Step |
Hyp |
Ref |
Expression |
1 |
|
evls1gsumadd.q |
⊢ 𝑄 = ( 𝑆 evalSub1 𝑅 ) |
2 |
|
evls1gsumadd.k |
⊢ 𝐾 = ( Base ‘ 𝑆 ) |
3 |
|
evls1gsumadd.w |
⊢ 𝑊 = ( Poly1 ‘ 𝑈 ) |
4 |
|
evls1gsumadd.0 |
⊢ 0 = ( 0g ‘ 𝑊 ) |
5 |
|
evls1gsumadd.u |
⊢ 𝑈 = ( 𝑆 ↾s 𝑅 ) |
6 |
|
evls1gsumadd.p |
⊢ 𝑃 = ( 𝑆 ↑s 𝐾 ) |
7 |
|
evls1gsumadd.b |
⊢ 𝐵 = ( Base ‘ 𝑊 ) |
8 |
|
evls1gsumadd.s |
⊢ ( 𝜑 → 𝑆 ∈ CRing ) |
9 |
|
evls1gsumadd.r |
⊢ ( 𝜑 → 𝑅 ∈ ( SubRing ‘ 𝑆 ) ) |
10 |
|
evls1gsumadd.y |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑁 ) → 𝑌 ∈ 𝐵 ) |
11 |
|
evls1gsumadd.n |
⊢ ( 𝜑 → 𝑁 ⊆ ℕ0 ) |
12 |
|
evls1gsumadd.f |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑁 ↦ 𝑌 ) finSupp 0 ) |
13 |
5
|
subrgring |
⊢ ( 𝑅 ∈ ( SubRing ‘ 𝑆 ) → 𝑈 ∈ Ring ) |
14 |
3
|
ply1ring |
⊢ ( 𝑈 ∈ Ring → 𝑊 ∈ Ring ) |
15 |
|
ringcmn |
⊢ ( 𝑊 ∈ Ring → 𝑊 ∈ CMnd ) |
16 |
9 13 14 15
|
4syl |
⊢ ( 𝜑 → 𝑊 ∈ CMnd ) |
17 |
|
crngring |
⊢ ( 𝑆 ∈ CRing → 𝑆 ∈ Ring ) |
18 |
8 17
|
syl |
⊢ ( 𝜑 → 𝑆 ∈ Ring ) |
19 |
2
|
fvexi |
⊢ 𝐾 ∈ V |
20 |
18 19
|
jctir |
⊢ ( 𝜑 → ( 𝑆 ∈ Ring ∧ 𝐾 ∈ V ) ) |
21 |
6
|
pwsring |
⊢ ( ( 𝑆 ∈ Ring ∧ 𝐾 ∈ V ) → 𝑃 ∈ Ring ) |
22 |
|
ringmnd |
⊢ ( 𝑃 ∈ Ring → 𝑃 ∈ Mnd ) |
23 |
20 21 22
|
3syl |
⊢ ( 𝜑 → 𝑃 ∈ Mnd ) |
24 |
|
nn0ex |
⊢ ℕ0 ∈ V |
25 |
24
|
a1i |
⊢ ( 𝜑 → ℕ0 ∈ V ) |
26 |
25 11
|
ssexd |
⊢ ( 𝜑 → 𝑁 ∈ V ) |
27 |
1 2 6 5 3
|
evls1rhm |
⊢ ( ( 𝑆 ∈ CRing ∧ 𝑅 ∈ ( SubRing ‘ 𝑆 ) ) → 𝑄 ∈ ( 𝑊 RingHom 𝑃 ) ) |
28 |
8 9 27
|
syl2anc |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝑊 RingHom 𝑃 ) ) |
29 |
|
rhmghm |
⊢ ( 𝑄 ∈ ( 𝑊 RingHom 𝑃 ) → 𝑄 ∈ ( 𝑊 GrpHom 𝑃 ) ) |
30 |
|
ghmmhm |
⊢ ( 𝑄 ∈ ( 𝑊 GrpHom 𝑃 ) → 𝑄 ∈ ( 𝑊 MndHom 𝑃 ) ) |
31 |
28 29 30
|
3syl |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝑊 MndHom 𝑃 ) ) |
32 |
7 4 16 23 26 31 10 12
|
gsummptmhm |
⊢ ( 𝜑 → ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ ( 𝑄 ‘ 𝑌 ) ) ) = ( 𝑄 ‘ ( 𝑊 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑌 ) ) ) ) |
33 |
32
|
eqcomd |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝑊 Σg ( 𝑥 ∈ 𝑁 ↦ 𝑌 ) ) ) = ( 𝑃 Σg ( 𝑥 ∈ 𝑁 ↦ ( 𝑄 ‘ 𝑌 ) ) ) ) |