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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 ) ) → ( 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 ) )