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Theorem grpopncan 20934
Description: Cancellation law for group division. (pncan 9073 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
grpopncan  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) D B )  =  A )

Proof of Theorem grpopncan
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  G  e.  GrpOp )
2 simp2 956 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
3 simp3 957 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  B  e.  X )
4 grpdivf.1 . . . 4  |-  X  =  ran  G
5 grpdivf.3 . . . 4  |-  D  =  (  /g  `  G
)
64, 5grpomuldivass 20932 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  B  e.  X )
)  ->  ( ( A G B ) D B )  =  ( A G ( B D B ) ) )
71, 2, 3, 3, 6syl13anc 1184 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) D B )  =  ( A G ( B D B ) ) )
8 eqid 2296 . . . . 5  |-  (GId `  G )  =  (GId
`  G )
94, 5, 8grpodivid 20933 . . . 4  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( B D B )  =  (GId `  G )
)
109oveq2d 5890 . . 3  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  ( A G ( B D B ) )  =  ( A G (GId
`  G ) ) )
11103adant2 974 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G ( B D B ) )  =  ( A G (GId
`  G ) ) )
124, 8grporid 20903 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G (GId `  G
) )  =  A )
13123adant3 975 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G (GId `  G
) )  =  A )
147, 11, 133eqtrd 2332 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A G B ) D B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ran crn 4706   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869  GIdcgi 20870    /g cgs 20872
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-grpo 20874  df-gid 20875  df-ginv 20876  df-gdiv 20877
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