Step |
Hyp |
Ref |
Expression |
1 |
|
ply1coefsupp.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
2 |
|
ply1coefsupp.x |
⊢ 𝑋 = ( var1 ‘ 𝑅 ) |
3 |
|
ply1coefsupp.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
4 |
|
ply1coefsupp.n |
⊢ · = ( ·𝑠 ‘ 𝑃 ) |
5 |
|
ply1coefsupp.m |
⊢ 𝑀 = ( mulGrp ‘ 𝑃 ) |
6 |
|
ply1coefsupp.e |
⊢ ↑ = ( .g ‘ 𝑀 ) |
7 |
|
ply1coefsupp.a |
⊢ 𝐴 = ( coe1 ‘ 𝐾 ) |
8 |
|
eqid |
⊢ ( Scalar ‘ 𝑃 ) = ( Scalar ‘ 𝑃 ) |
9 |
1
|
ply1lmod |
⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ LMod ) |
10 |
9
|
adantr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ) → 𝑃 ∈ LMod ) |
11 |
|
nn0ex |
⊢ ℕ0 ∈ V |
12 |
11
|
a1i |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ) → ℕ0 ∈ V ) |
13 |
1
|
ply1ring |
⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring ) |
14 |
5
|
ringmgp |
⊢ ( 𝑃 ∈ Ring → 𝑀 ∈ Mnd ) |
15 |
13 14
|
syl |
⊢ ( 𝑅 ∈ Ring → 𝑀 ∈ Mnd ) |
16 |
15
|
ad2antrr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ) ∧ 𝑘 ∈ ℕ0 ) → 𝑀 ∈ Mnd ) |
17 |
|
simpr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℕ0 ) |
18 |
2 1 3
|
vr1cl |
⊢ ( 𝑅 ∈ Ring → 𝑋 ∈ 𝐵 ) |
19 |
18
|
ad2antrr |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ) ∧ 𝑘 ∈ ℕ0 ) → 𝑋 ∈ 𝐵 ) |
20 |
5 3
|
mgpbas |
⊢ 𝐵 = ( Base ‘ 𝑀 ) |
21 |
20 6
|
mulgnn0cl |
⊢ ( ( 𝑀 ∈ Mnd ∧ 𝑘 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑘 ↑ 𝑋 ) ∈ 𝐵 ) |
22 |
16 17 19 21
|
syl3anc |
⊢ ( ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 ↑ 𝑋 ) ∈ 𝐵 ) |
23 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
24 |
7 3 1 23
|
coe1f |
⊢ ( 𝐾 ∈ 𝐵 → 𝐴 : ℕ0 ⟶ ( Base ‘ 𝑅 ) ) |
25 |
24
|
adantl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ) → 𝐴 : ℕ0 ⟶ ( Base ‘ 𝑅 ) ) |
26 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
27 |
7 3 1 26
|
coe1sfi |
⊢ ( 𝐾 ∈ 𝐵 → 𝐴 finSupp ( 0g ‘ 𝑅 ) ) |
28 |
27
|
adantl |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ) → 𝐴 finSupp ( 0g ‘ 𝑅 ) ) |
29 |
1
|
ply1sca |
⊢ ( 𝑅 ∈ Ring → 𝑅 = ( Scalar ‘ 𝑃 ) ) |
30 |
29
|
eqcomd |
⊢ ( 𝑅 ∈ Ring → ( Scalar ‘ 𝑃 ) = 𝑅 ) |
31 |
30
|
adantr |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ) → ( Scalar ‘ 𝑃 ) = 𝑅 ) |
32 |
31
|
fveq2d |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ) → ( 0g ‘ ( Scalar ‘ 𝑃 ) ) = ( 0g ‘ 𝑅 ) ) |
33 |
28 32
|
breqtrrd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ) → 𝐴 finSupp ( 0g ‘ ( Scalar ‘ 𝑃 ) ) ) |
34 |
3 8 4 10 12 22 25 33
|
mptscmfsuppd |
⊢ ( ( 𝑅 ∈ Ring ∧ 𝐾 ∈ 𝐵 ) → ( 𝑘 ∈ ℕ0 ↦ ( ( 𝐴 ‘ 𝑘 ) · ( 𝑘 ↑ 𝑋 ) ) ) finSupp ( 0g ‘ 𝑃 ) ) |