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Theorem grpoasscan1 23743
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 2443 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
3 grpasscan1.2 . . . . 5  |-  N  =  ( inv `  G
)
41, 2, 3grporinv 23735 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
543adant3 1008 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
65oveq1d 6125 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G ( N `  A ) ) G B )  =  ( (GId `  G ) G B ) )
71, 3grpoinvcl 23732 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
81grpoass 23709 . . . . . 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 1205 . . . . 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 429 . . . 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 1184 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G ( N `  A ) ) G B )  =  ( A G ( ( N `  A ) G B ) ) )
131, 2grpolid 23725 . . 3  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  (
(GId `  G ) G B )  =  B )
14133adant2 1007 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
(GId `  G ) G B )  =  B )
156, 12, 143eqtr3d 2483 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ran crn 4860   ` cfv 5437  (class class class)co 6110   GrpOpcgr 23692  GIdcgi 23693   invcgn 23694
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 4422  ax-sep 4432  ax-nul 4440  ax-pr 4550  ax-un 6391
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 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-grpo 23697  df-gid 23698  df-ginv 23699
This theorem is referenced by:  gxcom  23775  gxsuc  23778  ghgrp  23874
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