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Theorem grponpcan 21842
Description: Cancellation law for group division. (npcan 9316 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 2438 . . . 4  |-  ( inv `  G )  =  ( inv `  G )
3 grpdivf.3 . . . 4  |-  D  =  (  /g  `  G
)
41, 2, 3grpodivval 21833 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( ( inv `  G
) `  B )
) )
54oveq1d 6098 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B ) G B )  =  ( ( A G ( ( inv `  G ) `  B
) ) G B ) )
6 simp1 958 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  G  e.  GrpOp )
7 simp2 959 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
81, 2grpoinvcl 21816 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  (
( inv `  G
) `  B )  e.  X )
983adant2 977 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( inv `  G
) `  B )  e.  X )
10 simp3 960 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  B  e.  X )
111grpoass 21793 . . . 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 1187 . . 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 2438 . . . . . . 7  |-  (GId `  G )  =  (GId
`  G )
141, 13, 2grpolinv 21818 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  (
( ( inv `  G
) `  B ) G B )  =  (GId
`  G ) )
1514oveq2d 6099 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( A G ( ( ( inv `  G ) `
 B ) G B ) )  =  ( A G (GId
`  G ) ) )
16153adant2 977 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( ( ( inv `  G ) `
 B ) G B ) )  =  ( A G (GId
`  G ) ) )
171, 13grporid 21810 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
18173adant3 978 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G (GId `  G
) )  =  A )
1916, 18eqtrd 2470 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( ( ( inv `  G ) `
 B ) G B ) )  =  A )
2012, 19eqtrd 2470 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G ( ( inv `  G
) `  B )
) G B )  =  A )
215, 20eqtrd 2470 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B ) G B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   ran crn 4881   ` cfv 5456  (class class class)co 6083   GrpOpcgr 21776  GIdcgi 21777   invcgn 21778    /g cgs 21779
This theorem is referenced by:  grponpncan  21845  grpodiveq  21846  ablonnncan  21883  ghgrp  21958  grpoeqdivid  26558  ghomdiv  26561
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-grpo 21781  df-gid 21782  df-ginv 21783  df-gdiv 21784
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