Step |
Hyp |
Ref |
Expression |
1 |
|
imasvscaf.u |
⊢ ( 𝜑 → 𝑈 = ( 𝐹 “s 𝑅 ) ) |
2 |
|
imasvscaf.v |
⊢ ( 𝜑 → 𝑉 = ( Base ‘ 𝑅 ) ) |
3 |
|
imasvscaf.f |
⊢ ( 𝜑 → 𝐹 : 𝑉 –onto→ 𝐵 ) |
4 |
|
imasvscaf.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝑍 ) |
5 |
|
imasvscaf.g |
⊢ 𝐺 = ( Scalar ‘ 𝑅 ) |
6 |
|
imasvscaf.k |
⊢ 𝐾 = ( Base ‘ 𝐺 ) |
7 |
|
imasvscaf.q |
⊢ · = ( ·𝑠 ‘ 𝑅 ) |
8 |
|
imasvscaf.s |
⊢ ∙ = ( ·𝑠 ‘ 𝑈 ) |
9 |
|
imasvscaf.e |
⊢ ( ( 𝜑 ∧ ( 𝑝 ∈ 𝐾 ∧ 𝑎 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉 ) ) → ( ( 𝐹 ‘ 𝑎 ) = ( 𝐹 ‘ 𝑞 ) → ( 𝐹 ‘ ( 𝑝 · 𝑎 ) ) = ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ) ) |
10 |
|
imasvscaf.c |
⊢ ( ( 𝜑 ∧ ( 𝑝 ∈ 𝐾 ∧ 𝑞 ∈ 𝑉 ) ) → ( 𝑝 · 𝑞 ) ∈ 𝑉 ) |
11 |
1 2 3 4 5 6 7 8 9
|
imasvscafn |
⊢ ( 𝜑 → ∙ Fn ( 𝐾 × 𝐵 ) ) |
12 |
1 2 3 4 5 6 7 8
|
imasvsca |
⊢ ( 𝜑 → ∙ = ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ 𝐾 , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ) ) |
13 |
|
fof |
⊢ ( 𝐹 : 𝑉 –onto→ 𝐵 → 𝐹 : 𝑉 ⟶ 𝐵 ) |
14 |
3 13
|
syl |
⊢ ( 𝜑 → 𝐹 : 𝑉 ⟶ 𝐵 ) |
15 |
14
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ ( 𝑝 · 𝑞 ) ∈ 𝑉 ) → ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ∈ 𝐵 ) |
16 |
10 15
|
syldan |
⊢ ( ( 𝜑 ∧ ( 𝑝 ∈ 𝐾 ∧ 𝑞 ∈ 𝑉 ) ) → ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ∈ 𝐵 ) |
17 |
16
|
ralrimivw |
⊢ ( ( 𝜑 ∧ ( 𝑝 ∈ 𝐾 ∧ 𝑞 ∈ 𝑉 ) ) → ∀ 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ∈ 𝐵 ) |
18 |
17
|
anass1rs |
⊢ ( ( ( 𝜑 ∧ 𝑞 ∈ 𝑉 ) ∧ 𝑝 ∈ 𝐾 ) → ∀ 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ∈ 𝐵 ) |
19 |
18
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝑉 ) → ∀ 𝑝 ∈ 𝐾 ∀ 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ∈ 𝐵 ) |
20 |
|
eqid |
⊢ ( 𝑝 ∈ 𝐾 , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ) = ( 𝑝 ∈ 𝐾 , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ) |
21 |
20
|
fmpo |
⊢ ( ∀ 𝑝 ∈ 𝐾 ∀ 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ∈ 𝐵 ↔ ( 𝑝 ∈ 𝐾 , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ) : ( 𝐾 × { ( 𝐹 ‘ 𝑞 ) } ) ⟶ 𝐵 ) |
22 |
19 21
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝑉 ) → ( 𝑝 ∈ 𝐾 , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ) : ( 𝐾 × { ( 𝐹 ‘ 𝑞 ) } ) ⟶ 𝐵 ) |
23 |
|
fssxp |
⊢ ( ( 𝑝 ∈ 𝐾 , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ) : ( 𝐾 × { ( 𝐹 ‘ 𝑞 ) } ) ⟶ 𝐵 → ( 𝑝 ∈ 𝐾 , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ) ⊆ ( ( 𝐾 × { ( 𝐹 ‘ 𝑞 ) } ) × 𝐵 ) ) |
24 |
22 23
|
syl |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝑉 ) → ( 𝑝 ∈ 𝐾 , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ) ⊆ ( ( 𝐾 × { ( 𝐹 ‘ 𝑞 ) } ) × 𝐵 ) ) |
25 |
14
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝑉 ) → ( 𝐹 ‘ 𝑞 ) ∈ 𝐵 ) |
26 |
25
|
snssd |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝑉 ) → { ( 𝐹 ‘ 𝑞 ) } ⊆ 𝐵 ) |
27 |
|
xpss2 |
⊢ ( { ( 𝐹 ‘ 𝑞 ) } ⊆ 𝐵 → ( 𝐾 × { ( 𝐹 ‘ 𝑞 ) } ) ⊆ ( 𝐾 × 𝐵 ) ) |
28 |
|
xpss1 |
⊢ ( ( 𝐾 × { ( 𝐹 ‘ 𝑞 ) } ) ⊆ ( 𝐾 × 𝐵 ) → ( ( 𝐾 × { ( 𝐹 ‘ 𝑞 ) } ) × 𝐵 ) ⊆ ( ( 𝐾 × 𝐵 ) × 𝐵 ) ) |
29 |
26 27 28
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝑉 ) → ( ( 𝐾 × { ( 𝐹 ‘ 𝑞 ) } ) × 𝐵 ) ⊆ ( ( 𝐾 × 𝐵 ) × 𝐵 ) ) |
30 |
24 29
|
sstrd |
⊢ ( ( 𝜑 ∧ 𝑞 ∈ 𝑉 ) → ( 𝑝 ∈ 𝐾 , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ) ⊆ ( ( 𝐾 × 𝐵 ) × 𝐵 ) ) |
31 |
30
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑞 ∈ 𝑉 ( 𝑝 ∈ 𝐾 , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ) ⊆ ( ( 𝐾 × 𝐵 ) × 𝐵 ) ) |
32 |
|
iunss |
⊢ ( ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ 𝐾 , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ) ⊆ ( ( 𝐾 × 𝐵 ) × 𝐵 ) ↔ ∀ 𝑞 ∈ 𝑉 ( 𝑝 ∈ 𝐾 , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ) ⊆ ( ( 𝐾 × 𝐵 ) × 𝐵 ) ) |
33 |
31 32
|
sylibr |
⊢ ( 𝜑 → ∪ 𝑞 ∈ 𝑉 ( 𝑝 ∈ 𝐾 , 𝑥 ∈ { ( 𝐹 ‘ 𝑞 ) } ↦ ( 𝐹 ‘ ( 𝑝 · 𝑞 ) ) ) ⊆ ( ( 𝐾 × 𝐵 ) × 𝐵 ) ) |
34 |
12 33
|
eqsstrd |
⊢ ( 𝜑 → ∙ ⊆ ( ( 𝐾 × 𝐵 ) × 𝐵 ) ) |
35 |
|
dff2 |
⊢ ( ∙ : ( 𝐾 × 𝐵 ) ⟶ 𝐵 ↔ ( ∙ Fn ( 𝐾 × 𝐵 ) ∧ ∙ ⊆ ( ( 𝐾 × 𝐵 ) × 𝐵 ) ) ) |
36 |
11 34 35
|
sylanbrc |
⊢ ( 𝜑 → ∙ : ( 𝐾 × 𝐵 ) ⟶ 𝐵 ) |