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Theorem gidsn 23835
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 23702 . 2  |-  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
3 opex 4556 . . . . 5  |-  <. A ,  A >.  e.  _V
43rnsnop 5320 . . . 4  |-  ran  { <. <. A ,  A >. ,  A >. }  =  { A }
54eqcomi 2447 . . 3  |-  { A }  =  ran  { <. <. A ,  A >. ,  A >. }
6 eqid 2443 . . 3  |-  (GId `  { <. <. A ,  A >. ,  A >. } )  =  (GId `  { <. <. A ,  A >. ,  A >. } )
75, 6grpoidcl 23704 . 2  |-  ( {
<. <. A ,  A >. ,  A >. }  e.  GrpOp  ->  (GId `  { <. <. A ,  A >. ,  A >. } )  e.  { A } )
8 elsni 3902 . 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 1369    e. wcel 1756   _Vcvv 2972   {csn 3877   <.cop 3883   ran crn 4841   ` cfv 5418   GrpOpcgr 23673  GIdcgi 23674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-grpo 23678  df-gid 23679
This theorem is referenced by:  zrdivrng  23919
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