Step |
Hyp |
Ref |
Expression |
1 |
|
gsummptfsadd.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsummptfsadd.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
gsummptfsadd.p |
⊢ + = ( +g ‘ 𝐺 ) |
4 |
|
gsummptfsadd.g |
⊢ ( 𝜑 → 𝐺 ∈ CMnd ) |
5 |
|
gsummptfsadd.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
6 |
|
gsummptfsadd.c |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝐵 ) |
7 |
|
gsummptfsadd.d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐷 ∈ 𝐵 ) |
8 |
|
gsummptfsadd.f |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) |
9 |
|
gsummptfsadd.h |
⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ 𝐴 ↦ 𝐷 ) ) |
10 |
|
gsummptfsadd.w |
⊢ ( 𝜑 → 𝐹 finSupp 0 ) |
11 |
|
gsummptfsadd.v |
⊢ ( 𝜑 → 𝐻 finSupp 0 ) |
12 |
5 6 7 8 9
|
offval2 |
⊢ ( 𝜑 → ( 𝐹 ∘f + 𝐻 ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝐶 + 𝐷 ) ) ) |
13 |
12
|
eqcomd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( 𝐶 + 𝐷 ) ) = ( 𝐹 ∘f + 𝐻 ) ) |
14 |
13
|
oveq2d |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ ( 𝐶 + 𝐷 ) ) ) = ( 𝐺 Σg ( 𝐹 ∘f + 𝐻 ) ) ) |
15 |
8 6
|
fmpt3d |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
16 |
9 7
|
fmpt3d |
⊢ ( 𝜑 → 𝐻 : 𝐴 ⟶ 𝐵 ) |
17 |
1 2 3 4 5 15 16 10 11
|
gsumadd |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝐹 ∘f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) ) |
18 |
14 17
|
eqtrd |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ ( 𝐶 + 𝐷 ) ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) ) |