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 ∈ GrpOp ∧ 𝐴 ∈ 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 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) ∈ 𝑋 )
4		eqid	⊢ ( GId ‘ 𝐺 ) = ( GId ‘ 𝐺 )
5	1 4 2	grporinv	⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝑁 ‘ 𝐴 ) ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝐴 ) 𝐺 ( 𝑁 ‘ ( 𝑁 ‘ 𝐴 ) ) ) = ( GId ‘ 𝐺 ) )
6	3 5	syldan	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝐴 ) 𝐺 ( 𝑁 ‘ ( 𝑁 ‘ 𝐴 ) ) ) = ( GId ‘ 𝐺 ) )
7	1 4 2	grpolinv	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝐴 ) 𝐺 𝐴 ) = ( GId ‘ 𝐺 ) )
8	6 7	eqtr4d	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝐴 ) 𝐺 ( 𝑁 ‘ ( 𝑁 ‘ 𝐴 ) ) ) = ( ( 𝑁 ‘ 𝐴 ) 𝐺 𝐴 ) )
9	1 2	grpoinvcl	⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝑁 ‘ 𝐴 ) ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝐴 ) ) ∈ 𝑋 )
10	3 9	syldan	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝐴 ) ) ∈ 𝑋 )
11		simpr	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → 𝐴 ∈ 𝑋 )
12	10 11 3	3jca	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑁 ‘ ( 𝑁 ‘ 𝐴 ) ) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑁 ‘ 𝐴 ) ∈ 𝑋 ) )
13	1	grpolcan	⊢ ( ( 𝐺 ∈ GrpOp ∧ ( ( 𝑁 ‘ ( 𝑁 ‘ 𝐴 ) ) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ ( 𝑁 ‘ 𝐴 ) ∈ 𝑋 ) ) → ( ( ( 𝑁 ‘ 𝐴 ) 𝐺 ( 𝑁 ‘ ( 𝑁 ‘ 𝐴 ) ) ) = ( ( 𝑁 ‘ 𝐴 ) 𝐺 𝐴 ) ↔ ( 𝑁 ‘ ( 𝑁 ‘ 𝐴 ) ) = 𝐴 ) )
14	12 13	syldan	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( ( ( 𝑁 ‘ 𝐴 ) 𝐺 ( 𝑁 ‘ ( 𝑁 ‘ 𝐴 ) ) ) = ( ( 𝑁 ‘ 𝐴 ) 𝐺 𝐴 ) ↔ ( 𝑁 ‘ ( 𝑁 ‘ 𝐴 ) ) = 𝐴 ) )
15	8 14	mpbid	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ ( 𝑁 ‘ 𝐴 ) ) = 𝐴 )