Step |
Hyp |
Ref |
Expression |
1 |
|
gsummptfidmadd.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsummptfidmadd.p |
⊢ + = ( +g ‘ 𝐺 ) |
3 |
|
gsummptfidmadd.g |
⊢ ( 𝜑 → 𝐺 ∈ CMnd ) |
4 |
|
gsummptfidmadd.a |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
5 |
|
gsummptfidmadd.c |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝐵 ) |
6 |
|
gsummptfidmadd.d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐷 ∈ 𝐵 ) |
7 |
|
gsummptfidmadd.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) |
8 |
|
gsummptfidmadd.h |
⊢ 𝐻 = ( 𝑥 ∈ 𝐴 ↦ 𝐷 ) |
9 |
7
|
a1i |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) |
10 |
8
|
a1i |
⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ 𝐴 ↦ 𝐷 ) ) |
11 |
4 5 6 9 10
|
offval2 |
⊢ ( 𝜑 → ( 𝐹 ∘f + 𝐻 ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝐶 + 𝐷 ) ) ) |
12 |
11
|
oveq2d |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝐹 ∘f + 𝐻 ) ) = ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ ( 𝐶 + 𝐷 ) ) ) ) |
13 |
1 2 3 4 5 6 7 8
|
gsummptfidmadd |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ ( 𝐶 + 𝐷 ) ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) ) |
14 |
12 13
|
eqtrd |
⊢ ( 𝜑 → ( 𝐺 Σg ( 𝐹 ∘f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) ) |