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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 
|- ( ph -> G e. CMnd )
gsumcl.a 
|- ( ph -> A e. V )
gsumcl.f 
|- ( ph -> F : A --> B )
gsumcl2.w 
|- ( ph -> ( F supp .0. ) e. Fin )
Assertion gsumcl2 
|- ( ph -> ( G gsum F ) e. B )

Proof

Step Hyp Ref Expression
1 gsumcl.b 
 |-  B = ( Base ` G )
2 gsumcl.z 
 |-  .0. = ( 0g ` G )
3 gsumcl.g 
 |-  ( ph -> G e. CMnd )
4 gsumcl.a 
 |-  ( ph -> A e. V )
5 gsumcl.f 
 |-  ( ph -> F : A --> B )
6 gsumcl2.w 
 |-  ( ph -> ( F supp .0. ) e. Fin )
7 eqid 
 |-  ( Cntz ` G ) = ( Cntz ` G )
8 cmnmnd 
 |-  ( G e. CMnd -> G e. Mnd )
9 3 8 syl 
 |-  ( ph -> G e. Mnd )
10 1 7 3 5 cntzcmnf 
 |-  ( ph -> ran F C_ ( ( Cntz ` G ) ` ran F ) )
11 1 2 7 9 4 5 10 6 gsumzcl2 
 |-  ( ph -> ( G gsum F ) e. B )