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Theorem grpodivfo 25374
Description: A "division" maps onto the group's underlying set. (Contributed by FL, 21-Jun-2010.)
Hypotheses
Ref Expression
grpdivfo.1  |-  X  =  ran  G
grpdivfo.2  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
grpodivfo  |-  ( G  e.  GrpOp  ->  D :
( X  X.  X
) -onto-> X )

Proof of Theorem grpodivfo
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpdivfo.1 . . 3  |-  X  =  ran  G
2 grpdivfo.2 . . 3  |-  D  =  (  /g  `  G
)
31, 2grpodivf 20913 . 2  |-  ( G  e.  GrpOp  ->  D :
( X  X.  X
) --> X )
4 ffn 5389 . . . 4  |-  ( D : ( X  X.  X ) --> X  ->  D  Fn  ( X  X.  X ) )
5 fnrnov 5993 . . . 4  |-  ( D  Fn  ( X  X.  X )  ->  ran  D  =  { x  |  E. y  e.  X  E. z  e.  X  x  =  ( y D z ) } )
63, 4, 53syl 18 . . 3  |-  ( G  e.  GrpOp  ->  ran  D  =  { x  |  E. y  e.  X  E. z  e.  X  x  =  ( y D z ) } )
7 fovrn 5990 . . . . . . . . . . . 12  |-  ( ( D : ( X  X.  X ) --> X  /\  y  e.  X  /\  z  e.  X
)  ->  ( y D z )  e.  X )
8 eleq1a 2352 . . . . . . . . . . . 12  |-  ( ( y D z )  e.  X  ->  (
x  =  ( y D z )  ->  x  e.  X )
)
97, 8syl 15 . . . . . . . . . . 11  |-  ( ( D : ( X  X.  X ) --> X  /\  y  e.  X  /\  z  e.  X
)  ->  ( x  =  ( y D z )  ->  x  e.  X ) )
1093exp 1150 . . . . . . . . . 10  |-  ( D : ( X  X.  X ) --> X  -> 
( y  e.  X  ->  ( z  e.  X  ->  ( x  =  ( y D z )  ->  x  e.  X
) ) ) )
113, 10syl 15 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ( y  e.  X  ->  ( z  e.  X  ->  (
x  =  ( y D z )  ->  x  e.  X )
) ) )
1211com4l 78 . . . . . . . 8  |-  ( y  e.  X  ->  (
z  e.  X  -> 
( x  =  ( y D z )  ->  ( G  e. 
GrpOp  ->  x  e.  X
) ) ) )
1312rexlimdv 2666 . . . . . . 7  |-  ( y  e.  X  ->  ( E. z  e.  X  x  =  ( y D z )  -> 
( G  e.  GrpOp  ->  x  e.  X )
) )
1413rexlimiv 2661 . . . . . 6  |-  ( E. y  e.  X  E. z  e.  X  x  =  ( y D z )  ->  ( G  e.  GrpOp  ->  x  e.  X ) )
1514com12 27 . . . . 5  |-  ( G  e.  GrpOp  ->  ( E. y  e.  X  E. z  e.  X  x  =  ( y D z )  ->  x  e.  X ) )
16 eqid 2283 . . . . . . 7  |-  (GId `  G )  =  (GId
`  G )
171, 16grpoidcl 20884 . . . . . 6  |-  ( G  e.  GrpOp  ->  (GId `  G
)  e.  X )
181, 2, 16grpodivone 25373 . . . . . . . 8  |-  ( ( G  e.  GrpOp  /\  x  e.  X )  ->  (
x D (GId `  G ) )  =  x )
19 rspceov 5893 . . . . . . . . . . . 12  |-  ( ( x  e.  X  /\  (GId `  G )  e.  X  /\  x  =  ( x D (GId
`  G ) ) )  ->  E. y  e.  X  E. z  e.  X  x  =  ( y D z ) )
20193exp 1150 . . . . . . . . . . 11  |-  ( x  e.  X  ->  (
(GId `  G )  e.  X  ->  ( x  =  ( x D (GId `  G )
)  ->  E. y  e.  X  E. z  e.  X  x  =  ( y D z ) ) ) )
2120adantl 452 . . . . . . . . . 10  |-  ( ( G  e.  GrpOp  /\  x  e.  X )  ->  (
(GId `  G )  e.  X  ->  ( x  =  ( x D (GId `  G )
)  ->  E. y  e.  X  E. z  e.  X  x  =  ( y D z ) ) ) )
2221com3r 73 . . . . . . . . 9  |-  ( x  =  ( x D (GId `  G )
)  ->  ( ( G  e.  GrpOp  /\  x  e.  X )  ->  (
(GId `  G )  e.  X  ->  E. y  e.  X  E. z  e.  X  x  =  ( y D z ) ) ) )
2322eqcoms 2286 . . . . . . . 8  |-  ( ( x D (GId `  G ) )  =  x  ->  ( ( G  e.  GrpOp  /\  x  e.  X )  ->  (
(GId `  G )  e.  X  ->  E. y  e.  X  E. z  e.  X  x  =  ( y D z ) ) ) )
2418, 23mpcom 32 . . . . . . 7  |-  ( ( G  e.  GrpOp  /\  x  e.  X )  ->  (
(GId `  G )  e.  X  ->  E. y  e.  X  E. z  e.  X  x  =  ( y D z ) ) )
2524ex 423 . . . . . 6  |-  ( G  e.  GrpOp  ->  ( x  e.  X  ->  ( (GId
`  G )  e.  X  ->  E. y  e.  X  E. z  e.  X  x  =  ( y D z ) ) ) )
2617, 25mpid 37 . . . . 5  |-  ( G  e.  GrpOp  ->  ( x  e.  X  ->  E. y  e.  X  E. z  e.  X  x  =  ( y D z ) ) )
2715, 26impbid 183 . . . 4  |-  ( G  e.  GrpOp  ->  ( E. y  e.  X  E. z  e.  X  x  =  ( y D z )  <->  x  e.  X ) )
2827abbi1dv 2399 . . 3  |-  ( G  e.  GrpOp  ->  { x  |  E. y  e.  X  E. z  e.  X  x  =  ( y D z ) }  =  X )
296, 28eqtrd 2315 . 2  |-  ( G  e.  GrpOp  ->  ran  D  =  X )
30 dffo2 5455 . 2  |-  ( D : ( X  X.  X ) -onto-> X  <->  ( D : ( X  X.  X ) --> X  /\  ran  D  =  X ) )
313, 29, 30sylanbrc 645 1  |-  ( G  e.  GrpOp  ->  D :
( X  X.  X
) -onto-> X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544    X. cxp 4687   ran crn 4690    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853  GIdcgi 20854    /g cgs 20856
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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