Step |
Hyp |
Ref |
Expression |
1 |
|
prdsbasmpt.y |
⊢ 𝑌 = ( 𝑆 Xs 𝑅 ) |
2 |
|
prdsbasmpt.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
3 |
|
prdsvscaval.t |
⊢ · = ( ·𝑠 ‘ 𝑌 ) |
4 |
|
prdsvscaval.k |
⊢ 𝐾 = ( Base ‘ 𝑆 ) |
5 |
|
prdsvscaval.s |
⊢ ( 𝜑 → 𝑆 ∈ 𝑉 ) |
6 |
|
prdsvscaval.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
7 |
|
prdsvscaval.r |
⊢ ( 𝜑 → 𝑅 Fn 𝐼 ) |
8 |
|
prdsvscaval.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐾 ) |
9 |
|
prdsvscaval.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝐵 ) |
10 |
|
fnex |
⊢ ( ( 𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑊 ) → 𝑅 ∈ V ) |
11 |
7 6 10
|
syl2anc |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
12 |
7
|
fndmd |
⊢ ( 𝜑 → dom 𝑅 = 𝐼 ) |
13 |
1 5 11 2 12 4 3
|
prdsvsca |
⊢ ( 𝜑 → · = ( 𝑦 ∈ 𝐾 , 𝑧 ∈ 𝐵 ↦ ( 𝑥 ∈ 𝐼 ↦ ( 𝑦 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑧 ‘ 𝑥 ) ) ) ) ) |
14 |
|
id |
⊢ ( 𝑦 = 𝐹 → 𝑦 = 𝐹 ) |
15 |
|
fveq1 |
⊢ ( 𝑧 = 𝐺 → ( 𝑧 ‘ 𝑥 ) = ( 𝐺 ‘ 𝑥 ) ) |
16 |
14 15
|
oveqan12d |
⊢ ( ( 𝑦 = 𝐹 ∧ 𝑧 = 𝐺 ) → ( 𝑦 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑧 ‘ 𝑥 ) ) = ( 𝐹 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) |
17 |
16
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑦 = 𝐹 ∧ 𝑧 = 𝐺 ) ) → ( 𝑦 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑧 ‘ 𝑥 ) ) = ( 𝐹 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) |
18 |
17
|
mpteq2dv |
⊢ ( ( 𝜑 ∧ ( 𝑦 = 𝐹 ∧ 𝑧 = 𝐺 ) ) → ( 𝑥 ∈ 𝐼 ↦ ( 𝑦 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝑧 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝐹 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ) |
19 |
6
|
mptexd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( 𝐹 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ∈ V ) |
20 |
13 18 8 9 19
|
ovmpod |
⊢ ( 𝜑 → ( 𝐹 · 𝐺 ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝐹 ( ·𝑠 ‘ ( 𝑅 ‘ 𝑥 ) ) ( 𝐺 ‘ 𝑥 ) ) ) ) |