MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpoasscan1 Structured version   Unicode version

Theorem grpoasscan1 24901
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 2460 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
3 grpasscan1.2 . . . . 5  |-  N  =  ( inv `  G
)
41, 2, 3grporinv 24893 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
543adant3 1011 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( N `  A ) )  =  (GId `  G )
)
65oveq1d 6290 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G ( N `  A ) ) G B )  =  ( (GId `  G ) G B ) )
71, 3grpoinvcl 24890 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( N `  A )  e.  X )
81grpoass 24867 . . . . . 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 1209 . . . . 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 1188 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G ( N `  A ) ) G B )  =  ( A G ( ( N `  A ) G B ) ) )
131, 2grpolid 24883 . . 3  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  (
(GId `  G ) G B )  =  B )
14133adant2 1010 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
(GId `  G ) G B )  =  B )
156, 12, 143eqtr3d 2509 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 968    = wceq 1374    e. wcel 1762   ran crn 4993   ` cfv 5579  (class class class)co 6275   GrpOpcgr 24850  GIdcgi 24851   invcgn 24852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-grpo 24855  df-gid 24856  df-ginv 24857
This theorem is referenced by:  gxcom  24933  gxsuc  24936  ghgrp  25032
  Copyright terms: Public domain W3C validator