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Theorem gidsn 21015
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 20882 . 2  |-  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
3 opex 4237 . . . . 5  |-  <. A ,  A >.  e.  _V
43rnsnop 5153 . . . 4  |-  ran  { <. <. A ,  A >. ,  A >. }  =  { A }
54eqcomi 2287 . . 3  |-  { A }  =  ran  { <. <. A ,  A >. ,  A >. }
6 eqid 2283 . . 3  |-  (GId `  { <. <. A ,  A >. ,  A >. } )  =  (GId `  { <. <. A ,  A >. ,  A >. } )
75, 6grpoidcl 20884 . 2  |-  ( {
<. <. A ,  A >. ,  A >. }  e.  GrpOp  ->  (GId `  { <. <. A ,  A >. ,  A >. } )  e.  { A } )
8 elsni 3664 . 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 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   <.cop 3643   ran crn 4690   ` cfv 5255   GrpOpcgr 20853  GIdcgi 20854
This theorem is referenced by:  zrdivrng  21099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-grpo 20858  df-gid 20859
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