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 ) )