Metamath Proof Explorer


Theorem gsumzsubmcl

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

Ref Expression
Hypotheses gsumzsubmcl.0 0 = ( 0g𝐺 )
gsumzsubmcl.z 𝑍 = ( Cntz ‘ 𝐺 )
gsumzsubmcl.g ( 𝜑𝐺 ∈ Mnd )
gsumzsubmcl.a ( 𝜑𝐴𝑉 )
gsumzsubmcl.s ( 𝜑𝑆 ∈ ( SubMnd ‘ 𝐺 ) )
gsumzsubmcl.f ( 𝜑𝐹 : 𝐴𝑆 )
gsumzsubmcl.c ( 𝜑 → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) )
gsumzsubmcl.w ( 𝜑𝐹 finSupp 0 )
Assertion gsumzsubmcl ( 𝜑 → ( 𝐺 Σg 𝐹 ) ∈ 𝑆 )

Proof

Step Hyp Ref Expression
1 gsumzsubmcl.0 0 = ( 0g𝐺 )
2 gsumzsubmcl.z 𝑍 = ( Cntz ‘ 𝐺 )
3 gsumzsubmcl.g ( 𝜑𝐺 ∈ Mnd )
4 gsumzsubmcl.a ( 𝜑𝐴𝑉 )
5 gsumzsubmcl.s ( 𝜑𝑆 ∈ ( SubMnd ‘ 𝐺 ) )
6 gsumzsubmcl.f ( 𝜑𝐹 : 𝐴𝑆 )
7 gsumzsubmcl.c ( 𝜑 → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) )
8 gsumzsubmcl.w ( 𝜑𝐹 finSupp 0 )
9 eqid ( Base ‘ ( 𝐺s 𝑆 ) ) = ( Base ‘ ( 𝐺s 𝑆 ) )
10 eqid ( 0g ‘ ( 𝐺s 𝑆 ) ) = ( 0g ‘ ( 𝐺s 𝑆 ) )
11 eqid ( Cntz ‘ ( 𝐺s 𝑆 ) ) = ( Cntz ‘ ( 𝐺s 𝑆 ) )
12 eqid ( 𝐺s 𝑆 ) = ( 𝐺s 𝑆 )
13 12 submmnd ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) → ( 𝐺s 𝑆 ) ∈ Mnd )
14 5 13 syl ( 𝜑 → ( 𝐺s 𝑆 ) ∈ Mnd )
15 12 submbas ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) → 𝑆 = ( Base ‘ ( 𝐺s 𝑆 ) ) )
16 5 15 syl ( 𝜑𝑆 = ( Base ‘ ( 𝐺s 𝑆 ) ) )
17 16 feq3d ( 𝜑 → ( 𝐹 : 𝐴𝑆𝐹 : 𝐴 ⟶ ( Base ‘ ( 𝐺s 𝑆 ) ) ) )
18 6 17 mpbid ( 𝜑𝐹 : 𝐴 ⟶ ( Base ‘ ( 𝐺s 𝑆 ) ) )
19 6 frnd ( 𝜑 → ran 𝐹𝑆 )
20 7 19 ssind ( 𝜑 → ran 𝐹 ⊆ ( ( 𝑍 ‘ ran 𝐹 ) ∩ 𝑆 ) )
21 12 2 11 resscntz ( ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ∧ ran 𝐹𝑆 ) → ( ( Cntz ‘ ( 𝐺s 𝑆 ) ) ‘ ran 𝐹 ) = ( ( 𝑍 ‘ ran 𝐹 ) ∩ 𝑆 ) )
22 5 19 21 syl2anc ( 𝜑 → ( ( Cntz ‘ ( 𝐺s 𝑆 ) ) ‘ ran 𝐹 ) = ( ( 𝑍 ‘ ran 𝐹 ) ∩ 𝑆 ) )
23 20 22 sseqtrrd ( 𝜑 → ran 𝐹 ⊆ ( ( Cntz ‘ ( 𝐺s 𝑆 ) ) ‘ ran 𝐹 ) )
24 12 1 subm0 ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) → 0 = ( 0g ‘ ( 𝐺s 𝑆 ) ) )
25 5 24 syl ( 𝜑0 = ( 0g ‘ ( 𝐺s 𝑆 ) ) )
26 8 25 breqtrd ( 𝜑𝐹 finSupp ( 0g ‘ ( 𝐺s 𝑆 ) ) )
27 9 10 11 14 4 18 23 26 gsumzcl ( 𝜑 → ( ( 𝐺s 𝑆 ) Σg 𝐹 ) ∈ ( Base ‘ ( 𝐺s 𝑆 ) ) )
28 4 5 6 12 gsumsubm ( 𝜑 → ( 𝐺 Σg 𝐹 ) = ( ( 𝐺s 𝑆 ) Σg 𝐹 ) )
29 27 28 16 3eltr4d ( 𝜑 → ( 𝐺 Σg 𝐹 ) ∈ 𝑆 )