Theorem evpmid

Description: The identity is an even permutation. (Contributed by Thierry Arnoux, 18-Sep-2023)

	Ref	Expression
Hypothesis	evpmid.1	⊢ 𝑆 = ( SymGrp ‘ 𝐷 )
Assertion	evpmid	⊢ ( 𝐷 ∈ Fin → ( I ↾ 𝐷 ) ∈ ( pmEven ‘ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		evpmid.1	⊢ 𝑆 = ( SymGrp ‘ 𝐷 )
2	1	idresperm	⊢ ( 𝐷 ∈ Fin → ( I ↾ 𝐷 ) ∈ ( Base ‘ 𝑆 ) )
3		eqid	⊢ ( pmSgn ‘ 𝐷 ) = ( pmSgn ‘ 𝐷 )
4	3	psgnid	⊢ ( 𝐷 ∈ Fin → ( ( pmSgn ‘ 𝐷 ) ‘ ( I ↾ 𝐷 ) ) = 1 )
5		eqid	⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
6	1 5 3	psgnevpmb	⊢ ( 𝐷 ∈ Fin → ( ( I ↾ 𝐷 ) ∈ ( pmEven ‘ 𝐷 ) ↔ ( ( I ↾ 𝐷 ) ∈ ( Base ‘ 𝑆 ) ∧ ( ( pmSgn ‘ 𝐷 ) ‘ ( I ↾ 𝐷 ) ) = 1 ) ) )
7	2 4 6	mpbir2and	⊢ ( 𝐷 ∈ Fin → ( I ↾ 𝐷 ) ∈ ( pmEven ‘ 𝐷 ) )