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Metamath Proof Explorer

Theorem rngimcnv

Description: The converse of an isomorphism of non-unital rings is an isomorphism of non-unital rings. (Contributed by AV, 27-Feb-2025)

Ref Expression
Assertion rngimcnv ( 𝐹 ∈ ( 𝑆 RngIso 𝑇 ) → ◡ 𝐹 ∈ ( 𝑇 RngIso 𝑆 ) )

Proof

Step Hyp Ref Expression
1 rngimrcl ⊢ ( 𝐹 ∈ ( 𝑆 RngIso 𝑇 ) → ( 𝑆 ∈ V ∧ 𝑇 ∈ V ) )
2 isrngim ⊢ ( ( 𝑆 ∈ V ∧ 𝑇 ∈ V ) → ( 𝐹 ∈ ( 𝑆 RngIso 𝑇 ) ↔ ( 𝐹 ∈ ( 𝑆 RngHom 𝑇 ) ∧ ◡ 𝐹 ∈ ( 𝑇 RngHom 𝑆 ) ) ) )
3 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
4 eqid ⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 )
5 3 4 rnghmf ⊢ ( 𝐹 ∈ ( 𝑆 RngHom 𝑇 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) )
6 frel ⊢ ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) → Rel 𝐹 )
7 dfrel2 ⊢ ( Rel 𝐹 ↔ ◡ ◡ 𝐹 = 𝐹 )
8 6 7 sylib ⊢ ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) → ◡ ◡ 𝐹 = 𝐹 )
9 5 8 syl ⊢ ( 𝐹 ∈ ( 𝑆 RngHom 𝑇 ) → ◡ ◡ 𝐹 = 𝐹 )
10 id ⊢ ( 𝐹 ∈ ( 𝑆 RngHom 𝑇 ) → 𝐹 ∈ ( 𝑆 RngHom 𝑇 ) )
11 9 10 eqeltrd ⊢ ( 𝐹 ∈ ( 𝑆 RngHom 𝑇 ) → ◡ ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑇 ) )
12 11 anim1ci ⊢ ( ( 𝐹 ∈ ( 𝑆 RngHom 𝑇 ) ∧ ◡ 𝐹 ∈ ( 𝑇 RngHom 𝑆 ) ) → ( ◡ 𝐹 ∈ ( 𝑇 RngHom 𝑆 ) ∧ ◡ ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑇 ) ) )
13 isrngim ⊢ ( ( 𝑇 ∈ V ∧ 𝑆 ∈ V ) → ( ◡ 𝐹 ∈ ( 𝑇 RngIso 𝑆 ) ↔ ( ◡ 𝐹 ∈ ( 𝑇 RngHom 𝑆 ) ∧ ◡ ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑇 ) ) ) )
14 13 ancoms ⊢ ( ( 𝑆 ∈ V ∧ 𝑇 ∈ V ) → ( ◡ 𝐹 ∈ ( 𝑇 RngIso 𝑆 ) ↔ ( ◡ 𝐹 ∈ ( 𝑇 RngHom 𝑆 ) ∧ ◡ ◡ 𝐹 ∈ ( 𝑆 RngHom 𝑇 ) ) ) )
15 12 14 imbitrrid ⊢ ( ( 𝑆 ∈ V ∧ 𝑇 ∈ V ) → ( ( 𝐹 ∈ ( 𝑆 RngHom 𝑇 ) ∧ ◡ 𝐹 ∈ ( 𝑇 RngHom 𝑆 ) ) → ◡ 𝐹 ∈ ( 𝑇 RngIso 𝑆 ) ) )
16 2 15 sylbid ⊢ ( ( 𝑆 ∈ V ∧ 𝑇 ∈ V ) → ( 𝐹 ∈ ( 𝑆 RngIso 𝑇 ) → ◡ 𝐹 ∈ ( 𝑇 RngIso 𝑆 ) ) )
17 1 16 mpcom ⊢ ( 𝐹 ∈ ( 𝑆 RngIso 𝑇 ) → ◡ 𝐹 ∈ ( 𝑇 RngIso 𝑆 ) )