Step |
Hyp |
Ref |
Expression |
1 |
|
prdsplusgcl.y |
⊢ 𝑌 = ( 𝑆 Xs 𝑅 ) |
2 |
|
prdsplusgcl.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
3 |
|
prdsplusgcl.p |
⊢ + = ( +g ‘ 𝑌 ) |
4 |
|
prdsplusgcl.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑉 ) |
5 |
|
prdsplusgcl.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
6 |
|
prdsplusgcl.r |
⊢ ( 𝜑 → 𝑅 : 𝐼 ⟶ Mnd ) |
7 |
|
prdsplusgcl.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐵 ) |
8 |
|
prdsplusgcl.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐵 ) |
9 |
6
|
ffnd |
⊢ ( 𝜑 → 𝑅 Fn 𝐼 ) |
10 |
1 2 4 5 9 7 8 3
|
prdsplusgval |
⊢ ( 𝜑 → ( 𝐹 + 𝐺 ) = ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ) |
11 |
6
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑅 ‘ 𝑥 ) ∈ Mnd ) |
12 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝑆 ∈ 𝑉 ) |
13 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐼 ∈ 𝑊 ) |
14 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝑅 Fn 𝐼 ) |
15 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐹 ∈ 𝐵 ) |
16 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝑥 ∈ 𝐼 ) |
17 |
1 2 12 13 14 15 16
|
prdsbasprj |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ‘ 𝑥 ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
18 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐺 ∈ 𝐵 ) |
19 |
1 2 12 13 14 18 16
|
prdsbasprj |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
20 |
|
eqid |
⊢ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) = ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) |
21 |
|
eqid |
⊢ ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) = ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) |
22 |
20 21
|
mndcl |
⊢ ( ( ( 𝑅 ‘ 𝑥 ) ∈ Mnd ∧ ( 𝐹 ‘ 𝑥 ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ∧ ( 𝐺 ‘ 𝑥 ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
23 |
11 17 19 22
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
24 |
23
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
25 |
1 2 4 5 9
|
prdsbasmpt |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐼 ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) ) |
26 |
24 25
|
mpbird |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( ( 𝐹 ‘ 𝑥 ) ( +g ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ∈ 𝐵 ) |
27 |
10 26
|
eqeltrd |
⊢ ( 𝜑 → ( 𝐹 + 𝐺 ) ∈ 𝐵 ) |