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 ) )