Metamath Proof Explorer


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 ) ) → ( 𝐹𝐺 ) ∈ ( 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 ) ) → ( 𝐹𝐺 ) ∈ ( 𝑅 RingHom 𝑇 ) )
4 cnvco ( 𝐹𝐺 ) = ( 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 ) ) → ( 𝐹𝐺 ) ∈ ( 𝑇 RingHom 𝑅 ) )
8 3 7 anim12i ( ( ( 𝐹 ∈ ( 𝑆 RingHom 𝑇 ) ∧ 𝐺 ∈ ( 𝑅 RingHom 𝑆 ) ) ∧ ( 𝐹 ∈ ( 𝑇 RingHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RingHom 𝑅 ) ) ) → ( ( 𝐹𝐺 ) ∈ ( 𝑅 RingHom 𝑇 ) ∧ ( 𝐹𝐺 ) ∈ ( 𝑇 RingHom 𝑅 ) ) )
9 8 an4s ( ( ( 𝐹 ∈ ( 𝑆 RingHom 𝑇 ) ∧ 𝐹 ∈ ( 𝑇 RingHom 𝑆 ) ) ∧ ( 𝐺 ∈ ( 𝑅 RingHom 𝑆 ) ∧ 𝐺 ∈ ( 𝑆 RingHom 𝑅 ) ) ) → ( ( 𝐹𝐺 ) ∈ ( 𝑅 RingHom 𝑇 ) ∧ ( 𝐹𝐺 ) ∈ ( 𝑇 RingHom 𝑅 ) ) )
10 1 2 9 syl2anb ( ( 𝐹 ∈ ( 𝑆 RingIso 𝑇 ) ∧ 𝐺 ∈ ( 𝑅 RingIso 𝑆 ) ) → ( ( 𝐹𝐺 ) ∈ ( 𝑅 RingHom 𝑇 ) ∧ ( 𝐹𝐺 ) ∈ ( 𝑇 RingHom 𝑅 ) ) )
11 isrim0 ( ( 𝐹𝐺 ) ∈ ( 𝑅 RingIso 𝑇 ) ↔ ( ( 𝐹𝐺 ) ∈ ( 𝑅 RingHom 𝑇 ) ∧ ( 𝐹𝐺 ) ∈ ( 𝑇 RingHom 𝑅 ) ) )
12 10 11 sylibr ( ( 𝐹 ∈ ( 𝑆 RingIso 𝑇 ) ∧ 𝐺 ∈ ( 𝑅 RingIso 𝑆 ) ) → ( 𝐹𝐺 ) ∈ ( 𝑅 RingIso 𝑇 ) )