Step |
Hyp |
Ref |
Expression |
1 |
|
pwsvscaval.y |
⊢ 𝑌 = ( 𝑅 ↑s 𝐼 ) |
2 |
|
pwsvscaval.b |
⊢ 𝐵 = ( Base ‘ 𝑌 ) |
3 |
|
pwsvscaval.s |
⊢ · = ( ·𝑠 ‘ 𝑅 ) |
4 |
|
pwsvscaval.t |
⊢ ∙ = ( ·𝑠 ‘ 𝑌 ) |
5 |
|
pwsvscaval.f |
⊢ 𝐹 = ( Scalar ‘ 𝑅 ) |
6 |
|
pwsvscaval.k |
⊢ 𝐾 = ( Base ‘ 𝐹 ) |
7 |
|
pwsvscaval.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝑉 ) |
8 |
|
pwsvscaval.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
9 |
|
pwsvscaval.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝐾 ) |
10 |
|
pwsvscaval.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
11 |
1 5
|
pwsval |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → 𝑌 = ( 𝐹 Xs ( 𝐼 × { 𝑅 } ) ) ) |
12 |
7 8 11
|
syl2anc |
⊢ ( 𝜑 → 𝑌 = ( 𝐹 Xs ( 𝐼 × { 𝑅 } ) ) ) |
13 |
12
|
fveq2d |
⊢ ( 𝜑 → ( ·𝑠 ‘ 𝑌 ) = ( ·𝑠 ‘ ( 𝐹 Xs ( 𝐼 × { 𝑅 } ) ) ) ) |
14 |
4 13
|
eqtrid |
⊢ ( 𝜑 → ∙ = ( ·𝑠 ‘ ( 𝐹 Xs ( 𝐼 × { 𝑅 } ) ) ) ) |
15 |
14
|
oveqd |
⊢ ( 𝜑 → ( 𝐴 ∙ 𝑋 ) = ( 𝐴 ( ·𝑠 ‘ ( 𝐹 Xs ( 𝐼 × { 𝑅 } ) ) ) 𝑋 ) ) |
16 |
|
eqid |
⊢ ( 𝐹 Xs ( 𝐼 × { 𝑅 } ) ) = ( 𝐹 Xs ( 𝐼 × { 𝑅 } ) ) |
17 |
|
eqid |
⊢ ( Base ‘ ( 𝐹 Xs ( 𝐼 × { 𝑅 } ) ) ) = ( Base ‘ ( 𝐹 Xs ( 𝐼 × { 𝑅 } ) ) ) |
18 |
|
eqid |
⊢ ( ·𝑠 ‘ ( 𝐹 Xs ( 𝐼 × { 𝑅 } ) ) ) = ( ·𝑠 ‘ ( 𝐹 Xs ( 𝐼 × { 𝑅 } ) ) ) |
19 |
5
|
fvexi |
⊢ 𝐹 ∈ V |
20 |
19
|
a1i |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
21 |
|
fnconstg |
⊢ ( 𝑅 ∈ 𝑉 → ( 𝐼 × { 𝑅 } ) Fn 𝐼 ) |
22 |
7 21
|
syl |
⊢ ( 𝜑 → ( 𝐼 × { 𝑅 } ) Fn 𝐼 ) |
23 |
12
|
fveq2d |
⊢ ( 𝜑 → ( Base ‘ 𝑌 ) = ( Base ‘ ( 𝐹 Xs ( 𝐼 × { 𝑅 } ) ) ) ) |
24 |
2 23
|
eqtrid |
⊢ ( 𝜑 → 𝐵 = ( Base ‘ ( 𝐹 Xs ( 𝐼 × { 𝑅 } ) ) ) ) |
25 |
10 24
|
eleqtrd |
⊢ ( 𝜑 → 𝑋 ∈ ( Base ‘ ( 𝐹 Xs ( 𝐼 × { 𝑅 } ) ) ) ) |
26 |
16 17 18 6 20 8 22 9 25
|
prdsvscaval |
⊢ ( 𝜑 → ( 𝐴 ( ·𝑠 ‘ ( 𝐹 Xs ( 𝐼 × { 𝑅 } ) ) ) 𝑋 ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝐴 ( ·𝑠 ‘ ( ( 𝐼 × { 𝑅 } ) ‘ 𝑥 ) ) ( 𝑋 ‘ 𝑥 ) ) ) ) |
27 |
|
fvconst2g |
⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐼 × { 𝑅 } ) ‘ 𝑥 ) = 𝑅 ) |
28 |
7 27
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ( 𝐼 × { 𝑅 } ) ‘ 𝑥 ) = 𝑅 ) |
29 |
28
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ·𝑠 ‘ ( ( 𝐼 × { 𝑅 } ) ‘ 𝑥 ) ) = ( ·𝑠 ‘ 𝑅 ) ) |
30 |
29 3
|
eqtr4di |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( ·𝑠 ‘ ( ( 𝐼 × { 𝑅 } ) ‘ 𝑥 ) ) = · ) |
31 |
30
|
oveqd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝐴 ( ·𝑠 ‘ ( ( 𝐼 × { 𝑅 } ) ‘ 𝑥 ) ) ( 𝑋 ‘ 𝑥 ) ) = ( 𝐴 · ( 𝑋 ‘ 𝑥 ) ) ) |
32 |
31
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( 𝐴 ( ·𝑠 ‘ ( ( 𝐼 × { 𝑅 } ) ‘ 𝑥 ) ) ( 𝑋 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝐴 · ( 𝑋 ‘ 𝑥 ) ) ) ) |
33 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → 𝐴 ∈ 𝐾 ) |
34 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑥 ) ∈ V ) |
35 |
|
fconstmpt |
⊢ ( 𝐼 × { 𝐴 } ) = ( 𝑥 ∈ 𝐼 ↦ 𝐴 ) |
36 |
35
|
a1i |
⊢ ( 𝜑 → ( 𝐼 × { 𝐴 } ) = ( 𝑥 ∈ 𝐼 ↦ 𝐴 ) ) |
37 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
38 |
1 37 2 7 8 10
|
pwselbas |
⊢ ( 𝜑 → 𝑋 : 𝐼 ⟶ ( Base ‘ 𝑅 ) ) |
39 |
38
|
feqmptd |
⊢ ( 𝜑 → 𝑋 = ( 𝑥 ∈ 𝐼 ↦ ( 𝑋 ‘ 𝑥 ) ) ) |
40 |
8 33 34 36 39
|
offval2 |
⊢ ( 𝜑 → ( ( 𝐼 × { 𝐴 } ) ∘f · 𝑋 ) = ( 𝑥 ∈ 𝐼 ↦ ( 𝐴 · ( 𝑋 ‘ 𝑥 ) ) ) ) |
41 |
32 40
|
eqtr4d |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( 𝐴 ( ·𝑠 ‘ ( ( 𝐼 × { 𝑅 } ) ‘ 𝑥 ) ) ( 𝑋 ‘ 𝑥 ) ) ) = ( ( 𝐼 × { 𝐴 } ) ∘f · 𝑋 ) ) |
42 |
15 26 41
|
3eqtrd |
⊢ ( 𝜑 → ( 𝐴 ∙ 𝑋 ) = ( ( 𝐼 × { 𝐴 } ) ∘f · 𝑋 ) ) |