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Metamath Proof Explorer

Theorem gimco

Description: The composition of group isomorphisms is a group isomorphism. (Contributed by Mario Carneiro, 21-Apr-2016)

Ref Expression
Assertion gimco ( ( 𝐹 ∈ ( 𝑇 GrpIso 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 GrpIso 𝑇 ) ) → ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑆 GrpIso 𝑈 ) )

Proof

Step Hyp Ref Expression
1 isgim2 ⊢ ( 𝐹 ∈ ( 𝑇 GrpIso 𝑈 ) ↔ ( 𝐹 ∈ ( 𝑇 GrpHom 𝑈 ) ∧ ◡ 𝐹 ∈ ( 𝑈 GrpHom 𝑇 ) ) )
2 isgim2 ⊢ ( 𝐺 ∈ ( 𝑆 GrpIso 𝑇 ) ↔ ( 𝐺 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ ◡ 𝐺 ∈ ( 𝑇 GrpHom 𝑆 ) ) )
3 ghmco ⊢ ( ( 𝐹 ∈ ( 𝑇 GrpHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 GrpHom 𝑇 ) ) → ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑆 GrpHom 𝑈 ) )
4 cnvco ⊢ ◡ ( 𝐹 ∘ 𝐺 ) = ( ◡ 𝐺 ∘ ◡ 𝐹 )
5 ghmco ⊢ ( ( ◡ 𝐺 ∈ ( 𝑇 GrpHom 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑈 GrpHom 𝑇 ) ) → ( ◡ 𝐺 ∘ ◡ 𝐹 ) ∈ ( 𝑈 GrpHom 𝑆 ) )
6 5 ancoms ⊢ ( ( ◡ 𝐹 ∈ ( 𝑈 GrpHom 𝑇 ) ∧ ◡ 𝐺 ∈ ( 𝑇 GrpHom 𝑆 ) ) → ( ◡ 𝐺 ∘ ◡ 𝐹 ) ∈ ( 𝑈 GrpHom 𝑆 ) )
7 4 6 eqeltrid ⊢ ( ( ◡ 𝐹 ∈ ( 𝑈 GrpHom 𝑇 ) ∧ ◡ 𝐺 ∈ ( 𝑇 GrpHom 𝑆 ) ) → ◡ ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑈 GrpHom 𝑆 ) )
8 3 7 anim12i ⊢ ( ( ( 𝐹 ∈ ( 𝑇 GrpHom 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 GrpHom 𝑇 ) ) ∧ ( ◡ 𝐹 ∈ ( 𝑈 GrpHom 𝑇 ) ∧ ◡ 𝐺 ∈ ( 𝑇 GrpHom 𝑆 ) ) ) → ( ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑆 GrpHom 𝑈 ) ∧ ◡ ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑈 GrpHom 𝑆 ) ) )
9 8 an4s ⊢ ( ( ( 𝐹 ∈ ( 𝑇 GrpHom 𝑈 ) ∧ ◡ 𝐹 ∈ ( 𝑈 GrpHom 𝑇 ) ) ∧ ( 𝐺 ∈ ( 𝑆 GrpHom 𝑇 ) ∧ ◡ 𝐺 ∈ ( 𝑇 GrpHom 𝑆 ) ) ) → ( ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑆 GrpHom 𝑈 ) ∧ ◡ ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑈 GrpHom 𝑆 ) ) )
10 1 2 9 syl2anb ⊢ ( ( 𝐹 ∈ ( 𝑇 GrpIso 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 GrpIso 𝑇 ) ) → ( ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑆 GrpHom 𝑈 ) ∧ ◡ ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑈 GrpHom 𝑆 ) ) )
11 isgim2 ⊢ ( ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑆 GrpIso 𝑈 ) ↔ ( ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑆 GrpHom 𝑈 ) ∧ ◡ ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑈 GrpHom 𝑆 ) ) )
12 10 11 sylibr ⊢ ( ( 𝐹 ∈ ( 𝑇 GrpIso 𝑈 ) ∧ 𝐺 ∈ ( 𝑆 GrpIso 𝑇 ) ) → ( 𝐹 ∘ 𝐺 ) ∈ ( 𝑆 GrpIso 𝑈 ) )