Step |
Hyp |
Ref |
Expression |
1 |
|
dprdff.w |
⊢ 𝑊 = { ℎ ∈ X 𝑖 ∈ 𝐼 ( 𝑆 ‘ 𝑖 ) ∣ ℎ finSupp 0 } |
2 |
|
dprdff.1 |
⊢ ( 𝜑 → 𝐺 dom DProd 𝑆 ) |
3 |
|
dprdff.2 |
⊢ ( 𝜑 → dom 𝑆 = 𝐼 ) |
4 |
|
dprdff.3 |
⊢ ( 𝜑 → 𝐹 ∈ 𝑊 ) |
5 |
|
dprdff.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
6 |
1 2 3
|
dprdw |
⊢ ( 𝜑 → ( 𝐹 ∈ 𝑊 ↔ ( 𝐹 Fn 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ∈ ( 𝑆 ‘ 𝑥 ) ∧ 𝐹 finSupp 0 ) ) ) |
7 |
4 6
|
mpbid |
⊢ ( 𝜑 → ( 𝐹 Fn 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ∈ ( 𝑆 ‘ 𝑥 ) ∧ 𝐹 finSupp 0 ) ) |
8 |
7
|
simp1d |
⊢ ( 𝜑 → 𝐹 Fn 𝐼 ) |
9 |
7
|
simp2d |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ∈ ( 𝑆 ‘ 𝑥 ) ) |
10 |
2 3
|
dprdf2 |
⊢ ( 𝜑 → 𝑆 : 𝐼 ⟶ ( SubGrp ‘ 𝐺 ) ) |
11 |
10
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑆 ‘ 𝑥 ) ∈ ( SubGrp ‘ 𝐺 ) ) |
12 |
5
|
subgss |
⊢ ( ( 𝑆 ‘ 𝑥 ) ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑆 ‘ 𝑥 ) ⊆ 𝐵 ) |
13 |
11 12
|
syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑆 ‘ 𝑥 ) ⊆ 𝐵 ) |
14 |
13
|
sseld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐹 ‘ 𝑥 ) ∈ ( 𝑆 ‘ 𝑥 ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) ) |
15 |
14
|
ralimdva |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ∈ ( 𝑆 ‘ 𝑥 ) → ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) ) |
16 |
9 15
|
mpd |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) |
17 |
|
ffnfv |
⊢ ( 𝐹 : 𝐼 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐼 ∧ ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) ) |
18 |
8 16 17
|
sylanbrc |
⊢ ( 𝜑 → 𝐹 : 𝐼 ⟶ 𝐵 ) |