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Theorem gidsn 21031
Description: The identity element of the trivial group. (Contributed by FL, 21-Jun-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
ablsn.1  |-  A  e. 
_V
Assertion
Ref Expression
gidsn  |-  (GId `  { <. <. A ,  A >. ,  A >. } )  =  A

Proof of Theorem gidsn
StepHypRef Expression
1 ablsn.1 . . 3  |-  A  e. 
_V
21grposn 20898 . 2  |-  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
3 opex 4253 . . . . 5  |-  <. A ,  A >.  e.  _V
43rnsnop 5169 . . . 4  |-  ran  { <. <. A ,  A >. ,  A >. }  =  { A }
54eqcomi 2300 . . 3  |-  { A }  =  ran  { <. <. A ,  A >. ,  A >. }
6 eqid 2296 . . 3  |-  (GId `  { <. <. A ,  A >. ,  A >. } )  =  (GId `  { <. <. A ,  A >. ,  A >. } )
75, 6grpoidcl 20900 . 2  |-  ( {
<. <. A ,  A >. ,  A >. }  e.  GrpOp  ->  (GId `  { <. <. A ,  A >. ,  A >. } )  e.  { A } )
8 elsni 3677 . 2  |-  ( (GId
`  { <. <. A ,  A >. ,  A >. } )  e.  { A }  ->  (GId `  { <. <. A ,  A >. ,  A >. } )  =  A )
92, 7, 8mp2b 9 1  |-  (GId `  { <. <. A ,  A >. ,  A >. } )  =  A
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   <.cop 3656   ran crn 4706   ` cfv 5271   GrpOpcgr 20869  GIdcgi 20870
This theorem is referenced by:  zrdivrng  21115
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-riota 6320  df-grpo 20874  df-gid 20875
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