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Metamath Proof Explorer

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 ffvelcdmda ⊢ ( ( 𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ) → ( ( *rf ‘ 𝑅 ) ‘ 𝑋 ) ∈ 𝐵 )
10 5 9 eqeltrrd ⊢ ( ( 𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ) → ( ∗ ‘ 𝑋 ) ∈ 𝐵 )