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Theorem subgoinv 21028
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 21023 . . . . . 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 2316 . . . . . 6  |-  (GId `  H )  =  (GId
`  H )
5 subgoinv.3 . . . . . 6  |-  N  =  ( inv `  H
)
63, 4, 5grporinv 20949 . . . . 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 20946 . . . . . 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 21025 . . . . 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 2316 . . . . . 6  |-  (GId `  G )  =  (GId
`  G )
1514, 4subgoid 21027 . . . . 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 2356 . . 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 2316 . . . . . 6  |-  ran  G  =  ran  G
2120, 3subgornss 21026 . . . . 5  |-  ( H  e.  ( SubGrpOp `  G
)  ->  W  C_  ran  G )
2221sselda 3214 . . . 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 3215 . . . 4  |-  ( ( H  e.  ( SubGrpOp `  G )  /\  A  e.  W )  ->  ( N `  A )  e.  ran  G )
25 subgoinv.2 . . . . 5  |-  M  =  ( inv `  G
)
2620, 14, 25grpoinvid1 20950 . . . 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 2321 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 1633    e. wcel 1701    C_ wss 3186   ran crn 4727   ` cfv 5292  (class class class)co 5900   GrpOpcgr 20906  GIdcgi 20907   invcgn 20908   SubGrpOpcsubgo 21021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-riota 6346  df-grpo 20911  df-gid 20912  df-ginv 20913  df-subgo 21022
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