Theorem srngcl

Description: The involution function in a star ring is closed in the ring. (Contributed by Mario Carneiro, 6-Oct-2015)

	Ref	Expression
Hypotheses	srngcl.i	⊢ ∗ = ( *𝑟 ‘ 𝑅 )
	srngcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
Assertion	srngcl	⊢ ( ( 𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ) → ( ∗ ‘ 𝑋 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		srngcl.i	⊢ ∗ = ( *𝑟 ‘ 𝑅 )
2		srngcl.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
3		eqid	⊢ ( *rf ‘ 𝑅 ) = ( *rf ‘ 𝑅 )
4	2 1 3	stafval	⊢ ( 𝑋 ∈ 𝐵 → ( ( *rf ‘ 𝑅 ) ‘ 𝑋 ) = ( ∗ ‘ 𝑋 ) )
5	4	adantl	⊢ ( ( 𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ) → ( ( *rf ‘ 𝑅 ) ‘ 𝑋 ) = ( ∗ ‘ 𝑋 ) )
6	3 2	srngf1o	⊢ ( 𝑅 ∈ *-Ring → ( *rf ‘ 𝑅 ) : 𝐵 –1-1-onto→ 𝐵 )
7		f1of	⊢ ( ( *rf ‘ 𝑅 ) : 𝐵 –1-1-onto→ 𝐵 → ( *rf ‘ 𝑅 ) : 𝐵 ⟶ 𝐵 )
8	6 7	syl	⊢ ( 𝑅 ∈ *-Ring → ( *rf ‘ 𝑅 ) : 𝐵 ⟶ 𝐵 )
9	8	ffvelrnda	⊢ ( ( 𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ) → ( ( *rf ‘ 𝑅 ) ‘ 𝑋 ) ∈ 𝐵 )
10	5 9	eqeltrrd	⊢ ( ( 𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ) → ( ∗ ‘ 𝑋 ) ∈ 𝐵 )