Theorem gsumcl2

Description: Closure of a finite group sum. This theorem has a weaker hypothesis than gsumcl , because it is not required that F is a function (actually, the hypothesis always holds for any proper class F ). (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 )
	gsumcl2.w	⊢ ( 𝜑 → ( 𝐹 supp 0 ) ∈ Fin )
Assertion	gsumcl2	⊢ ( 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		gsumcl2.w	⊢ ( 𝜑 → ( 𝐹 supp 0 ) ∈ Fin )
7		eqid	⊢ ( Cntz ‘ 𝐺 ) = ( Cntz ‘ 𝐺 )
8		cmnmnd	⊢ ( 𝐺 ∈ CMnd → 𝐺 ∈ Mnd )
9	3 8	syl	⊢ ( 𝜑 → 𝐺 ∈ Mnd )
10	1 7 3 5	cntzcmnf	⊢ ( 𝜑 → ran 𝐹 ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ran 𝐹 ) )
11	1 2 7 9 4 5 10 6	gsumzcl2	⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) ∈ 𝐵 )