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Theorem subgoinv 23800
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 23795 . . . . . 6  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
21simp2bi 1004 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  GrpOp
)
3 subgoinv.1 . . . . . 6  |-  W  =  ran  H
4 eqid 2443 . . . . . 6  |-  (GId `  H )  =  (GId
`  H )
5 subgoinv.3 . . . . . 6  |-  N  =  ( inv `  H
)
63, 4, 5grporinv 23721 . . . . 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 23718 . . . . . 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 23797 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  ( A  e.  W  /\  ( N `  A )  e.  W ) )  ->  ( A H ( N `  A
) )  =  ( A G ( N `
 A ) ) )
138, 9, 11, 12syl12anc 1216 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( A H ( N `  A ) )  =  ( A G ( N `  A ) ) )
14 eqid 2443 . . . . . 6  |-  (GId `  G )  =  (GId
`  G )
1514, 4subgoid 23799 . . . . 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 2483 . . 3  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
181simp1bi 1003 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  GrpOp
)
1918adantr 465 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  G  e.  GrpOp )
20 eqid 2443 . . . . . 6  |-  ran  G  =  ran  G
2120, 3subgornss 23798 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  W  C_  ran  G )
2221sselda 3361 . . . 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 3362 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( N `  A )  e.  ran  G )
25 subgoinv.2 . . . . 5  |-  M  =  ( inv `  G
)
2620, 14, 25grpoinvid1 23722 . . . 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 1218 . . 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 2448 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 1369    e. wcel 1756    C_ wss 3333   ran crn 4846   ` cfv 5423  (class class class)co 6096   GrpOpcgr 23678  GIdcgi 23679   invcgn 23680   SubGrpOpcsubgo 23793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  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 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-grpo 23683  df-gid 23684  df-ginv 23685  df-subgo 23794
This theorem is referenced by: (None)
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