Step |
Hyp |
Ref |
Expression |
1 |
|
prdsvscacl.y |
⊢ 𝑌 = ( 𝑆 Xs 𝑅 ) |
2 |
|
prdsvscacl.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
3 |
|
prdsvscacl.t |
⊢ · = ( ·𝑠 ‘ 𝑌 ) |
4 |
|
prdsvscacl.k |
⊢ 𝐾 = ( Base ‘ 𝑆 ) |
5 |
|
prdsvscacl.s |
⊢ ( 𝜑 → 𝑆 ∈ Ring ) |
6 |
|
prdsvscacl.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
7 |
|
prdsvscacl.r |
⊢ ( 𝜑 → 𝑅 : 𝐼 ⟶ LMod ) |
8 |
|
prdsvscacl.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐾 ) |
9 |
|
prdsvscacl.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐵 ) |
10 |
|
prdsvscacl.sr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( Scalar ‘ ( 𝑅 ‘ 𝑥 ) ) = 𝑆 ) |
11 |
7
|
ffnd |
⊢ ( 𝜑 → 𝑅 Fn 𝐼 ) |
12 |
1 2 3 4 5 6 11 8 9
|
prdsvscaval |
⊢ ( 𝜑 → ( 𝐹 · 𝐺 ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝐹 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ) |
13 |
7
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑅 ‘ 𝑥 ) ∈ LMod ) |
14 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐹 ∈ 𝐾 ) |
15 |
10
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( Base ‘ ( Scalar ‘ ( 𝑅 ‘ 𝑥 ) ) ) = ( Base ‘ 𝑆 ) ) |
16 |
15 4
|
eqtr4di |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( Base ‘ ( Scalar ‘ ( 𝑅 ‘ 𝑥 ) ) ) = 𝐾 ) |
17 |
14 16
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐹 ∈ ( Base ‘ ( Scalar ‘ ( 𝑅 ‘ 𝑥 ) ) ) ) |
18 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝑆 ∈ Ring ) |
19 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐼 ∈ 𝑊 ) |
20 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝑅 Fn 𝐼 ) |
21 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐺 ∈ 𝐵 ) |
22 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝑥 ∈ 𝐼 ) |
23 |
1 2 18 19 20 21 22
|
prdsbasprj |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐺 ‘ 𝑥 ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
24 |
|
eqid |
⊢ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) = ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) |
25 |
|
eqid |
⊢ ( Scalar ‘ ( 𝑅 ‘ 𝑥 ) ) = ( Scalar ‘ ( 𝑅 ‘ 𝑥 ) ) |
26 |
|
eqid |
⊢ ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) = ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) |
27 |
|
eqid |
⊢ ( Base ‘ ( Scalar ‘ ( 𝑅 ‘ 𝑥 ) ) ) = ( Base ‘ ( Scalar ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
28 |
24 25 26 27
|
lmodvscl |
⊢ ( ( ( 𝑅 ‘ 𝑥 ) ∈ LMod ∧ 𝐹 ∈ ( Base ‘ ( Scalar ‘ ( 𝑅 ‘ 𝑥 ) ) ) ∧ ( 𝐺 ‘ 𝑥 ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) → ( 𝐹 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
29 |
13 17 23 28
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐹 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
30 |
29
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 ( 𝐹 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) |
31 |
1 2 5 6 11
|
prdsbasmpt |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ ( 𝐹 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐼 ( 𝐹 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) ) |
32 |
30 31
|
mpbird |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( 𝐹 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ∈ 𝐵 ) |
33 |
12 32
|
eqeltrd |
⊢ ( 𝜑 → ( 𝐹 · 𝐺 ) ∈ 𝐵 ) |