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 = ( 0g ‘ 𝐺 )
gsumzsubmcl.z ⊢ 𝑍 = ( Cntz ‘ 𝐺 )
gsumzsubmcl.g ⊢ ( 𝜑 → 𝐺 ∈ Mnd )
gsumzsubmcl.a ⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
gsumzsubmcl.s ⊢ ( 𝜑 → 𝑆 ∈ ( SubMnd ‘ 𝐺 ) )
gsumzsubmcl.f ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 )
gsumzsubmcl.c ⊢ ( 𝜑 → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) )
gsumzsubmcl.w ⊢ ( 𝜑 → 𝐹 finSupp 0 )
Assertion gsumzsubmcl ( 𝜑 → ( 𝐺 Σg 𝐹 ) ∈ 𝑆 )

Proof

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