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Metamath Proof Explorer

Theorem grpodivf

Description: Mapping for group division. (Contributed by NM, 10-Apr-2008) (Revised by Mario Carneiro, 15-Dec-2013) (New usage is discouraged.)

Ref Expression
Hypotheses grpdivf.1 
|- X = ran G
grpdivf.3 
|- D = ( /g ` G )
Assertion grpodivf 
|- ( G e. GrpOp -> D : ( X X. X ) --> X )

Proof

Step Hyp Ref Expression
1 grpdivf.1 
 |-  X = ran G
2 grpdivf.3 
 |-  D = ( /g ` G )
3 eqid 
 |-  ( inv ` G ) = ( inv ` G )
4 1 3 grpoinvcl 
 |-  ( ( G e. GrpOp /\ y e. X ) -> ( ( inv ` G ) ` y ) e. X )
5 4 3adant2 
 |-  ( ( G e. GrpOp /\ x e. X /\ y e. X ) -> ( ( inv ` G ) ` y ) e. X )
6 1 grpocl 
 |-  ( ( G e. GrpOp /\ x e. X /\ ( ( inv ` G ) ` y ) e. X ) -> ( x G ( ( inv ` G ) ` y ) ) e. X )
7 5 6 syld3an3 
 |-  ( ( G e. GrpOp /\ x e. X /\ y e. X ) -> ( x G ( ( inv ` G ) ` y ) ) e. X )
8 7 3expib 
 |-  ( G e. GrpOp -> ( ( x e. X /\ y e. X ) -> ( x G ( ( inv ` G ) ` y ) ) e. X ) )
9 8 ralrimivv 
 |-  ( G e. GrpOp -> A. x e. X A. y e. X ( x G ( ( inv ` G ) ` y ) ) e. X )
10 eqid 
 |-  ( x e. X , y e. X |-> ( x G ( ( inv ` G ) ` y ) ) ) = ( x e. X , y e. X |-> ( x G ( ( inv ` G ) ` y ) ) )
11 10 fmpo 
 |-  ( 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 )
12 9 11 sylib 
 |-  ( G e. GrpOp -> ( x e. X , y e. X |-> ( x G ( ( inv ` G ) ` y ) ) ) : ( X X. X ) --> X )
13 1 3 2 grpodivfval 
 |-  ( G e. GrpOp -> D = ( x e. X , y e. X |-> ( x G ( ( inv ` G ) ` y ) ) ) )
14 13 feq1d 
 |-  ( G e. GrpOp -> ( D : ( X X. X ) --> X <-> ( x e. X , y e. X |-> ( x G ( ( inv ` G ) ` y ) ) ) : ( X X. X ) --> X ) )
15 12 14 mpbird 
 |-  ( G e. GrpOp -> D : ( X X. X ) --> X )