Step |
Hyp |
Ref |
Expression |
1 |
|
evl1gsumadd.q |
⊢ 𝑄 = ( eval1 ‘ 𝑅 ) |
2 |
|
evl1gsumadd.k |
⊢ 𝐾 = ( Base ‘ 𝑅 ) |
3 |
|
evl1gsumadd.w |
⊢ 𝑊 = ( Poly1 ‘ 𝑅 ) |
4 |
|
evl1gsumadd.p |
⊢ 𝑃 = ( 𝑅 ↑s 𝐾 ) |
5 |
|
evl1gsumadd.b |
⊢ 𝐵 = ( Base ‘ 𝑊 ) |
6 |
|
evl1gsumadd.r |
⊢ ( 𝜑 → 𝑅 ∈ CRing ) |
7 |
|
evl1gsumadd.y |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → 𝑌 ∈ 𝐵 ) |
8 |
|
evl1gsumadd.n |
⊢ ( 𝜑 → 𝑁 ⊆ ℕ0 ) |
9 |
|
evl1gsumadd.0 |
⊢ 0 = ( 0g ‘ 𝑊 ) |
10 |
|
evl1gsumadd.f |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑁 ↦ 𝑌 ) finSupp 0 ) |
11 |
1 2
|
evl1fval1 |
⊢ 𝑄 = ( 𝑅 evalSub1 𝐾 ) |
12 |
11
|
a1i |
⊢ ( 𝜑 → 𝑄 = ( 𝑅 evalSub1 𝐾 ) ) |
13 |
12
|
fveq1d |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝑊 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑌 ) ) ) = ( ( 𝑅 evalSub1 𝐾 ) ‘ ( 𝑊 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑌 ) ) ) ) |
14 |
2
|
ressid |
⊢ ( 𝑅 ∈ CRing → ( 𝑅 ↾s 𝐾 ) = 𝑅 ) |
15 |
6 14
|
syl |
⊢ ( 𝜑 → ( 𝑅 ↾s 𝐾 ) = 𝑅 ) |
16 |
15
|
eqcomd |
⊢ ( 𝜑 → 𝑅 = ( 𝑅 ↾s 𝐾 ) ) |
17 |
16
|
fveq2d |
⊢ ( 𝜑 → ( Poly1 ‘ 𝑅 ) = ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) ) |
18 |
3 17
|
eqtrid |
⊢ ( 𝜑 → 𝑊 = ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) ) |
19 |
18
|
fvoveq1d |
⊢ ( 𝜑 → ( ( 𝑅 evalSub1 𝐾 ) ‘ ( 𝑊 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑌 ) ) ) = ( ( 𝑅 evalSub1 𝐾 ) ‘ ( ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) Σg ( 𝑥 ∈ 𝑁 ↦ 𝑌 ) ) ) ) |
20 |
|
eqid |
⊢ ( 𝑅 evalSub1 𝐾 ) = ( 𝑅 evalSub1 𝐾 ) |
21 |
|
eqid |
⊢ ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) = ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) |
22 |
|
eqid |
⊢ ( 0g ‘ ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) ) = ( 0g ‘ ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) ) |
23 |
|
eqid |
⊢ ( 𝑅 ↾s 𝐾 ) = ( 𝑅 ↾s 𝐾 ) |
24 |
|
eqid |
⊢ ( Base ‘ ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) ) = ( Base ‘ ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) ) |
25 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
26 |
2
|
subrgid |
⊢ ( 𝑅 ∈ Ring → 𝐾 ∈ ( SubRing ‘ 𝑅 ) ) |
27 |
6 25 26
|
3syl |
⊢ ( 𝜑 → 𝐾 ∈ ( SubRing ‘ 𝑅 ) ) |
28 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → 𝑊 = ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) ) |
29 |
28
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → ( Base ‘ 𝑊 ) = ( Base ‘ ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) ) ) |
30 |
5 29
|
eqtrid |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → 𝐵 = ( Base ‘ ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) ) ) |
31 |
7 30
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → 𝑌 ∈ ( Base ‘ ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) ) ) |
32 |
18
|
eqcomd |
⊢ ( 𝜑 → ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) = 𝑊 ) |
33 |
32
|
fveq2d |
⊢ ( 𝜑 → ( 0g ‘ ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) ) = ( 0g ‘ 𝑊 ) ) |
34 |
33 9
|
eqtr4di |
⊢ ( 𝜑 → ( 0g ‘ ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) ) = 0 ) |
35 |
10 34
|
breqtrrd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑁 ↦ 𝑌 ) finSupp ( 0g ‘ ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) ) ) |
36 |
20 2 21 22 23 4 24 6 27 31 8 35
|
evls1gsumadd |
⊢ ( 𝜑 → ( ( 𝑅 evalSub1 𝐾 ) ‘ ( ( Poly1 ‘ ( 𝑅 ↾s 𝐾 ) ) Σg ( 𝑥 ∈ 𝑁 ↦ 𝑌 ) ) ) = ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( 𝑅 evalSub1 𝐾 ) ‘ 𝑌 ) ) ) ) |
37 |
19 36
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑅 evalSub1 𝐾 ) ‘ ( 𝑊 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑌 ) ) ) = ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( 𝑅 evalSub1 𝐾 ) ‘ 𝑌 ) ) ) ) |
38 |
12
|
fveq1d |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝑌 ) = ( ( 𝑅 evalSub1 𝐾 ) ‘ 𝑌 ) ) |
39 |
38
|
eqcomd |
⊢ ( 𝜑 → ( ( 𝑅 evalSub1 𝐾 ) ‘ 𝑌 ) = ( 𝑄 ‘ 𝑌 ) ) |
40 |
39
|
mpteq2dv |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑁 ↦ ( ( 𝑅 evalSub1 𝐾 ) ‘ 𝑌 ) ) = ( 𝑥 ∈ 𝑁 ↦ ( 𝑄 ‘ 𝑌 ) ) ) |
41 |
40
|
oveq2d |
⊢ ( 𝜑 → ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ ( ( 𝑅 evalSub1 𝐾 ) ‘ 𝑌 ) ) ) = ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ ( 𝑄 ‘ 𝑌 ) ) ) ) |
42 |
13 37 41
|
3eqtrd |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝑊 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑌 ) ) ) = ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ ( 𝑄 ‘ 𝑌 ) ) ) ) |