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Theorem rngosn 23896
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 23839 . . . 4  |-  { <. <. A ,  A >. ,  A >. }  e.  AbelOp
32a1i 11 . . 3  |-  ( T. 
->  { <. <. A ,  A >. ,  A >. }  e.  AbelOp )
4 opex 4561 . . . . . 6  |-  <. A ,  A >.  e.  _V
54rnsnop 5325 . . . . 5  |-  ran  { <. <. A ,  A >. ,  A >. }  =  { A }
65eqcomi 2447 . . . 4  |-  { A }  =  ran  { <. <. A ,  A >. ,  A >. }
76a1i 11 . . 3  |-  ( T. 
->  { A }  =  ran  { <. <. A ,  A >. ,  A >. } )
8 ablogrpo 23776 . . . 4  |-  ( {
<. <. A ,  A >. ,  A >. }  e.  AbelOp  ->  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
)
96grpofo 23691 . . . 4  |-  ( {
<. <. A ,  A >. ,  A >. }  e.  GrpOp  ->  { <. <. A ,  A >. ,  A >. } :
( { A }  X.  { A } )
-onto-> { A } )
10 fof 5625 . . . 4  |-  ( {
<. <. A ,  A >. ,  A >. } :
( { A }  X.  { A } )
-onto-> { A }  ->  {
<. <. A ,  A >. ,  A >. } :
( { A }  X.  { A } ) --> { A } )
113, 8, 9, 104syl 21 . . 3  |-  ( T. 
->  { <. <. A ,  A >. ,  A >. } :
( { A }  X.  { A } ) --> { A } )
12 elsni 3907 . . . . . 6  |-  ( x  e.  { A }  ->  x  =  A )
13 elsni 3907 . . . . . 6  |-  ( y  e.  { A }  ->  y  =  A )
14 elsni 3907 . . . . . 6  |-  ( z  e.  { A }  ->  z  =  A )
1512, 13, 143anim123i 1173 . . . . 5  |-  ( ( x  e.  { A }  /\  y  e.  { A }  /\  z  e.  { A } )  ->  ( x  =  A  /\  y  =  A  /\  z  =  A ) )
1615adantl 466 . . . 4  |-  ( ( T.  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  z  e.  { A } ) )  -> 
( x  =  A  /\  y  =  A  /\  z  =  A ) )
17 simp1 988 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  x  =  A )
18 simp2 989 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  y  =  A )
1917, 18oveq12d 6114 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
20 df-ov 6099 . . . . . . . 8  |-  ( A { <. <. A ,  A >. ,  A >. } A
)  =  ( {
<. <. A ,  A >. ,  A >. } `  <. A ,  A >. )
214, 1fvsn 5916 . . . . . . . 8  |-  ( {
<. <. A ,  A >. ,  A >. } `  <. A ,  A >. )  =  A
2220, 21eqtri 2463 . . . . . . 7  |-  ( A { <. <. A ,  A >. ,  A >. } A
)  =  A
2319, 22syl6eq 2491 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  A )
2423, 17eqtr4d 2478 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  x )
25 simp3 990 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  z  =  A )
2618, 25oveq12d 6114 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } z )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
2726, 22syl6eq 2491 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } z )  =  A )
2825, 27eqtr4d 2478 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  z  =  ( y { <. <. A ,  A >. ,  A >. } z ) )
2924, 28oveq12d 6114 . . . 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 ) ) )
3016, 29syl 16 . . 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 ) ) )
3117, 23eqtr4d 2478 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  x  =  ( x { <. <. A ,  A >. ,  A >. } y ) )
3218, 17eqtr4d 2478 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  y  =  x )
3332oveq1d 6111 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } z )  =  ( x { <. <. A ,  A >. ,  A >. } z ) )
3431, 33oveq12d 6114 . . . 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 ) ) )
3516, 34syl 16 . . 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 ) ) )
3618, 25eqtr4d 2478 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  y  =  z )
3736oveq2d 6112 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  ( x { <. <. A ,  A >. ,  A >. } z ) )
3837, 28oveq12d 6114 . . . 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 ) ) )
3916, 38syl 16 . . 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 ) ) )
401snid 3910 . . . 4  |-  A  e. 
{ A }
4140a1i 11 . . 3  |-  ( T. 
->  A  e.  { A } )
4213oveq2d 6112 . . . . . 6  |-  ( y  e.  { A }  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
4342, 22syl6eq 2491 . . . . 5  |-  ( y  e.  { A }  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  A )
4443, 13eqtr4d 2478 . . . 4  |-  ( y  e.  { A }  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  y )
4544adantl 466 . . 3  |-  ( ( T.  /\  y  e. 
{ A } )  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  y )
4613oveq1d 6111 . . . . . 6  |-  ( y  e.  { A }  ->  ( y { <. <. A ,  A >. ,  A >. } A )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
4746, 22syl6eq 2491 . . . . 5  |-  ( y  e.  { A }  ->  ( y { <. <. A ,  A >. ,  A >. } A )  =  A )
4847, 13eqtr4d 2478 . . . 4  |-  ( y  e.  { A }  ->  ( y { <. <. A ,  A >. ,  A >. } A )  =  y )
4948adantl 466 . . 3  |-  ( ( T.  /\  y  e. 
{ A } )  ->  ( y {
<. <. A ,  A >. ,  A >. } A
)  =  y )
503, 7, 11, 30, 35, 39, 41, 45, 49isrngod 23871 . 2  |-  ( T. 
->  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. }
>.  e.  RingOps )
5150trud 1378 1  |-  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. } >.  e.  RingOps
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 965    = wceq 1369   T. wtru 1370    e. wcel 1756   _Vcvv 2977   {csn 3882   <.cop 3888    X. cxp 4843   ran crn 4846   -->wf 5419   -onto->wfo 5421   ` cfv 5423  (class class class)co 6096   GrpOpcgr 23678   AbelOpcablo 23773   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-rep 4408  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-pw 3867  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-res 4857  df-ima 4858  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-grpo 23683  df-ablo 23774  df-rngo 23868
This theorem is referenced by: (None)
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