  z  =  ( y { <. <. A ,  A >. ,  A >. } z ) )
3025, 29oveq12d 5892 . . . 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 2331 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  x  =  ( x { <. <. A ,  A >. ,  A >. } y ) )
3319, 18eqtr4d 2331 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  y  =  x )
3433oveq1d 5889 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } z )  =  ( x { <. <. A ,  A >. ,  A >. } z ) )
3532, 34oveq12d 5892 . . . 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 2331 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  y  =  z )
3837oveq2d 5890 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  ( x { <. <. A ,  A >. ,  A >. } z ) )
3938, 29oveq12d 5892 . . . 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 3680 . . . 4  |-  A  e. 
{ A }
4241a1i 10 . . 3  |-  (  T. 
->  A  e.  { A } )
4314oveq2d 5890 . . . . . 6  |-  ( y  e.  { A }  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
4443, 23syl6eq 2344 . . . . 5  |-  ( y  e.  { A }  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  A )
4544, 14eqtr4d 2331 . . . 4  |-  ( y  e.  { A }  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  y )
4645adantl 452 . . 3  |-  ( (  T.  /\  y  e. 
{ A } )  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  y )
4714oveq1d 5889 . . . . . 6  |-  ( y  e.  { A }  ->  ( y { <. <. A ,  A >. ,  A >. } A )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
4847, 23syl6eq 2344 . . . . 5  |-  ( y  e.  { A }  ->  ( y { <. <. A ,  A >. ,  A >. } A )  =  A )
4948, 14eqtr4d 2331 . . . 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 21062 . 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 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   <.cop 3656    X. cxp 4703   ran crn 4706   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869   AbelOpcablo 20964   RingOpscrngo 21058
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-rep 4147  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-pw 3640  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-res 4717  df-ima 4718  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-grpo 20874  df-ablo 20965  df-rngo 21059
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