This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )