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Theorem grpodivone 25373
Description: "Division" by the neutral element of a group. (Contributed by FL, 21-Jun-2010.)
Hypotheses
Ref Expression
grpdivfo.1  |-  X  =  ran  G
grpdivfo.2  |-  D  =  (  /g  `  G
)
grpdivfo.3  |-  U  =  (GId `  G )
Assertion
Ref Expression
grpodivone  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A D U )  =  A )

Proof of Theorem grpodivone
StepHypRef Expression
1 grpdivfo.1 . . . 4  |-  X  =  ran  G
2 grpdivfo.3 . . . 4  |-  U  =  (GId `  G )
31, 2grpoidcl 20884 . . 3  |-  ( G  e.  GrpOp  ->  U  e.  X )
4 eqid 2283 . . . . . 6  |-  ( inv `  G )  =  ( inv `  G )
5 grpdivfo.2 . . . . . 6  |-  D  =  (  /g  `  G
)
61, 4, 5grpodivval 20910 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  U  e.  X )  ->  ( A D U )  =  ( A G ( ( inv `  G
) `  U )
) )
72, 4grpoinvid 20899 . . . . . . . . 9  |-  ( G  e.  GrpOp  ->  ( ( inv `  G ) `  U )  =  U )
87oveq2d 5874 . . . . . . . 8  |-  ( G  e.  GrpOp  ->  ( A G ( ( inv `  G ) `  U
) )  =  ( A G U ) )
9 eqtr 2300 . . . . . . . . . . 11  |-  ( ( ( A D U )  =  ( A G ( ( inv `  G ) `  U
) )  /\  ( A G ( ( inv `  G ) `  U
) )  =  ( A G U ) )  ->  ( A D U )  =  ( A G U ) )
109ex 423 . . . . . . . . . 10  |-  ( ( A D U )  =  ( A G ( ( inv `  G
) `  U )
)  ->  ( ( A G ( ( inv `  G ) `  U
) )  =  ( A G U )  ->  ( A D U )  =  ( A G U ) ) )
111, 2grporid 20887 . . . . . . . . . . . . 13  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A G U )  =  A )
12 eqtr 2300 . . . . . . . . . . . . . 14  |-  ( ( ( A D U )  =  ( A G U )  /\  ( A G U )  =  A )  -> 
( A D U )  =  A )
1312ex 423 . . . . . . . . . . . . 13  |-  ( ( A D U )  =  ( A G U )  ->  (
( A G U )  =  A  -> 
( A D U )  =  A ) )
1411, 13syl5com 26 . . . . . . . . . . . 12  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  (
( A D U )  =  ( A G U )  -> 
( A D U )  =  A ) )
15143adant3 975 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  U  e.  X )  ->  (
( A D U )  =  ( A G U )  -> 
( A D U )  =  A ) )
1615com12 27 . . . . . . . . . 10  |-  ( ( A D U )  =  ( A G U )  ->  (
( G  e.  GrpOp  /\  A  e.  X  /\  U  e.  X )  ->  ( A D U )  =  A ) )
1710, 16syl6 29 . . . . . . . . 9  |-  ( ( A D U )  =  ( A G ( ( inv `  G
) `  U )
)  ->  ( ( A G ( ( inv `  G ) `  U
) )  =  ( A G U )  ->  ( ( G  e.  GrpOp  /\  A  e.  X  /\  U  e.  X
)  ->  ( A D U )  =  A ) ) )
1817com3l 75 . . . . . . . 8  |-  ( ( A G ( ( inv `  G ) `
 U ) )  =  ( A G U )  ->  (
( G  e.  GrpOp  /\  A  e.  X  /\  U  e.  X )  ->  ( ( A D U )  =  ( A G ( ( inv `  G ) `
 U ) )  ->  ( A D U )  =  A ) ) )
198, 18syl 15 . . . . . . 7  |-  ( G  e.  GrpOp  ->  ( ( G  e.  GrpOp  /\  A  e.  X  /\  U  e.  X )  ->  (
( A D U )  =  ( A G ( ( inv `  G ) `  U
) )  ->  ( A D U )  =  A ) ) )
20193ad2ant1 976 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  U  e.  X )  ->  (
( G  e.  GrpOp  /\  A  e.  X  /\  U  e.  X )  ->  ( ( A D U )  =  ( A G ( ( inv `  G ) `
 U ) )  ->  ( A D U )  =  A ) ) )
2120pm2.43i 43 . . . . 5  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  U  e.  X )  ->  (
( A D U )  =  ( A G ( ( inv `  G ) `  U
) )  ->  ( A D U )  =  A ) )
226, 21mpd 14 . . . 4  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  U  e.  X )  ->  ( A D U )  =  A )
23223exp 1150 . . 3  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( U  e.  X  ->  ( A D U )  =  A ) ) )
243, 23mpid 37 . 2  |-  ( G  e.  GrpOp  ->  ( A  e.  X  ->  ( A D U )  =  A ) )
2524imp 418 1  |-  ( ( G  e.  GrpOp  /\  A  e.  X )  ->  ( A D U )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855    /g cgs 20856
This theorem is referenced by:  grpodivfo  25374
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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