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Theorem grpodivf 20929
Description: Mapping for group division. (Contributed by NM, 10-Apr-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
grpdivf.1  |-  X  =  ran  G
grpdivf.3  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
grpodivf  |-  ( G  e.  GrpOp  ->  D :
( X  X.  X
) --> X )

Proof of Theorem grpodivf
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpdivf.1 . . . . . . . 8  |-  X  =  ran  G
2 eqid 2296 . . . . . . . 8  |-  ( inv `  G )  =  ( inv `  G )
31, 2grpoinvcl 20909 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  y  e.  X )  ->  (
( inv `  G
) `  y )  e.  X )
433adant2 974 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  x  e.  X  /\  y  e.  X )  ->  (
( inv `  G
) `  y )  e.  X )
51grpocl 20883 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  x  e.  X  /\  (
( inv `  G
) `  y )  e.  X )  ->  (
x G ( ( inv `  G ) `
 y ) )  e.  X )
64, 5syld3an3 1227 . . . . 5  |-  ( ( G  e.  GrpOp  /\  x  e.  X  /\  y  e.  X )  ->  (
x G ( ( inv `  G ) `
 y ) )  e.  X )
763expib 1154 . . . 4  |-  ( G  e.  GrpOp  ->  ( (
x  e.  X  /\  y  e.  X )  ->  ( x G ( ( inv `  G
) `  y )
)  e.  X ) )
87ralrimivv 2647 . . 3  |-  ( G  e.  GrpOp  ->  A. x  e.  X  A. y  e.  X  ( x G ( ( inv `  G ) `  y
) )  e.  X
)
9 eqid 2296 . . . 4  |-  ( x  e.  X ,  y  e.  X  |->  ( x G ( ( inv `  G ) `  y
) ) )  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( ( inv `  G
) `  y )
) )
109fmpt2 6207 . . 3  |-  ( A. x  e.  X  A. y  e.  X  (
x G ( ( inv `  G ) `
 y ) )  e.  X  <->  ( x  e.  X ,  y  e.  X  |->  ( x G ( ( inv `  G
) `  y )
) ) : ( X  X.  X ) --> X )
118, 10sylib 188 . 2  |-  ( G  e.  GrpOp  ->  ( x  e.  X ,  y  e.  X  |->  ( x G ( ( inv `  G
) `  y )
) ) : ( X  X.  X ) --> X )
12 grpdivf.3 . . . 4  |-  D  =  (  /g  `  G
)
131, 2, 12grpodivfval 20925 . . 3  |-  ( G  e.  GrpOp  ->  D  =  ( x  e.  X ,  y  e.  X  |->  ( x G ( ( inv `  G
) `  y )
) ) )
1413feq1d 5395 . 2  |-  ( G  e.  GrpOp  ->  ( D : ( X  X.  X ) --> X  <->  ( x  e.  X ,  y  e.  X  |->  ( x G ( ( inv `  G
) `  y )
) ) : ( X  X.  X ) --> X ) )
1511, 14mpbird 223 1  |-  ( G  e.  GrpOp  ->  D :
( X  X.  X
) --> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556    X. cxp 4703   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   GrpOpcgr 20869   invcgn 20871    /g cgs 20872
This theorem is referenced by:  grpodivcl  20930  grpodivfo  25477
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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