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 |
|
evl1gsummul.1 |
⊢ 1 = ( 1r ‘ 𝑊 ) |
10 |
|
evl1gsummul.g |
⊢ 𝐺 = ( mulGrp ‘ 𝑊 ) |
11 |
|
evl1gsummul.h |
⊢ 𝐻 = ( mulGrp ‘ 𝑃 ) |
12 |
|
evl1gsummul.f |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑁 ↦ 𝑌 ) finSupp 1 ) |
13 |
10 5
|
mgpbas |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
14 |
10 9
|
ringidval |
⊢ 1 = ( 0g ‘ 𝐺 ) |
15 |
3
|
ply1crng |
⊢ ( 𝑅 ∈ CRing → 𝑊 ∈ CRing ) |
16 |
10
|
crngmgp |
⊢ ( 𝑊 ∈ CRing → 𝐺 ∈ CMnd ) |
17 |
6 15 16
|
3syl |
⊢ ( 𝜑 → 𝐺 ∈ CMnd ) |
18 |
|
crngring |
⊢ ( 𝑅 ∈ CRing → 𝑅 ∈ Ring ) |
19 |
6 18
|
syl |
⊢ ( 𝜑 → 𝑅 ∈ Ring ) |
20 |
2
|
fvexi |
⊢ 𝐾 ∈ V |
21 |
19 20
|
jctir |
⊢ ( 𝜑 → ( 𝑅 ∈ Ring ∧ 𝐾 ∈ V ) ) |
22 |
4
|
pwsring |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ V ) → 𝑃 ∈ Ring ) |
23 |
11
|
ringmgp |
⊢ ( 𝑃 ∈ Ring → 𝐻 ∈ Mnd ) |
24 |
21 22 23
|
3syl |
⊢ ( 𝜑 → 𝐻 ∈ Mnd ) |
25 |
|
nn0ex |
⊢ ℕ0 ∈ V |
26 |
25
|
a1i |
⊢ ( 𝜑 → ℕ0 ∈ V ) |
27 |
26 8
|
ssexd |
⊢ ( 𝜑 → 𝑁 ∈ V ) |
28 |
1 3 4 2
|
evl1rhm |
⊢ ( 𝑅 ∈ CRing → 𝑄 ∈ ( 𝑊 RingHom 𝑃 ) ) |
29 |
10 11
|
rhmmhm |
⊢ ( 𝑄 ∈ ( 𝑊 RingHom 𝑃 ) → 𝑄 ∈ ( 𝐺 MndHom 𝐻 ) ) |
30 |
6 28 29
|
3syl |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝐺 MndHom 𝐻 ) ) |
31 |
13 14 17 24 27 30 7 12
|
gsummptmhm |
⊢ ( 𝜑 → ( 𝐻 Σg ( 𝑥 ∈ 𝑁 ↦ ( 𝑄 ‘ 𝑌 ) ) ) = ( 𝑄 ‘ ( 𝐺 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑌 ) ) ) ) |
32 |
31
|
eqcomd |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐺 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑌 ) ) ) = ( 𝐻 Σg ( 𝑥 ∈ 𝑁 ↦ ( 𝑄 ‘ 𝑌 ) ) ) ) |