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Theorem grpoinvid1 20897
Description: The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 27-Oct-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpinv.1  |-  X  =  ran  G
grpinv.2  |-  U  =  (GId `  G )
grpinv.3  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
grpoinvid1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  A
)  =  B  <->  ( A G B )  =  U ) )

Proof of Theorem grpoinvid1
StepHypRef Expression
1 oveq2 5866 . . . 4  |-  ( ( N `  A )  =  B  ->  ( A G ( N `  A ) )  =  ( A G B ) )
21adantl 452 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( N `  A
)  =  B )  ->  ( A G ( N `  A
) )  =  ( A G B ) )
3 grpinv.1 . . . . . 6  |-  X  =  ran  G
4 grpinv.2 . . . . . 6  |-  U  =  (GId `  G )
5 grpinv.3 . . . . . 6  |-  N  =  ( inv `  G
)
63, 4, 5grporinv 20896 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( N `  A ) )  =  U )
763adant3 975 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( N `  A ) )  =  U )
87adantr 451 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( N `  A
)  =  B )  ->  ( A G ( N `  A
) )  =  U )
92, 8eqtr3d 2317 . 2  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( N `  A
)  =  B )  ->  ( A G B )  =  U )
10 oveq2 5866 . . . 4  |-  ( ( A G B )  =  U  ->  (
( N `  A
) G ( A G B ) )  =  ( ( N `
 A ) G U ) )
1110adantl 452 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( A G B )  =  U )  ->  ( ( N `
 A ) G ( A G B ) )  =  ( ( N `  A
) G U ) )
123, 4, 5grpolinv 20895 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G A )  =  U )
1312oveq1d 5873 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( N `  A ) G A ) G B )  =  ( U G B ) )
14133adant3 975 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( N `  A ) G A ) G B )  =  ( U G B ) )
153, 5grpoinvcl 20893 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
1615adantrr 697 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( N `  A )  e.  X
)
17 simprl 732 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  A  e.  X )
18 simprr 733 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  B  e.  X )
1916, 17, 183jca 1132 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( N `  A )  e.  X  /\  A  e.  X  /\  B  e.  X ) )
203grpoass 20870 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  (
( N `  A
)  e.  X  /\  A  e.  X  /\  B  e.  X )
)  ->  ( (
( N `  A
) G A ) G B )  =  ( ( N `  A ) G ( A G B ) ) )
2119, 20syldan 456 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( (
( N `  A
) G A ) G B )  =  ( ( N `  A ) G ( A G B ) ) )
22213impb 1147 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( N `  A ) G A ) G B )  =  ( ( N `
 A ) G ( A G B ) ) )
2314, 22eqtr3d 2317 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( U G B )  =  ( ( N `  A ) G ( A G B ) ) )
243, 4grpolid 20886 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( U G B )  =  B )
25243adant2 974 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( U G B )  =  B )
2623, 25eqtr3d 2317 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  A
) G ( A G B ) )  =  B )
2726adantr 451 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( A G B )  =  U )  ->  ( ( N `
 A ) G ( A G B ) )  =  B )
283, 4grporid 20887 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X )  ->  (
( N `  A
) G U )  =  ( N `  A ) )
2915, 28syldan 456 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G U )  =  ( N `  A ) )
30293adant3 975 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  A
) G U )  =  ( N `  A ) )
3130adantr 451 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( A G B )  =  U )  ->  ( ( N `
 A ) G U )  =  ( N `  A ) )
3211, 27, 313eqtr3rd 2324 . 2  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( A G B )  =  U )  ->  ( N `  A )  =  B )
339, 32impbida 805 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  A
)  =  B  <->  ( A G B )  =  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855
This theorem is referenced by:  grpoinvid  20899  grpoinvop  20908  subgoinv  20975  ghomgrpilem2  23993  grpodivzer  25377  multinv  25422  multinvb  25423  rngonegmn1l  26580
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-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-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
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