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Theorem subgoid 24982
Description: The identity element of a subgroup is the same as its parent's. (Contributed by Paul Chapman, 3-Mar-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
subgoid.1  |-  U  =  (GId `  G )
subgoid.2  |-  T  =  (GId `  H )
Assertion
Ref Expression
subgoid  |-  ( H  e.  ( SubGrpOp `  G
)  ->  T  =  U )

Proof of Theorem subgoid
StepHypRef Expression
1 id 22 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  ( SubGrpOp `  G )
)
2 issubgo 24978 . . . . . 6  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
32simp2bi 1012 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  GrpOp
)
4 eqid 2467 . . . . . 6  |-  ran  H  =  ran  H
5 subgoid.2 . . . . . 6  |-  T  =  (GId `  H )
64, 5grpoidcl 24892 . . . . 5  |-  ( H  e.  GrpOp  ->  T  e.  ran  H )
73, 6syl 16 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  T  e.  ran  H )
84subgoov 24980 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  ( T  e.  ran  H  /\  T  e.  ran  H ) )  ->  ( T H T )  =  ( T G T ) )
91, 7, 7, 8syl12anc 1226 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( T H T )  =  ( T G T ) )
104, 5grpolid 24894 . . . 4  |-  ( ( H  e.  GrpOp  /\  T  e.  ran  H )  -> 
( T H T )  =  T )
113, 7, 10syl2anc 661 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( T H T )  =  T )
129, 11eqtr3d 2510 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( T G T )  =  T )
132simp1bi 1011 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  GrpOp
)
14 eqid 2467 . . . . 5  |-  ran  G  =  ran  G
1514, 4subgornss 24981 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  C_  ran  G )
1615, 7sseldd 3505 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  T  e.  ran  G )
17 subgoid.1 . . . 4  |-  U  =  (GId `  G )
1814, 17grpoid 24898 . . 3  |-  ( ( G  e.  GrpOp  /\  T  e.  ran  G )  -> 
( T  =  U  <-> 
( T G T )  =  T ) )
1913, 16, 18syl2anc 661 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( T  =  U  <->  ( T G T )  =  T ) )
2012, 19mpbird 232 1  |-  ( H  e.  ( SubGrpOp `  G
)  ->  T  =  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379    e. wcel 1767    C_ wss 3476   ran crn 5000   ` cfv 5586  (class class class)co 6282   GrpOpcgr 24861  GIdcgi 24862   SubGrpOpcsubgo 24976
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-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-fo 5592  df-fv 5594  df-riota 6243  df-ov 6285  df-grpo 24866  df-gid 24867  df-subgo 24977
This theorem is referenced by:  subgoinv  24983
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