Step |
Hyp |
Ref |
Expression |
1 |
|
gsummptmhm.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
gsummptmhm.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
3 |
|
gsummptmhm.g |
⊢ ( 𝜑 → 𝐺 ∈ CMnd ) |
4 |
|
gsummptmhm.h |
⊢ ( 𝜑 → 𝐻 ∈ Mnd ) |
5 |
|
gsummptmhm.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
6 |
|
gsummptmhm.k |
⊢ ( 𝜑 → 𝐾 ∈ ( 𝐺 MndHom 𝐻 ) ) |
7 |
|
gsummptmhm.c |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝐵 ) |
8 |
|
gsummptmhm.w |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) finSupp 0 ) |
9 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) |
10 |
|
eqid |
⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 ) |
11 |
1 10
|
mhmf |
⊢ ( 𝐾 ∈ ( 𝐺 MndHom 𝐻 ) → 𝐾 : 𝐵 ⟶ ( Base ‘ 𝐻 ) ) |
12 |
|
ffn |
⊢ ( 𝐾 : 𝐵 ⟶ ( Base ‘ 𝐻 ) → 𝐾 Fn 𝐵 ) |
13 |
6 11 12
|
3syl |
⊢ ( 𝜑 → 𝐾 Fn 𝐵 ) |
14 |
|
dffn5 |
⊢ ( 𝐾 Fn 𝐵 ↔ 𝐾 = ( 𝑦 ∈ 𝐵 ↦ ( 𝐾 ‘ 𝑦 ) ) ) |
15 |
13 14
|
sylib |
⊢ ( 𝜑 → 𝐾 = ( 𝑦 ∈ 𝐵 ↦ ( 𝐾 ‘ 𝑦 ) ) ) |
16 |
|
fveq2 |
⊢ ( 𝑦 = 𝐶 → ( 𝐾 ‘ 𝑦 ) = ( 𝐾 ‘ 𝐶 ) ) |
17 |
7 9 15 16
|
fmptco |
⊢ ( 𝜑 → ( 𝐾 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝐾 ‘ 𝐶 ) ) ) |
18 |
17
|
oveq2d |
⊢ ( 𝜑 → ( 𝐻 Σg ( 𝐾 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) ) = ( 𝐻 Σg ( 𝑥 ∈ 𝐴 ↦ ( 𝐾 ‘ 𝐶 ) ) ) ) |
19 |
7
|
fmpttd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) : 𝐴 ⟶ 𝐵 ) |
20 |
1 2 3 4 5 6 19 8
|
gsummhm |
⊢ ( 𝜑 → ( 𝐻 Σg ( 𝐾 ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) ) = ( 𝐾 ‘ ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) ) ) |
21 |
18 20
|
eqtr3d |
⊢ ( 𝜑 → ( 𝐻 Σg ( 𝑥 ∈ 𝐴 ↦ ( 𝐾 ‘ 𝐶 ) ) ) = ( 𝐾 ‘ ( 𝐺 Σg ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) ) ) |