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Metamath Proof Explorer

Theorem srngnvl

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

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

Proof

Step Hyp Ref Expression
1 srngcl.i ⊢ ∗ = ( *𝑟 ‘ 𝑅 )
2 srngcl.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
3 1 2 srngcl ⊢ ( ( 𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ) → ( ∗ ‘ 𝑋 ) ∈ 𝐵 )
4 eqid ⊢ ( *rf ‘ 𝑅 ) = ( *rf ‘ 𝑅 )
5 2 1 4 stafval ⊢ ( ( ∗ ‘ 𝑋 ) ∈ 𝐵 → ( ( *rf ‘ 𝑅 ) ‘ ( ∗ ‘ 𝑋 ) ) = ( ∗ ‘ ( ∗ ‘ 𝑋 ) ) )
6 3 5 syl ⊢ ( ( 𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ) → ( ( *rf ‘ 𝑅 ) ‘ ( ∗ ‘ 𝑋 ) ) = ( ∗ ‘ ( ∗ ‘ 𝑋 ) ) )
7 4 srngcnv ⊢ ( 𝑅 ∈ *-Ring → ( *rf ‘ 𝑅 ) = ◡ ( *rf ‘ 𝑅 ) )
8 7 adantr ⊢ ( ( 𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ) → ( *rf ‘ 𝑅 ) = ◡ ( *rf ‘ 𝑅 ) )
9 8 fveq1d ⊢ ( ( 𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ) → ( ( *rf ‘ 𝑅 ) ‘ ( ( *rf ‘ 𝑅 ) ‘ 𝑋 ) ) = ( ◡ ( *rf ‘ 𝑅 ) ‘ ( ( *rf ‘ 𝑅 ) ‘ 𝑋 ) ) )
10 2 1 4 stafval ⊢ ( 𝑋 ∈ 𝐵 → ( ( *rf ‘ 𝑅 ) ‘ 𝑋 ) = ( ∗ ‘ 𝑋 ) )
11 10 adantl ⊢ ( ( 𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ) → ( ( *rf ‘ 𝑅 ) ‘ 𝑋 ) = ( ∗ ‘ 𝑋 ) )
12 11 fveq2d ⊢ ( ( 𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ) → ( ( *rf ‘ 𝑅 ) ‘ ( ( *rf ‘ 𝑅 ) ‘ 𝑋 ) ) = ( ( *rf ‘ 𝑅 ) ‘ ( ∗ ‘ 𝑋 ) ) )
13 4 2 srngf1o ⊢ ( 𝑅 ∈ *-Ring → ( *rf ‘ 𝑅 ) : 𝐵 –1-1-onto→ 𝐵 )
14 f1ocnvfv1 ⊢ ( ( ( *rf ‘ 𝑅 ) : 𝐵 –1-1-onto→ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( ◡ ( *rf ‘ 𝑅 ) ‘ ( ( *rf ‘ 𝑅 ) ‘ 𝑋 ) ) = 𝑋 )
15 13 14 sylan ⊢ ( ( 𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ) → ( ◡ ( *rf ‘ 𝑅 ) ‘ ( ( *rf ‘ 𝑅 ) ‘ 𝑋 ) ) = 𝑋 )
16 9 12 15 3eqtr3d ⊢ ( ( 𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ) → ( ( *rf ‘ 𝑅 ) ‘ ( ∗ ‘ 𝑋 ) ) = 𝑋 )
17 6 16 eqtr3d ⊢ ( ( 𝑅 ∈ *-Ring ∧ 𝑋 ∈ 𝐵 ) → ( ∗ ‘ ( ∗ ‘ 𝑋 ) ) = 𝑋 )