Metamath Proof Explorer

Theorem rimisrngim

Description: Each unital ring isomorphism is a non-unital ring isomorphism. (Contributed by AV, 30-Mar-2025)

Ref Expression
Assertion rimisrngim ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) )

Proof

Step Hyp Ref Expression
1 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
2 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
3 1 2 isrim ⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) )
4 rhmisrnghm ⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) )
5 4 anim1i ⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) → ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) )
6 3 5 sylbi ⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) )
7 rimrcl ⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → ( 𝑅 ∈ V ∧ 𝑆 ∈ V ) )
8 1 2 isrngim2 ⊢ ( ( 𝑅 ∈ V ∧ 𝑆 ∈ V ) → ( 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) ) )
9 7 8 syl ⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → ( 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RngHom 𝑆 ) ∧ 𝐹 : ( Base ‘ 𝑅 ) –1-1-onto→ ( Base ‘ 𝑆 ) ) ) )
10 6 9 mpbird ⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → 𝐹 ∈ ( 𝑅 RngIso 𝑆 ) )