Metamath Proof Explorer
Description: Closure of a finite group sum. (Contributed by Mario Carneiro, 24-Apr-2016) (Revised by AV, 1-Jun-2019)
|
|
Ref |
Expression |
|
Hypotheses |
gsumzcl.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
|
|
gsumzcl.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
|
|
gsumzcl.z |
⊢ 𝑍 = ( Cntz ‘ 𝐺 ) |
|
|
gsumzcl.g |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
|
|
gsumzcl.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
|
|
gsumzcl.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
|
|
gsumzcl.c |
⊢ ( 𝜑 → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) ) |
|
|
gsumzcl.w |
⊢ ( 𝜑 → 𝐹 finSupp 0 ) |
|
Assertion |
gsumzcl |
⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) ∈ 𝐵 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
gsumzcl.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsumzcl.0 |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
gsumzcl.z |
⊢ 𝑍 = ( Cntz ‘ 𝐺 ) |
4 |
|
gsumzcl.g |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
5 |
|
gsumzcl.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
6 |
|
gsumzcl.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
7 |
|
gsumzcl.c |
⊢ ( 𝜑 → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) ) |
8 |
|
gsumzcl.w |
⊢ ( 𝜑 → 𝐹 finSupp 0 ) |
9 |
8
|
fsuppimpd |
⊢ ( 𝜑 → ( 𝐹 supp 0 ) ∈ Fin ) |
10 |
1 2 3 4 5 6 7 9
|
gsumzcl2 |
⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) ∈ 𝐵 ) |