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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 )