Metamath Proof Explorer
Description: Closure of a group sum in a subgroup. (Contributed by Mario Carneiro, 15-Dec-2014) (Revised by AV, 3-Jun-2019)
|
|
Ref |
Expression |
|
Hypotheses |
gsumsubgcl.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
|
|
gsumsubgcl.g |
⊢ ( 𝜑 → 𝐺 ∈ Abel ) |
|
|
gsumsubgcl.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
|
|
gsumsubgcl.s |
⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
|
|
gsumsubgcl.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 ) |
|
|
gsumsubgcl.w |
⊢ ( 𝜑 → 𝐹 finSupp 0 ) |
|
Assertion |
gsumsubgcl |
⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) ∈ 𝑆 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
gsumsubgcl.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
2 |
|
gsumsubgcl.g |
⊢ ( 𝜑 → 𝐺 ∈ Abel ) |
3 |
|
gsumsubgcl.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
4 |
|
gsumsubgcl.s |
⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ) |
5 |
|
gsumsubgcl.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 ) |
6 |
|
gsumsubgcl.w |
⊢ ( 𝜑 → 𝐹 finSupp 0 ) |
7 |
|
ablcmn |
⊢ ( 𝐺 ∈ Abel → 𝐺 ∈ CMnd ) |
8 |
2 7
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ CMnd ) |
9 |
|
subgsubm |
⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ) |
10 |
4 9
|
syl |
⊢ ( 𝜑 → 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ) |
11 |
1 8 3 10 5 6
|
gsumsubmcl |
⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) ∈ 𝑆 ) |