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Theorem rngosn 21994
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 21937 . . . 4  |-  { <. <. A ,  A >. ,  A >. }  e.  AbelOp
32a1i 11 . . 3  |-  (  T. 
->  { <. <. A ,  A >. ,  A >. }  e.  AbelOp )
4 opex 4429 . . . . . 6  |-  <. A ,  A >.  e.  _V
54rnsnop 5352 . . . . 5  |-  ran  { <. <. A ,  A >. ,  A >. }  =  { A }
65eqcomi 2442 . . . 4  |-  { A }  =  ran  { <. <. A ,  A >. ,  A >. }
76a1i 11 . . 3  |-  (  T. 
->  { A }  =  ran  { <. <. A ,  A >. ,  A >. } )
8 ablogrpo 21874 . . . 4  |-  ( {
<. <. A ,  A >. ,  A >. }  e.  AbelOp  ->  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
)
96grpofo 21789 . . . 4  |-  ( {
<. <. A ,  A >. ,  A >. }  e.  GrpOp  ->  { <. <. A ,  A >. ,  A >. } :
( { A }  X.  { A } )
-onto-> { A } )
10 fof 5655 . . . 4  |-  ( {
<. <. A ,  A >. ,  A >. } :
( { A }  X.  { A } )
-onto-> { A }  ->  {
<. <. A ,  A >. ,  A >. } :
( { A }  X.  { A } ) --> { A } )
113, 8, 9, 104syl 20 . . 3  |-  (  T. 
->  { <. <. A ,  A >. ,  A >. } :
( { A }  X.  { A } ) --> { A } )
12 elsni 3840 . . . . . 6  |-  ( x  e.  { A }  ->  x  =  A )
13 elsni 3840 . . . . . 6  |-  ( y  e.  { A }  ->  y  =  A )
14 elsni 3840 . . . . . 6  |-  ( z  e.  { A }  ->  z  =  A )
1512, 13, 143anim123i 1140 . . . . 5  |-  ( ( x  e.  { A }  /\  y  e.  { A }  /\  z  e.  { A } )  ->  ( x  =  A  /\  y  =  A  /\  z  =  A ) )
1615adantl 454 . . . 4  |-  ( (  T.  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  z  e.  { A } ) )  -> 
( x  =  A  /\  y  =  A  /\  z  =  A ) )
17 simp1 958 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  x  =  A )
18 simp2 959 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  y  =  A )
1917, 18oveq12d 6101 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
20 df-ov 6086 . . . . . . . 8  |-  ( A { <. <. A ,  A >. ,  A >. } A
)  =  ( {
<. <. A ,  A >. ,  A >. } `  <. A ,  A >. )
214, 1fvsn 5928 . . . . . . . 8  |-  ( {
<. <. A ,  A >. ,  A >. } `  <. A ,  A >. )  =  A
2220, 21eqtri 2458 . . . . . . 7  |-  ( A { <. <. A ,  A >. ,  A >. } A
)  =  A
2319, 22syl6eq 2486 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  A )
2423, 17eqtr4d 2473 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  x )
25 simp3 960 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  z  =  A )
2618, 25oveq12d 6101 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } z )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
2726, 22syl6eq 2486 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } z )  =  A )
2825, 27eqtr4d 2473 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  z  =  ( y { <. <. A ,  A >. ,  A >. } z ) )
2924, 28oveq12d 6101 . . . 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 2473 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  x  =  ( x { <. <. A ,  A >. ,  A >. } y ) )
3218, 17eqtr4d 2473 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  y  =  x )
3332oveq1d 6098 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } z )  =  ( x { <. <. A ,  A >. ,  A >. } z ) )
3431, 33oveq12d 6101 . . . 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 2473 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  y  =  z )
3736oveq2d 6099 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  ( x { <. <. A ,  A >. ,  A >. } z ) )
3837, 28oveq12d 6101 . . . 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 3843 . . . 4  |-  A  e. 
{ A }
4140a1i 11 . . 3  |-  (  T. 
->  A  e.  { A } )
4213oveq2d 6099 . . . . . 6  |-  ( y  e.  { A }  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
4342, 22syl6eq 2486 . . . . 5  |-  ( y  e.  { A }  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  A )
4443, 13eqtr4d 2473 . . . 4  |-  ( y  e.  { A }  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  y )
4544adantl 454 . . 3  |-  ( (  T.  /\  y  e. 
{ A } )  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  y )
4613oveq1d 6098 . . . . . 6  |-  ( y  e.  { A }  ->  ( y { <. <. A ,  A >. ,  A >. } A )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
4746, 22syl6eq 2486 . . . . 5  |-  ( y  e.  { A }  ->  ( y { <. <. A ,  A >. ,  A >. } A )  =  A )
4847, 13eqtr4d 2473 . . . 4  |-  ( y  e.  { A }  ->  ( y { <. <. A ,  A >. ,  A >. } A )  =  y )
4948adantl 454 . . 3  |-  ( (  T.  /\  y  e. 
{ A } )  ->  ( y {
<. <. A ,  A >. ,  A >. } A
)  =  y )
503, 7, 11, 30, 35, 39, 41, 45, 49isrngod 21969 . 2  |-  (  T. 
->  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. }
>.  e.  RingOps )
5150trud 1333 1  |-  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. } >.  e.  RingOps
Colors of variables: wff set class
Syntax hints:    /\ wa 360    /\ w3a 937    T. wtru 1326    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816   <.cop 3819    X. cxp 4878   ran crn 4881   -->wf 5452   -onto->wfo 5454   ` cfv 5456  (class class class)co 6083   GrpOpcgr 21776   AbelOpcablo 21871   RingOpscrngo 21965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-grpo 21781  df-ablo 21872  df-rngo 21966
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