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Theorem subgoinv 24983
Description: The inverse of a subgroup element is the same as its inverse in the parent group. (Contributed by Mario Carneiro, 8-Jul-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
subgoinv.1  |-  W  =  ran  H
subgoinv.2  |-  M  =  ( inv `  G
)
subgoinv.3  |-  N  =  ( inv `  H
)
Assertion
Ref Expression
subgoinv  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( N `  A )  =  ( M `  A ) )

Proof of Theorem subgoinv
StepHypRef Expression
1 issubgo 24978 . . . . . 6  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
21simp2bi 1012 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  GrpOp
)
3 subgoinv.1 . . . . . 6  |-  W  =  ran  H
4 eqid 2467 . . . . . 6  |-  (GId `  H )  =  (GId
`  H )
5 subgoinv.3 . . . . . 6  |-  N  =  ( inv `  H
)
63, 4, 5grporinv 24904 . . . . 5  |-  ( ( H  e.  GrpOp  /\  A  e.  W )  ->  ( A H ( N `  A ) )  =  (GId `  H )
)
72, 6sylan 471 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( A H ( N `  A ) )  =  (GId `  H )
)
8 simpl 457 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  H  e.  ( SubGrpOp `  G )
)
9 simpr 461 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  A  e.  W )
103, 5grpoinvcl 24901 . . . . . 6  |-  ( ( H  e.  GrpOp  /\  A  e.  W )  ->  ( N `  A )  e.  W )
112, 10sylan 471 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( N `  A )  e.  W )
123subgoov 24980 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  ( A  e.  W  /\  ( N `  A )  e.  W ) )  ->  ( A H ( N `  A
) )  =  ( A G ( N `
 A ) ) )
138, 9, 11, 12syl12anc 1226 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( A H ( N `  A ) )  =  ( A G ( N `  A ) ) )
14 eqid 2467 . . . . . 6  |-  (GId `  G )  =  (GId
`  G )
1514, 4subgoid 24982 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  (GId `  H
)  =  (GId `  G ) )
1615adantr 465 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  (GId `  H )  =  (GId
`  G ) )
177, 13, 163eqtr3d 2516 . . 3  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
181simp1bi 1011 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  GrpOp
)
1918adantr 465 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  G  e.  GrpOp )
20 eqid 2467 . . . . . 6  |-  ran  G  =  ran  G
2120, 3subgornss 24981 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  W  C_  ran  G )
2221sselda 3504 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  A  e.  ran  G )
2321adantr 465 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  W  C_ 
ran  G )
2423, 11sseldd 3505 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( N `  A )  e.  ran  G )
25 subgoinv.2 . . . . 5  |-  M  =  ( inv `  G
)
2620, 14, 25grpoinvid1 24905 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  ran  G  /\  ( N `  A )  e.  ran  G )  -> 
( ( M `  A )  =  ( N `  A )  <-> 
( A G ( N `  A ) )  =  (GId `  G ) ) )
2719, 22, 24, 26syl3anc 1228 . . 3  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  (
( M `  A
)  =  ( N `
 A )  <->  ( A G ( N `  A ) )  =  (GId `  G )
) )
2817, 27mpbird 232 . 2  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( M `  A )  =  ( N `  A ) )
2928eqcomd 2475 1  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( N `  A )  =  ( M `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476   ran crn 5000   ` cfv 5586  (class class class)co 6282   GrpOpcgr 24861  GIdcgi 24862   invcgn 24863   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-rep 4558  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-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-grpo 24866  df-gid 24867  df-ginv 24868  df-subgo 24977
This theorem is referenced by: (None)
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