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Theorem subgoid 21900
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 21 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  ( SubGrpOp `  G )
)
2 issubgo 21896 . . . . . 6  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
32simp2bi 974 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  GrpOp
)
4 eqid 2438 . . . . . 6  |-  ran  H  =  ran  H
5 subgoid.2 . . . . . 6  |-  T  =  (GId `  H )
64, 5grpoidcl 21810 . . . . 5  |-  ( H  e.  GrpOp  ->  T  e.  ran  H )
73, 6syl 16 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  T  e.  ran  H )
84subgoov 21898 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  ( T  e.  ran  H  /\  T  e.  ran  H ) )  ->  ( T H T )  =  ( T G T ) )
91, 7, 7, 8syl12anc 1183 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( T H T )  =  ( T G T ) )
104, 5grpolid 21812 . . . 4  |-  ( ( H  e.  GrpOp  /\  T  e.  ran  H )  -> 
( T H T )  =  T )
113, 7, 10syl2anc 644 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( T H T )  =  T )
129, 11eqtr3d 2472 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( T G T )  =  T )
132simp1bi 973 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  GrpOp
)
14 eqid 2438 . . . . 5  |-  ran  G  =  ran  G
1514, 4subgornss 21899 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  C_  ran  G )
1615, 7sseldd 3351 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  T  e.  ran  G )
17 subgoid.1 . . . 4  |-  U  =  (GId `  G )
1814, 17grpoid 21816 . . 3  |-  ( ( G  e.  GrpOp  /\  T  e.  ran  G )  -> 
( T  =  U  <-> 
( T G T )  =  T ) )
1913, 16, 18syl2anc 644 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( T  =  U  <->  ( T G T )  =  T ) )
2012, 19mpbird 225 1  |-  ( H  e.  ( SubGrpOp `  G
)  ->  T  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    = wceq 1653    e. wcel 1726    C_ wss 3322   ran crn 4882   ` cfv 5457  (class class class)co 6084   GrpOpcgr 21779  GIdcgi 21780   SubGrpOpcsubgo 21894
This theorem is referenced by:  subgoinv  21901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fo 5463  df-fv 5465  df-ov 6087  df-riota 6552  df-grpo 21784  df-gid 21785  df-subgo 21895
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