Step |
Hyp |
Ref |
Expression |
1 |
|
gsumsubmcl.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
2 |
|
gsumsubmcl.g |
⊢ ( 𝜑 → 𝐺 ∈ CMnd ) |
3 |
|
gsumsubmcl.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
4 |
|
gsumsubmcl.s |
⊢ ( 𝜑 → 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ) |
5 |
|
gsumsubmcl.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝑆 ) |
6 |
|
gsumsubmcl.w |
⊢ ( 𝜑 → 𝐹 finSupp 0 ) |
7 |
|
eqid |
⊢ ( Cntz ‘ 𝐺 ) = ( Cntz ‘ 𝐺 ) |
8 |
|
cmnmnd |
⊢ ( 𝐺 ∈ CMnd → 𝐺 ∈ Mnd ) |
9 |
2 8
|
syl |
⊢ ( 𝜑 → 𝐺 ∈ Mnd ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) |
11 |
10
|
submss |
⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) → 𝑆 ⊆ ( Base ‘ 𝐺 ) ) |
12 |
4 11
|
syl |
⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝐺 ) ) |
13 |
5 12
|
fssd |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ( Base ‘ 𝐺 ) ) |
14 |
10 7 2 13
|
cntzcmnf |
⊢ ( 𝜑 → ran 𝐹 ⊆ ( ( Cntz ‘ 𝐺 ) ‘ ran 𝐹 ) ) |
15 |
1 7 9 3 4 5 14 6
|
gsumzsubmcl |
⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) ∈ 𝑆 ) |