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Theorem subgoid 23941
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 23937 . . . . . 6  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
32simp2bi 1004 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  GrpOp
)
4 eqid 2452 . . . . . 6  |-  ran  H  =  ran  H
5 subgoid.2 . . . . . 6  |-  T  =  (GId `  H )
64, 5grpoidcl 23851 . . . . 5  |-  ( H  e.  GrpOp  ->  T  e.  ran  H )
73, 6syl 16 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  T  e.  ran  H )
84subgoov 23939 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  ( T  e.  ran  H  /\  T  e.  ran  H ) )  ->  ( T H T )  =  ( T G T ) )
91, 7, 7, 8syl12anc 1217 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( T H T )  =  ( T G T ) )
104, 5grpolid 23853 . . . 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 2495 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( T G T )  =  T )
132simp1bi 1003 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  GrpOp
)
14 eqid 2452 . . . . 5  |-  ran  G  =  ran  G
1514, 4subgornss 23940 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  C_  ran  G )
1615, 7sseldd 3460 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  T  e.  ran  G )
17 subgoid.1 . . . 4  |-  U  =  (GId `  G )
1814, 17grpoid 23857 . . 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 1370    e. wcel 1758    C_ wss 3431   ran crn 4944   ` cfv 5521  (class class class)co 6195   GrpOpcgr 23820  GIdcgi 23821   SubGrpOpcsubgo 23935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-fo 5527  df-fv 5529  df-riota 6156  df-ov 6198  df-grpo 23825  df-gid 23826  df-subgo 23936
This theorem is referenced by:  subgoinv  23942
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