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Metamath Proof Explorer

Theorem grpo2inv

Description: Double inverse law for groups. Lemma 2.2.1(c) of Herstein p. 55. (Contributed by NM, 27-Oct-2006) (New usage is discouraged.)

Ref Expression
Hypotheses grpasscan1.1 
|- X = ran G
grpasscan1.2 
|- N = ( inv ` G )
Assertion grpo2inv 
|- ( ( G e. GrpOp /\ A e. X ) -> ( N ` ( N ` A ) ) = A )

Proof

Step Hyp Ref Expression
1 grpasscan1.1 
 |-  X = ran G
2 grpasscan1.2 
 |-  N = ( inv ` G )
3 1 2 grpoinvcl 
 |-  ( ( G e. GrpOp /\ A e. X ) -> ( N ` A ) e. X )
4 eqid 
 |-  ( GId ` G ) = ( GId ` G )
5 1 4 2 grporinv 
 |-  ( ( G e. GrpOp /\ ( N ` A ) e. X ) -> ( ( N ` A ) G ( N ` ( N ` A ) ) ) = ( GId ` G ) )
6 3 5 syldan 
 |-  ( ( G e. GrpOp /\ A e. X ) -> ( ( N ` A ) G ( N ` ( N ` A ) ) ) = ( GId ` G ) )
7 1 4 2 grpolinv 
 |-  ( ( G e. GrpOp /\ A e. X ) -> ( ( N ` A ) G A ) = ( GId ` G ) )
8 6 7 eqtr4d 
 |-  ( ( G e. GrpOp /\ A e. X ) -> ( ( N ` A ) G ( N ` ( N ` A ) ) ) = ( ( N ` A ) G A ) )
9 1 2 grpoinvcl 
 |-  ( ( G e. GrpOp /\ ( N ` A ) e. X ) -> ( N ` ( N ` A ) ) e. X )
10 3 9 syldan 
 |-  ( ( G e. GrpOp /\ A e. X ) -> ( N ` ( N ` A ) ) e. X )
11 simpr 
 |-  ( ( G e. GrpOp /\ A e. X ) -> A e. X )
12 10 11 3 3jca 
 |-  ( ( G e. GrpOp /\ A e. X ) -> ( ( N ` ( N ` A ) ) e. X /\ A e. X /\ ( N ` A ) e. X ) )
13 1 grpolcan 
 |-  ( ( G e. GrpOp /\ ( ( N ` ( N ` A ) ) e. X /\ A e. X /\ ( N ` A ) e. X ) ) -> ( ( ( N ` A ) G ( N ` ( N ` A ) ) ) = ( ( N ` A ) G A ) <-> ( N ` ( N ` A ) ) = A ) )
14 12 13 syldan 
 |-  ( ( G e. GrpOp /\ A e. X ) -> ( ( ( N ` A ) G ( N ` ( N ` A ) ) ) = ( ( N ` A ) G A ) <-> ( N ` ( N ` A ) ) = A ) )
15 8 14 mpbid 
 |-  ( ( G e. GrpOp /\ A e. X ) -> ( N ` ( N ` A ) ) = A )