Theorem rimco

Description: The composition of ring isomorphisms is a ring isomorphism. (Contributed by SN, 17-Jan-2025)

	Ref	Expression
Assertion	rimco	⊢ ( ( 𝐹 ∈ ( 𝑆 RingIso 𝑇 ) ∧ 𝐺 ∈ ( 𝑅 RingIso 𝑆 ) ) → ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑅 RingIso 𝑇 ) )

Proof

Step	Hyp	Ref	Expression
1		isrim0	⊢ ( 𝐹 ∈ ( 𝑆 RingIso 𝑇 ) ↔ ( 𝐹 ∈ ( 𝑆 RingHom 𝑇 ) ∧ ◡ 𝐹 ∈ ( 𝑇 RingHom 𝑆 ) ) )
2		isrim0	⊢ ( 𝐺 ∈ ( 𝑅 RingIso 𝑆 ) ↔ ( 𝐺 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ◡ 𝐺 ∈ ( 𝑆 RingHom 𝑅 ) ) )
3		rhmco	⊢ ( ( 𝐹 ∈ ( 𝑆 RingHom 𝑇 ) ∧ 𝐺 ∈ ( 𝑅 RingHom 𝑆 ) ) → ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑅 RingHom 𝑇 ) )
4		cnvco	⊢ ◡ ( 𝐹 ∘ 𝐺 ) = ( ◡ 𝐺 ∘ ◡ 𝐹 )
5		rhmco	⊢ ( ( ◡ 𝐺 ∈ ( 𝑆 RingHom 𝑅 ) ∧ ◡ 𝐹 ∈ ( 𝑇 RingHom 𝑆 ) ) → ( ◡ 𝐺 ∘ ◡ 𝐹 ) ∈ ( 𝑇 RingHom 𝑅 ) )
6	5	ancoms	⊢ ( ( ◡ 𝐹 ∈ ( 𝑇 RingHom 𝑆 ) ∧ ◡ 𝐺 ∈ ( 𝑆 RingHom 𝑅 ) ) → ( ◡ 𝐺 ∘ ◡ 𝐹 ) ∈ ( 𝑇 RingHom 𝑅 ) )
7	4 6	eqeltrid	⊢ ( ( ◡ 𝐹 ∈ ( 𝑇 RingHom 𝑆 ) ∧ ◡ 𝐺 ∈ ( 𝑆 RingHom 𝑅 ) ) → ◡ ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑇 RingHom 𝑅 ) )
8	3 7	anim12i	⊢ ( ( ( 𝐹 ∈ ( 𝑆 RingHom 𝑇 ) ∧ 𝐺 ∈ ( 𝑅 RingHom 𝑆 ) ) ∧ ( ◡ 𝐹 ∈ ( 𝑇 RingHom 𝑆 ) ∧ ◡ 𝐺 ∈ ( 𝑆 RingHom 𝑅 ) ) ) → ( ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑅 RingHom 𝑇 ) ∧ ◡ ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑇 RingHom 𝑅 ) ) )
9	8	an4s	⊢ ( ( ( 𝐹 ∈ ( 𝑆 RingHom 𝑇 ) ∧ ◡ 𝐹 ∈ ( 𝑇 RingHom 𝑆 ) ) ∧ ( 𝐺 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ◡ 𝐺 ∈ ( 𝑆 RingHom 𝑅 ) ) ) → ( ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑅 RingHom 𝑇 ) ∧ ◡ ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑇 RingHom 𝑅 ) ) )
10	1 2 9	syl2anb	⊢ ( ( 𝐹 ∈ ( 𝑆 RingIso 𝑇 ) ∧ 𝐺 ∈ ( 𝑅 RingIso 𝑆 ) ) → ( ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑅 RingHom 𝑇 ) ∧ ◡ ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑇 RingHom 𝑅 ) ) )
11		isrim0	⊢ ( ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑅 RingIso 𝑇 ) ↔ ( ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑅 RingHom 𝑇 ) ∧ ◡ ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑇 RingHom 𝑅 ) ) )
12	10 11	sylibr	⊢ ( ( 𝐹 ∈ ( 𝑆 RingIso 𝑇 ) ∧ 𝐺 ∈ ( 𝑅 RingIso 𝑆 ) ) → ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑅 RingIso 𝑇 ) )