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Theorem grpoinvid1 20913
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 5882 . . . 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 20912 . . . . 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 2330 . 2  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( N `  A
)  =  B )  ->  ( A G B )  =  U )
10 oveq2 5882 . . . 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 20911 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G A )  =  U )
1312oveq1d 5889 . . . . . . 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 20909 . . . . . . . . . 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 20886 . . . . . . . 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 2330 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( U G B )  =  ( ( N `  A ) G ( A G B ) ) )
243, 4grpolid 20902 . . . . . 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 2330 . . . 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 20903 . . . . . 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 2337 . 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 1632    e. wcel 1696   ran crn 4706   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869  GIdcgi 20870   invcgn 20871
This theorem is referenced by:  grpoinvid  20915  grpoinvop  20924  subgoinv  20991  ghomgrpilem2  24008  grpodivzer  25480  multinv  25525  multinvb  25526  rngonegmn1l  26683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876
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