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Theorem gidsn 25173
Description: Obsolete as of 23-Jan-2020. Use mnd1id 15836 instead. 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 25040 . 2  |-  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
3 opex 4717 . . . . 5  |-  <. A ,  A >.  e.  _V
43rnsnop 5495 . . . 4  |-  ran  { <. <. A ,  A >. ,  A >. }  =  { A }
54eqcomi 2480 . . 3  |-  { A }  =  ran  { <. <. A ,  A >. ,  A >. }
6 eqid 2467 . . 3  |-  (GId `  { <. <. A ,  A >. ,  A >. } )  =  (GId `  { <. <. A ,  A >. ,  A >. } )
75, 6grpoidcl 25042 . 2  |-  ( {
<. <. A ,  A >. ,  A >. }  e.  GrpOp  ->  (GId `  { <. <. A ,  A >. ,  A >. } )  e.  { A } )
8 elsni 4058 . 2  |-  ( (GId
`  { <. <. A ,  A >. ,  A >. } )  e.  { A }  ->  (GId `  { <. <. A ,  A >. ,  A >. } )  =  A )
92, 7, 8mp2b 10 1  |-  (GId `  { <. <. A ,  A >. ,  A >. } )  =  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   _Vcvv 3118   {csn 4033   <.cop 4039   ran crn 5006   ` cfv 5594   GrpOpcgr 25011  GIdcgi 25012
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-grpo 25016  df-gid 25017
This theorem is referenced by:  zrdivrng  25257
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