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Theorem rngosn 25082
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 25025 . . . 4  |-  { <. <. A ,  A >. ,  A >. }  e.  AbelOp
32a1i 11 . . 3  |-  ( T. 
->  { <. <. A ,  A >. ,  A >. }  e.  AbelOp )
4 opex 4711 . . . . . 6  |-  <. A ,  A >.  e.  _V
54rnsnop 5487 . . . . 5  |-  ran  { <. <. A ,  A >. ,  A >. }  =  { A }
65eqcomi 2480 . . . 4  |-  { A }  =  ran  { <. <. A ,  A >. ,  A >. }
76a1i 11 . . 3  |-  ( T. 
->  { A }  =  ran  { <. <. A ,  A >. ,  A >. } )
8 ablogrpo 24962 . . . 4  |-  ( {
<. <. A ,  A >. ,  A >. }  e.  AbelOp  ->  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
)
96grpofo 24877 . . . 4  |-  ( {
<. <. A ,  A >. ,  A >. }  e.  GrpOp  ->  { <. <. A ,  A >. ,  A >. } :
( { A }  X.  { A } )
-onto-> { A } )
10 fof 5793 . . . 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 4052 . . . . . 6  |-  ( x  e.  { A }  ->  x  =  A )
13 elsni 4052 . . . . . 6  |-  ( y  e.  { A }  ->  y  =  A )
14 elsni 4052 . . . . . 6  |-  ( z  e.  { A }  ->  z  =  A )
1512, 13, 143anim123i 1181 . . . . 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 996 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  x  =  A )
18 simp2 997 . . . . . . . 8  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  y  =  A )
1917, 18oveq12d 6300 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
20 df-ov 6285 . . . . . . . 8  |-  ( A { <. <. A ,  A >. ,  A >. } A
)  =  ( {
<. <. A ,  A >. ,  A >. } `  <. A ,  A >. )
214, 1fvsn 6092 . . . . . . . 8  |-  ( {
<. <. A ,  A >. ,  A >. } `  <. A ,  A >. )  =  A
2220, 21eqtri 2496 . . . . . . 7  |-  ( A { <. <. A ,  A >. ,  A >. } A
)  =  A
2319, 22syl6eq 2524 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  A )
2423, 17eqtr4d 2511 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  x )
25 simp3 998 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  z  =  A )
2618, 25oveq12d 6300 . . . . . . 7  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } z )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
2726, 22syl6eq 2524 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } z )  =  A )
2825, 27eqtr4d 2511 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  z  =  ( y { <. <. A ,  A >. ,  A >. } z ) )
2924, 28oveq12d 6300 . . . 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 2511 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  x  =  ( x { <. <. A ,  A >. ,  A >. } y ) )
3218, 17eqtr4d 2511 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  y  =  x )
3332oveq1d 6297 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( y { <. <. A ,  A >. ,  A >. } z )  =  ( x { <. <. A ,  A >. ,  A >. } z ) )
3431, 33oveq12d 6300 . . . 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 2511 . . . . . 6  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  y  =  z )
3736oveq2d 6298 . . . . 5  |-  ( ( x  =  A  /\  y  =  A  /\  z  =  A )  ->  ( x { <. <. A ,  A >. ,  A >. } y )  =  ( x { <. <. A ,  A >. ,  A >. } z ) )
3837, 28oveq12d 6300 . . . 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 4055 . . . 4  |-  A  e. 
{ A }
4140a1i 11 . . 3  |-  ( T. 
->  A  e.  { A } )
4213oveq2d 6298 . . . . . 6  |-  ( y  e.  { A }  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
4342, 22syl6eq 2524 . . . . 5  |-  ( y  e.  { A }  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  A )
4443, 13eqtr4d 2511 . . . 4  |-  ( y  e.  { A }  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  y )
4544adantl 466 . . 3  |-  ( ( T.  /\  y  e. 
{ A } )  ->  ( A { <. <. A ,  A >. ,  A >. } y )  =  y )
4613oveq1d 6297 . . . . . 6  |-  ( y  e.  { A }  ->  ( y { <. <. A ,  A >. ,  A >. } A )  =  ( A { <. <. A ,  A >. ,  A >. } A
) )
4746, 22syl6eq 2524 . . . . 5  |-  ( y  e.  { A }  ->  ( y { <. <. A ,  A >. ,  A >. } A )  =  A )
4847, 13eqtr4d 2511 . . . 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 25057 . 2  |-  ( T. 
->  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. }
>.  e.  RingOps )
5150trud 1388 1  |-  <. { <. <. A ,  A >. ,  A >. } ,  { <. <. A ,  A >. ,  A >. } >.  e.  RingOps
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 973    = wceq 1379   T. wtru 1380    e. wcel 1767   _Vcvv 3113   {csn 4027   <.cop 4033    X. cxp 4997   ran crn 5000   -->wf 5582   -onto->wfo 5584   ` cfv 5586  (class class class)co 6282   GrpOpcgr 24864   AbelOpcablo 24959   RingOpscrngo 25053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-grpo 24869  df-ablo 24960  df-rngo 25054
This theorem is referenced by: (None)
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