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

Theorem grponpcan 24916
Description: Cancellation law for group division. (npcan 9818 analog.) (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdivf.1  |-  X  =  ran  G
grpdivf.3  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
grponpcan  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B ) G B )  =  A )

Proof of Theorem grponpcan
StepHypRef Expression
1 grpdivf.1 . . . 4  |-  X  =  ran  G
2 eqid 2460 . . . 4  |-  ( inv `  G )  =  ( inv `  G )
3 grpdivf.3 . . . 4  |-  D  =  (  /g  `  G
)
41, 2, 3grpodivval 24907 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( ( inv `  G
) `  B )
) )
54oveq1d 6290 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B ) G B )  =  ( ( A G ( ( inv `  G ) `  B
) ) G B ) )
6 simp1 991 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  G  e.  GrpOp )
7 simp2 992 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
81, 2grpoinvcl 24890 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  (
( inv `  G
) `  B )  e.  X )
983adant2 1010 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( inv `  G
) `  B )  e.  X )
10 simp3 993 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  B  e.  X )
111grpoass 24867 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  ( ( inv `  G
) `  B )  e.  X  /\  B  e.  X ) )  -> 
( ( A G ( ( inv `  G
) `  B )
) G B )  =  ( A G ( ( ( inv `  G ) `  B
) G B ) ) )
126, 7, 9, 10, 11syl13anc 1225 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G ( ( inv `  G
) `  B )
) G B )  =  ( A G ( ( ( inv `  G ) `  B
) G B ) ) )
13 eqid 2460 . . . . . . 7  |-  (GId `  G )  =  (GId
`  G )
141, 13, 2grpolinv 24892 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  (
( ( inv `  G
) `  B ) G B )  =  (GId
`  G ) )
1514oveq2d 6291 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( A G ( ( ( inv `  G ) `
 B ) G B ) )  =  ( A G (GId
`  G ) ) )
16153adant2 1010 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( ( ( inv `  G ) `
 B ) G B ) )  =  ( A G (GId
`  G ) ) )
171, 13grporid 24884 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
18173adant3 1011 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G (GId `  G
) )  =  A )
1916, 18eqtrd 2501 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( ( ( inv `  G ) `
 B ) G B ) )  =  A )
2012, 19eqtrd 2501 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G ( ( inv `  G
) `  B )
) G B )  =  A )
215, 20eqtrd 2501 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B ) G B )  =  A )
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    /g cgs 24853
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-pow 4618  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-pw 4005  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-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-grpo 24855  df-gid 24856  df-ginv 24857  df-gdiv 24858
This theorem is referenced by:  grponpncan  24919  grpodiveq  24920  ablonnncan  24957  ghgrp  25032  grpoeqdivid  29933  ghomdiv  29936
  Copyright terms: Public domain W3C validator