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Theorem grpoinvid1 25358
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 6304 . . . 4  |-  ( ( N `  A )  =  B  ->  ( A G ( N `  A ) )  =  ( A G B ) )
21adantl 466 . . 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 25357 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( N `  A ) )  =  U )
763adant3 1016 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( N `  A ) )  =  U )
87adantr 465 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( N `  A
)  =  B )  ->  ( A G ( N `  A
) )  =  U )
92, 8eqtr3d 2500 . 2  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( N `  A
)  =  B )  ->  ( A G B )  =  U )
10 oveq2 6304 . . . 4  |-  ( ( A G B )  =  U  ->  (
( N `  A
) G ( A G B ) )  =  ( ( N `
 A ) G U ) )
1110adantl 466 . . 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 25356 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G A )  =  U )
1312oveq1d 6311 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( N `  A ) G A ) G B )  =  ( U G B ) )
14133adant3 1016 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( N `  A ) G A ) G B )  =  ( U G B ) )
153, 5grpoinvcl 25354 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
1615adantrr 716 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( N `  A )  e.  X
)
17 simprl 756 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  A  e.  X )
18 simprr 757 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  B  e.  X )
1916, 17, 183jca 1176 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( N `  A )  e.  X  /\  A  e.  X  /\  B  e.  X ) )
203grpoass 25331 . . . . . . . 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 470 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( (
( N `  A
) G A ) G B )  =  ( ( N `  A ) G ( A G B ) ) )
22213impb 1192 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( N `  A ) G A ) G B )  =  ( ( N `
 A ) G ( A G B ) ) )
2314, 22eqtr3d 2500 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( U G B )  =  ( ( N `  A ) G ( A G B ) ) )
243, 4grpolid 25347 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( U G B )  =  B )
25243adant2 1015 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( U G B )  =  B )
2623, 25eqtr3d 2500 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  A
) G ( A G B ) )  =  B )
2726adantr 465 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( A G B )  =  U )  ->  ( ( N `
 A ) G ( A G B ) )  =  B )
283, 4grporid 25348 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X )  ->  (
( N `  A
) G U )  =  ( N `  A ) )
2915, 28syldan 470 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G U )  =  ( N `  A ) )
30293adant3 1016 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  A
) G U )  =  ( N `  A ) )
3130adantr 465 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( A G B )  =  U )  ->  ( ( N `
 A ) G U )  =  ( N `  A ) )
3211, 27, 313eqtr3rd 2507 . 2  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( A G B )  =  U )  ->  ( N `  A )  =  B )
339, 32impbida 832 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  A
)  =  B  <->  ( A G B )  =  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   ran crn 5009   ` cfv 5594  (class class class)co 6296   GrpOpcgr 25314  GIdcgi 25315   invcgn 25316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-grpo 25319  df-gid 25320  df-ginv 25321
This theorem is referenced by:  grpoinvid  25360  grpoinvop  25369  subgoinv  25436  ghomgrpilem2  29201  rngonegmn1l  30514
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