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Theorem grpodivzer 25480
Description: Condition for a "subtraction" (or "division") value to be equal to the identity element. (Contributed by FL, 14-Sep-2010.)
Hypotheses
Ref Expression
grpdivzer.1  |-  X  =  ran  G
grpdivzer.2  |-  U  =  (GId `  G )
grpdivzer.3  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
grpodivzer  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B )  =  U  <->  A  =  B ) )

Proof of Theorem grpodivzer
StepHypRef Expression
1 grpdivzer.1 . . . 4  |-  X  =  ran  G
2 eqid 2296 . . . 4  |-  ( inv `  G )  =  ( inv `  G )
3 grpdivzer.3 . . . 4  |-  D  =  (  /g  `  G
)
41, 2, 3grpodivval 20926 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( ( inv `  G
) `  B )
) )
54eqeq1d 2304 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B )  =  U  <->  ( A G ( ( inv `  G ) `  B
) )  =  U ) )
61, 2grpoinvcl 20909 . . . 4  |-  ( ( G  e.  GrpOp  /\  B  e.  X )  ->  (
( inv `  G
) `  B )  e.  X )
763adant2 974 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( inv `  G
) `  B )  e.  X )
8 grpdivzer.2 . . . 4  |-  U  =  (GId `  G )
91, 8, 2grpoinvid1 20913 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  (
( inv `  G
) `  B )  e.  X )  ->  (
( ( inv `  G
) `  A )  =  ( ( inv `  G ) `  B
)  <->  ( A G ( ( inv `  G
) `  B )
)  =  U ) )
107, 9syld3an3 1227 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( inv `  G
) `  A )  =  ( ( inv `  G ) `  B
)  <->  ( A G ( ( inv `  G
) `  B )
)  =  U ) )
111, 2grpoinvf 20923 . . . . 5  |-  ( G  e.  GrpOp  ->  ( inv `  G ) : X -1-1-onto-> X
)
12 f1of1 5487 . . . . 5  |-  ( ( inv `  G ) : X -1-1-onto-> X  ->  ( inv `  G ) : X -1-1-> X )
1311, 12syl 15 . . . 4  |-  ( G  e.  GrpOp  ->  ( inv `  G ) : X -1-1-> X )
14133ad2ant1 976 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( inv `  G ) : X -1-1-> X )
15 3simpc 954 . . 3  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A  e.  X  /\  B  e.  X )
)
16 f1fveq 5802 . . 3  |-  ( ( ( inv `  G
) : X -1-1-> X  /\  ( A  e.  X  /\  B  e.  X
) )  ->  (
( ( inv `  G
) `  A )  =  ( ( inv `  G ) `  B
)  <->  A  =  B
) )
1714, 15, 16syl2anc 642 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( ( inv `  G
) `  A )  =  ( ( inv `  G ) `  B
)  <->  A  =  B
) )
185, 10, 173bitr2d 272 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B )  =  U  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ran crn 4706   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   GrpOpcgr 20869  GIdcgi 20870   invcgn 20871    /g cgs 20872
This theorem is referenced by:  svli2  25587
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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