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Theorem subgoid 21856
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 20 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  ( SubGrpOp `  G )
)
2 issubgo 21852 . . . . . 6  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
32simp2bi 973 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  GrpOp
)
4 eqid 2412 . . . . . 6  |-  ran  H  =  ran  H
5 subgoid.2 . . . . . 6  |-  T  =  (GId `  H )
64, 5grpoidcl 21766 . . . . 5  |-  ( H  e.  GrpOp  ->  T  e.  ran  H )
73, 6syl 16 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  T  e.  ran  H )
84subgoov 21854 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  ( T  e.  ran  H  /\  T  e.  ran  H ) )  ->  ( T H T )  =  ( T G T ) )
91, 7, 7, 8syl12anc 1182 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( T H T )  =  ( T G T ) )
104, 5grpolid 21768 . . . 4  |-  ( ( H  e.  GrpOp  /\  T  e.  ran  H )  -> 
( T H T )  =  T )
113, 7, 10syl2anc 643 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( T H T )  =  T )
129, 11eqtr3d 2446 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( T G T )  =  T )
132simp1bi 972 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  GrpOp
)
14 eqid 2412 . . . . 5  |-  ran  G  =  ran  G
1514, 4subgornss 21855 . . . 4  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ran  H  C_  ran  G )
1615, 7sseldd 3317 . . 3  |-  ( H  e.  ( SubGrpOp `  G
)  ->  T  e.  ran  G )
17 subgoid.1 . . . 4  |-  U  =  (GId `  G )
1814, 17grpoid 21772 . . 3  |-  ( ( G  e.  GrpOp  /\  T  e.  ran  G )  -> 
( T  =  U  <-> 
( T G T )  =  T ) )
1913, 16, 18syl2anc 643 . 2  |-  ( H  e.  ( SubGrpOp `  G
)  ->  ( T  =  U  <->  ( T G T )  =  T ) )
2012, 19mpbird 224 1  |-  ( H  e.  ( SubGrpOp `  G
)  ->  T  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721    C_ wss 3288   ran crn 4846   ` cfv 5421  (class class class)co 6048   GrpOpcgr 21735  GIdcgi 21736   SubGrpOpcsubgo 21850
This theorem is referenced by:  subgoinv  21857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fo 5427  df-fv 5429  df-ov 6051  df-riota 6516  df-grpo 21740  df-gid 21741  df-subgo 21851
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