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Theorem grpoasscan1 25533
Description: An associative cancellation law for groups. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpasscan1.1  |-  X  =  ran  G
grpasscan1.2  |-  N  =  ( inv `  G
)
Assertion
Ref Expression
grpoasscan1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( ( N `
 A ) G B ) )  =  B )

Proof of Theorem grpoasscan1
StepHypRef Expression
1 grpasscan1.1 . . . . 5  |-  X  =  ran  G
2 eqid 2402 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
3 grpasscan1.2 . . . . 5  |-  N  =  ( inv `  G
)
41, 2, 3grporinv 25525 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
543adant3 1017 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
65oveq1d 6249 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G ( N `  A ) ) G B )  =  ( (GId `  G ) G B ) )
71, 3grpoinvcl 25522 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
81grpoass 25499 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  ( N `  A )  e.  X  /\  B  e.  X ) )  -> 
( ( A G ( N `  A
) ) G B )  =  ( A G ( ( N `
 A ) G B ) ) )
983exp2 1215 . . . . 5  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( ( N `  A )  e.  X  ->  ( B  e.  X  ->  ( ( A G ( N `  A ) ) G B )  =  ( A G ( ( N `  A ) G B ) ) ) ) ) )
109imp 427 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( N `  A
)  e.  X  -> 
( B  e.  X  ->  ( ( A G ( N `  A
) ) G B )  =  ( A G ( ( N `
 A ) G B ) ) ) ) )
117, 10mpd 15 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( B  e.  X  ->  ( ( A G ( N `  A ) ) G B )  =  ( A G ( ( N `  A ) G B ) ) ) )
12113impia 1194 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G ( N `  A ) ) G B )  =  ( A G ( ( N `  A ) G B ) ) )
131, 2grpolid 25515 . . 3  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  (
(GId `  G ) G B )  =  B )
14133adant2 1016 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
(GId `  G ) G B )  =  B )
156, 12, 143eqtr3d 2451 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( ( N `
 A ) G B ) )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ran crn 4943   ` cfv 5525  (class class class)co 6234   GrpOpcgr 25482  GIdcgi 25483   invcgn 25484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-grpo 25487  df-gid 25488  df-ginv 25489
This theorem is referenced by:  gxcom  25565  gxsuc  25568  ghgrpOLD  25664
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