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Theorem grpodrcan 25375
Description: Right cancellation law for group "subtraction" (or "division"). (Contributed by FL, 14-Sep-2010.)
Hypotheses
Ref Expression
grpdrcan.1  |-  X  =  ran  G
grpdrcan.2  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
grpodrcan  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D C )  =  ( B D C )  <->  A  =  B
) )

Proof of Theorem grpodrcan
StepHypRef Expression
1 grpdrcan.1 . . . 4  |-  X  =  ran  G
2 eqid 2283 . . . 4  |-  ( inv `  G )  =  ( inv `  G )
3 grpdrcan.2 . . . 4  |-  D  =  (  /g  `  G
)
41, 2, 3grpodivval 20910 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  C  e.  X )  ->  ( A D C )  =  ( A G ( ( inv `  G
) `  C )
) )
543adant3r2 1161 . 2  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D C )  =  ( A G ( ( inv `  G ) `
 C ) ) )
61, 2, 3grpodivval 20910 . . . . 5  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B D C )  =  ( B G ( ( inv `  G
) `  C )
) )
763adant3r1 1160 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( B D C )  =  ( B G ( ( inv `  G ) `
 C ) ) )
8 simpr1 961 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
9 simpr2 962 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
101, 2grpoinvcl 20893 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  C  e.  X )  ->  (
( inv `  G
) `  C )  e.  X )
11103ad2antr3 1122 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( inv `  G ) `  C )  e.  X
)
128, 9, 113jca 1132 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A  e.  X  /\  B  e.  X  /\  ( ( inv `  G ) `
 C )  e.  X ) )
131grporcan 20888 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  ( ( inv `  G
) `  C )  e.  X ) )  -> 
( ( A G ( ( inv `  G
) `  C )
)  =  ( B G ( ( inv `  G ) `  C
) )  <->  A  =  B ) )
1412, 13syldan 456 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G ( ( inv `  G ) `  C
) )  =  ( B G ( ( inv `  G ) `
 C ) )  <-> 
A  =  B ) )
15 eqeq2 2292 . . . . . 6  |-  ( ( B D C )  =  ( B G ( ( inv `  G
) `  C )
)  ->  ( ( A G ( ( inv `  G ) `  C
) )  =  ( B D C )  <-> 
( A G ( ( inv `  G
) `  C )
)  =  ( B G ( ( inv `  G ) `  C
) ) ) )
1615bibi1d 310 . . . . 5  |-  ( ( B D C )  =  ( B G ( ( inv `  G
) `  C )
)  ->  ( (
( A G ( ( inv `  G
) `  C )
)  =  ( B D C )  <->  A  =  B )  <->  ( ( A G ( ( inv `  G ) `  C
) )  =  ( B G ( ( inv `  G ) `
 C ) )  <-> 
A  =  B ) ) )
1714, 16syl5ibr 212 . . . 4  |-  ( ( B D C )  =  ( B G ( ( inv `  G
) `  C )
)  ->  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G ( ( inv `  G ) `  C
) )  =  ( B D C )  <-> 
A  =  B ) ) )
187, 17mpcom 32 . . 3  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A G ( ( inv `  G ) `  C
) )  =  ( B D C )  <-> 
A  =  B ) )
19 eqeq1 2289 . . . 4  |-  ( ( A D C )  =  ( A G ( ( inv `  G
) `  C )
)  ->  ( ( A D C )  =  ( B D C )  <->  ( A G ( ( inv `  G
) `  C )
)  =  ( B D C ) ) )
2019bibi1d 310 . . 3  |-  ( ( A D C )  =  ( A G ( ( inv `  G
) `  C )
)  ->  ( (
( A D C )  =  ( B D C )  <->  A  =  B )  <->  ( ( A G ( ( inv `  G ) `  C
) )  =  ( B D C )  <-> 
A  =  B ) ) )
2118, 20syl5ibr 212 . 2  |-  ( ( A D C )  =  ( A G ( ( inv `  G
) `  C )
)  ->  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D C )  =  ( B D C )  <->  A  =  B
) ) )
225, 21mpcom 32 1  |-  ( ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( A D C )  =  ( B D C )  <->  A  =  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853   invcgn 20855    /g cgs 20856
This theorem is referenced by:  vecsrcan  25469  mvecrtol  25473  mvecrtol2  25477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861
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