Metamath Proof Explorer


Theorem gsumsubmcl

Description: Closure of a group sum in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015) (Revised by Mario Carneiro, 24-Apr-2016) (Revised by AV, 3-Jun-2019)

Ref Expression
Hypotheses gsumsubmcl.z
|- .0. = ( 0g ` G )
gsumsubmcl.g
|- ( ph -> G e. CMnd )
gsumsubmcl.a
|- ( ph -> A e. V )
gsumsubmcl.s
|- ( ph -> S e. ( SubMnd ` G ) )
gsumsubmcl.f
|- ( ph -> F : A --> S )
gsumsubmcl.w
|- ( ph -> F finSupp .0. )
Assertion gsumsubmcl
|- ( ph -> ( G gsum F ) e. S )

Proof

Step Hyp Ref Expression
1 gsumsubmcl.z
 |-  .0. = ( 0g ` G )
2 gsumsubmcl.g
 |-  ( ph -> G e. CMnd )
3 gsumsubmcl.a
 |-  ( ph -> A e. V )
4 gsumsubmcl.s
 |-  ( ph -> S e. ( SubMnd ` G ) )
5 gsumsubmcl.f
 |-  ( ph -> F : A --> S )
6 gsumsubmcl.w
 |-  ( ph -> F finSupp .0. )
7 eqid
 |-  ( Cntz ` G ) = ( Cntz ` G )
8 cmnmnd
 |-  ( G e. CMnd -> G e. Mnd )
9 2 8 syl
 |-  ( ph -> G e. Mnd )
10 eqid
 |-  ( Base ` G ) = ( Base ` G )
11 10 submss
 |-  ( S e. ( SubMnd ` G ) -> S C_ ( Base ` G ) )
12 4 11 syl
 |-  ( ph -> S C_ ( Base ` G ) )
13 5 12 fssd
 |-  ( ph -> F : A --> ( Base ` G ) )
14 10 7 2 13 cntzcmnf
 |-  ( ph -> ran F C_ ( ( Cntz ` G ) ` ran F ) )
15 1 7 9 3 4 5 14 6 gsumzsubmcl
 |-  ( ph -> ( G gsum F ) e. S )