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Theorem subgoinv 20975
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 20970 . . . . . 6  |-  ( H  e.  ( SubGrpOp `  G
)  <->  ( G  e. 
GrpOp  /\  H  e.  GrpOp  /\  H  C_  G )
)
21simp2bi 971 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  H  e.  GrpOp
)
3 subgoinv.1 . . . . . 6  |-  W  =  ran  H
4 eqid 2283 . . . . . 6  |-  (GId `  H )  =  (GId
`  H )
5 subgoinv.3 . . . . . 6  |-  N  =  ( inv `  H
)
63, 4, 5grporinv 20896 . . . . 5  |-  ( ( H  e.  GrpOp  /\  A  e.  W )  ->  ( A H ( N `  A ) )  =  (GId `  H )
)
72, 6sylan 457 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( A H ( N `  A ) )  =  (GId `  H )
)
8 simpl 443 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  H  e.  ( SubGrpOp `  G )
)
9 simpr 447 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  A  e.  W )
103, 5grpoinvcl 20893 . . . . . 6  |-  ( ( H  e.  GrpOp  /\  A  e.  W )  ->  ( N `  A )  e.  W )
112, 10sylan 457 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( N `  A )  e.  W )
123subgoov 20972 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  ( A  e.  W  /\  ( N `  A )  e.  W ) )  ->  ( A H ( N `  A
) )  =  ( A G ( N `
 A ) ) )
138, 9, 11, 12syl12anc 1180 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( A H ( N `  A ) )  =  ( A G ( N `  A ) ) )
14 eqid 2283 . . . . . 6  |-  (GId `  G )  =  (GId
`  G )
1514, 4subgoid 20974 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  (GId `  H
)  =  (GId `  G ) )
1615adantr 451 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  (GId `  H )  =  (GId
`  G ) )
177, 13, 163eqtr3d 2323 . . 3  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
181simp1bi 970 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  G  e.  GrpOp
)
1918adantr 451 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  G  e.  GrpOp )
20 eqid 2283 . . . . . 6  |-  ran  G  =  ran  G
2120, 3subgornss 20973 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  W  C_  ran  G )
2221sselda 3180 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  A  e.  ran  G )
2321adantr 451 . . . . 5  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  W  C_ 
ran  G )
2423, 11sseldd 3181 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( N `  A )  e.  ran  G )
25 subgoinv.2 . . . . 5  |-  M  =  ( inv `  G
)
2620, 14, 25grpoinvid1 20897 . . . 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 1182 . . 3  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  (
( M `  A
)  =  ( N `
 A )  <->  ( A G ( N `  A ) )  =  (GId `  G )
) )
2817, 27mpbird 223 . 2  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( M `  A )  =  ( N `  A ) )
2928eqcomd 2288 1  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( N `  A )  =  ( M `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   ran crn 4690   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855   SubGrpOpcsubgo 20968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-subgo 20969
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