Metamath Proof Explorer

Theorem rimcnv

Description: The converse of a ring isomorphism is a ring isomorphism. (Contributed by SN, 10-Jan-2025)

Ref Expression
Assertion rimcnv ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → ◡ 𝐹 ∈ ( 𝑆 RingIso 𝑅 ) )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
2 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
3 1 2 rhmf ⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) )
4 frel ⊢ ( 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) → Rel 𝐹 )
5 dfrel2 ⊢ ( Rel 𝐹 ↔ ◡ ◡ 𝐹 = 𝐹 )
6 4 5 sylib ⊢ ( 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) → ◡ ◡ 𝐹 = 𝐹 )
7 3 6 syl ⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ◡ ◡ 𝐹 = 𝐹 )
8 id ⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) )
9 7 8 eqeltrd ⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ◡ ◡ 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) )
10 9 anim1ci ⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑆 RingHom 𝑅 ) ) → ( ◡ 𝐹 ∈ ( 𝑆 RingHom 𝑅 ) ∧ ◡ ◡ 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) )
11 isrim0 ⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑆 RingHom 𝑅 ) ) )
12 isrim0 ⊢ ( ◡ 𝐹 ∈ ( 𝑆 RingIso 𝑅 ) ↔ ( ◡ 𝐹 ∈ ( 𝑆 RingHom 𝑅 ) ∧ ◡ ◡ 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) )
13 10 11 12 3imtr4i ⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → ◡ 𝐹 ∈ ( 𝑆 RingIso 𝑅 ) )