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Metamath Proof Explorer

Theorem grpoinvval

Description: The inverse of a group element. (Contributed by NM, 26-Oct-2006) (Revised by Mario Carneiro, 15-Dec-2013) (New usage is discouraged.)

Ref Expression
Hypotheses grpinvfval.1 
|- X = ran G
grpinvfval.2 
|- U = ( GId ` G )
grpinvfval.3 
|- N = ( inv ` G )
Assertion grpoinvval 
|- ( ( G e. GrpOp /\ A e. X ) -> ( N ` A ) = ( iota_ y e. X ( y G A ) = U ) )

Proof

Step Hyp Ref Expression
1 grpinvfval.1 
 |-  X = ran G
2 grpinvfval.2 
 |-  U = ( GId ` G )
3 grpinvfval.3 
 |-  N = ( inv ` G )
4 1 2 3 grpoinvfval 
 |-  ( G e. GrpOp -> N = ( x e. X |-> ( iota_ y e. X ( y G x ) = U ) ) )
5 4 fveq1d 
 |-  ( G e. GrpOp -> ( N ` A ) = ( ( x e. X |-> ( iota_ y e. X ( y G x ) = U ) ) ` A ) )
6 oveq2 
 |-  ( x = A -> ( y G x ) = ( y G A ) )
7 6 eqeq1d 
 |-  ( x = A -> ( ( y G x ) = U <-> ( y G A ) = U ) )
8 7 riotabidv 
 |-  ( x = A -> ( iota_ y e. X ( y G x ) = U ) = ( iota_ y e. X ( y G A ) = U ) )
9 eqid 
 |-  ( x e. X |-> ( iota_ y e. X ( y G x ) = U ) ) = ( x e. X |-> ( iota_ y e. X ( y G x ) = U ) )
10 riotaex 
 |-  ( iota_ y e. X ( y G A ) = U ) e. _V
11 8 9 10 fvmpt 
 |-  ( A e. X -> ( ( x e. X |-> ( iota_ y e. X ( y G x ) = U ) ) ` A ) = ( iota_ y e. X ( y G A ) = U ) )
12 5 11 sylan9eq 
 |-  ( ( G e. GrpOp /\ A e. X ) -> ( N ` A ) = ( iota_ y e. X ( y G A ) = U ) )