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Theorem grpoinvid2 23717
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
grpoinvid2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  A
)  =  B  <->  ( B G A )  =  U ) )

Proof of Theorem grpoinvid2
StepHypRef Expression
1 oveq1 6097 . . . 4  |-  ( ( N `  A )  =  B  ->  (
( N `  A
) G A )  =  ( B G A ) )
21adantl 466 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( N `  A
)  =  B )  ->  ( ( N `
 A ) G A )  =  ( B G A ) )
3 grpinv.1 . . . . . 6  |-  X  =  ran  G
4 grpinv.2 . . . . . 6  |-  U  =  (GId `  G )
5 grpinv.3 . . . . . 6  |-  N  =  ( inv `  G
)
63, 4, 5grpolinv 23714 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
) G A )  =  U )
763adant3 1008 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  A
) G A )  =  U )
87adantr 465 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( N `  A
)  =  B )  ->  ( ( N `
 A ) G A )  =  U )
92, 8eqtr3d 2476 . 2  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( N `  A
)  =  B )  ->  ( B G A )  =  U )
103, 5grpoinvcl 23712 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
113, 4grpolid 23705 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( N `  A )  e.  X )  ->  ( U G ( N `  A ) )  =  ( N `  A
) )
1210, 11syldan 470 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( U G ( N `  A ) )  =  ( N `  A
) )
13123adant3 1008 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( U G ( N `  A ) )  =  ( N `  A
) )
1413eqcomd 2447 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  A )  =  ( U G ( N `  A
) ) )
1514adantr 465 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( B G A )  =  U )  ->  ( N `  A )  =  ( U G ( N `
 A ) ) )
16 oveq1 6097 . . . 4  |-  ( ( B G A )  =  U  ->  (
( B G A ) G ( N `
 A ) )  =  ( U G ( N `  A
) ) )
1716adantl 466 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( B G A )  =  U )  ->  ( ( B G A ) G ( N `  A
) )  =  ( U G ( N `
 A ) ) )
18 simprr 756 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  B  e.  X )
19 simprl 755 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  A  e.  X )
2010adantrr 716 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( N `  A )  e.  X
)
2118, 19, 203jca 1168 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( B  e.  X  /\  A  e.  X  /\  ( N `
 A )  e.  X ) )
223grpoass 23689 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  A  e.  X  /\  ( N `  A )  e.  X ) )  ->  ( ( B G A ) G ( N `  A
) )  =  ( B G ( A G ( N `  A ) ) ) )
2321, 22syldan 470 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( ( B G A ) G ( N `  A
) )  =  ( B G ( A G ( N `  A ) ) ) )
24233impb 1183 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( B G A ) G ( N `
 A ) )  =  ( B G ( A G ( N `  A ) ) ) )
253, 4, 5grporinv 23715 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( N `  A ) )  =  U )
2625oveq2d 6106 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( B G ( A G ( N `  A
) ) )  =  ( B G U ) )
27263adant3 1008 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( B G ( A G ( N `  A
) ) )  =  ( B G U ) )
283, 4grporid 23706 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( B G U )  =  B )
29283adant2 1007 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( B G U )  =  B )
3024, 27, 293eqtrd 2478 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( B G A ) G ( N `
 A ) )  =  B )
3130adantr 465 . . 3  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( B G A )  =  U )  ->  ( ( B G A ) G ( N `  A
) )  =  B )
3215, 17, 313eqtr2d 2480 . 2  |-  ( ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  /\  ( B G A )  =  U )  ->  ( N `  A )  =  B )
339, 32impbida 828 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( N `  A
)  =  B  <->  ( B G A )  =  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ran crn 4840   ` cfv 5417  (class class class)co 6090   GrpOpcgr 23672  GIdcgi 23673   invcgn 23674
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 4402  ax-sep 4412  ax-nul 4420  ax-pr 4530  ax-un 6371
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-grpo 23677  df-gid 23678  df-ginv 23679
This theorem is referenced by:  ghomf1olem  27312  rngonegmn1r  28754
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