MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpoinvid1 Structured version   Unicode version

Theorem grpoinvid1 23722
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 6104 . . . 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 23721 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( N `  A ) )  =  U )
763adant3 1008 . . . 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 2477 . 2  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( N `  A
)  =  B )  ->  ( A G B )  =  U )
10 oveq2 6104 . . . 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 23720 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G A )  =  U )
1312oveq1d 6111 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( ( N `  A ) G A ) G B )  =  ( U G B ) )
14133adant3 1008 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( N `  A ) G A ) G B )  =  ( U G B ) )
153, 5grpoinvcl 23718 . . . . . . . . . 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 755 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  A  e.  X )
18 simprr 756 . . . . . . . . 9  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  B  e.  X )
1916, 17, 183jca 1168 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( N `  A )  e.  X  /\  A  e.  X  /\  B  e.  X ) )
203grpoass 23695 . . . . . . . 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 1183 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( N `  A ) G A ) G B )  =  ( ( N `
 A ) G ( A G B ) ) )
2314, 22eqtr3d 2477 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( U G B )  =  ( ( N `  A ) G ( A G B ) ) )
243, 4grpolid 23711 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( U G B )  =  B )
25243adant2 1007 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( U G B )  =  B )
2623, 25eqtr3d 2477 . . . 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 23712 . . . . . 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 1008 . . . 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 2484 . 2  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( A G B )  =  U )  ->  ( N `  A )  =  B )
339, 32impbida 828 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 965    = wceq 1369    e. wcel 1756   ran crn 4846   ` cfv 5423  (class class class)co 6096   GrpOpcgr 23678  GIdcgi 23679   invcgn 23680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-grpo 23683  df-gid 23684  df-ginv 23685
This theorem is referenced by:  grpoinvid  23724  grpoinvop  23733  subgoinv  23800  ghomgrpilem2  27310  rngonegmn1l  28760
  Copyright terms: Public domain W3C validator