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Theorem subgoinv 26028
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 26023 . . . . . 6  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
21simp2bi 1022 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  GrpOp
)
3 subgoinv.1 . . . . . 6  |-  W  =  ran  H
4 eqid 2423 . . . . . 6  |-  (GId `  H )  =  (GId
`  H )
5 subgoinv.3 . . . . . 6  |-  N  =  ( inv `  H
)
63, 4, 5grporinv 25949 . . . . 5  |-  ( ( H  e.  GrpOp  /\  A  e.  W )  ->  ( A H ( N `  A ) )  =  (GId `  H )
)
72, 6sylan 474 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( A H ( N `  A ) )  =  (GId `  H )
)
8 simpl 459 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  H  e.  ( SubGrpOp `  G )
)
9 simpr 463 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  A  e.  W )
103, 5grpoinvcl 25946 . . . . . 6  |-  ( ( H  e.  GrpOp  /\  A  e.  W )  ->  ( N `  A )  e.  W )
112, 10sylan 474 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( N `  A )  e.  W )
123subgoov 26025 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  ( A  e.  W  /\  ( N `  A )  e.  W ) )  ->  ( A H ( N `  A
) )  =  ( A G ( N `
 A ) ) )
138, 9, 11, 12syl12anc 1263 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( A H ( N `  A ) )  =  ( A G ( N `  A ) ) )
14 eqid 2423 . . . . . 6  |-  (GId `  G )  =  (GId
`  G )
1514, 4subgoid 26027 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  (GId `  H
)  =  (GId `  G ) )
1615adantr 467 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  (GId `  H )  =  (GId
`  G ) )
177, 13, 163eqtr3d 2472 . . 3  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
181simp1bi 1021 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  GrpOp
)
1918adantr 467 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  G  e.  GrpOp )
20 eqid 2423 . . . . . 6  |-  ran  G  =  ran  G
2120, 3subgornss 26026 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  W  C_  ran  G )
2221sselda 3465 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  A  e.  ran  G )
2321adantr 467 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  W  C_ 
ran  G )
2423, 11sseldd 3466 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( N `  A )  e.  ran  G )
25 subgoinv.2 . . . . 5  |-  M  =  ( inv `  G
)
2620, 14, 25grpoinvid1 25950 . . . 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 1265 . . 3  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  (
( M `  A
)  =  ( N `
 A )  <->  ( A G ( N `  A ) )  =  (GId `  G )
) )
2817, 27mpbird 236 . 2  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( M `  A )  =  ( N `  A ) )
2928eqcomd 2431 1  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( N `  A )  =  ( M `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1438    e. wcel 1869    C_ wss 3437   ran crn 4852   ` cfv 5599  (class class class)co 6303   GrpOpcgr 25906  GIdcgi 25907   invcgn 25908   SubGrpOpcsubgo 26021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-grpo 25911  df-gid 25912  df-ginv 25913  df-subgo 26022
This theorem is referenced by: (None)
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