Theorem gsumcl

Description: Closure of a finite group sum. (Contributed by Mario Carneiro, 15-Dec-2014) (Revised by Mario Carneiro, 24-Apr-2016) (Revised by AV, 3-Jun-2019)

	Ref	Expression
Hypotheses	gsumcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
	gsumcl.z	⊢ 0 = ( 0g ‘ 𝐺 )
	gsumcl.g	⊢ ( 𝜑 → 𝐺 ∈ CMnd )
	gsumcl.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	gsumcl.f	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
	gsumcl.w	⊢ ( 𝜑 → 𝐹 finSupp 0 )
Assertion	gsumcl	⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		gsumcl.b	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		gsumcl.z	⊢ 0 = ( 0g ‘ 𝐺 )
3		gsumcl.g	⊢ ( 𝜑 → 𝐺 ∈ CMnd )
4		gsumcl.a	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
5		gsumcl.f	⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
6		gsumcl.w	⊢ ( 𝜑 → 𝐹 finSupp 0 )
7	6	fsuppimpd	⊢ ( 𝜑 → ( 𝐹 supp 0 ) ∈ Fin )
8	1 2 3 4 5 7	gsumcl2	⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) ∈ 𝐵 )