Step |
Hyp |
Ref |
Expression |
1 |
|
mdeg0.d |
⊢ 𝐷 = ( 𝐼 mDeg 𝑅 ) |
2 |
|
mdeg0.p |
⊢ 𝑃 = ( 𝐼 mPoly 𝑅 ) |
3 |
|
mdeg0.z |
⊢ 0 = ( 0g ‘ 𝑃 ) |
4 |
|
ringgrp |
⊢ ( 𝑅 ∈ Ring → 𝑅 ∈ Grp ) |
5 |
2
|
mplgrp |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Grp ) → 𝑃 ∈ Grp ) |
6 |
4 5
|
sylan2 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → 𝑃 ∈ Grp ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝑃 ) = ( Base ‘ 𝑃 ) |
8 |
7 3
|
grpidcl |
⊢ ( 𝑃 ∈ Grp → 0 ∈ ( Base ‘ 𝑃 ) ) |
9 |
|
eqid |
⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 ) |
10 |
|
eqid |
⊢ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } = { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } |
11 |
|
eqid |
⊢ ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) = ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) |
12 |
1 2 7 9 10 11
|
mdegval |
⊢ ( 0 ∈ ( Base ‘ 𝑃 ) → ( 𝐷 ‘ 0 ) = sup ( ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) “ ( 0 supp ( 0g ‘ 𝑅 ) ) ) , ℝ* , < ) ) |
13 |
6 8 12
|
3syl |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( 𝐷 ‘ 0 ) = sup ( ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) “ ( 0 supp ( 0g ‘ 𝑅 ) ) ) , ℝ* , < ) ) |
14 |
|
simpl |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → 𝐼 ∈ 𝑉 ) |
15 |
4
|
adantl |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → 𝑅 ∈ Grp ) |
16 |
2 10 9 3 14 15
|
mpl0 |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → 0 = ( { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } × { ( 0g ‘ 𝑅 ) } ) ) |
17 |
|
fvex |
⊢ ( 0g ‘ 𝑅 ) ∈ V |
18 |
|
fnconstg |
⊢ ( ( 0g ‘ 𝑅 ) ∈ V → ( { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } × { ( 0g ‘ 𝑅 ) } ) Fn { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ) |
19 |
17 18
|
mp1i |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } × { ( 0g ‘ 𝑅 ) } ) Fn { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ) |
20 |
16
|
fneq1d |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( 0 Fn { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↔ ( { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } × { ( 0g ‘ 𝑅 ) } ) Fn { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ) ) |
21 |
19 20
|
mpbird |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → 0 Fn { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ) |
22 |
|
ovex |
⊢ ( ℕ0 ↑m 𝐼 ) ∈ V |
23 |
22
|
rabex |
⊢ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ∈ V |
24 |
23
|
a1i |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ∈ V ) |
25 |
17
|
a1i |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( 0g ‘ 𝑅 ) ∈ V ) |
26 |
|
fnsuppeq0 |
⊢ ( ( 0 Fn { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ∧ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ∈ V ∧ ( 0g ‘ 𝑅 ) ∈ V ) → ( ( 0 supp ( 0g ‘ 𝑅 ) ) = ∅ ↔ 0 = ( { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } × { ( 0g ‘ 𝑅 ) } ) ) ) |
27 |
21 24 25 26
|
syl3anc |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( ( 0 supp ( 0g ‘ 𝑅 ) ) = ∅ ↔ 0 = ( { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } × { ( 0g ‘ 𝑅 ) } ) ) ) |
28 |
16 27
|
mpbird |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( 0 supp ( 0g ‘ 𝑅 ) ) = ∅ ) |
29 |
28
|
imaeq2d |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) “ ( 0 supp ( 0g ‘ 𝑅 ) ) ) = ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) “ ∅ ) ) |
30 |
|
ima0 |
⊢ ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) “ ∅ ) = ∅ |
31 |
29 30
|
eqtrdi |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) “ ( 0 supp ( 0g ‘ 𝑅 ) ) ) = ∅ ) |
32 |
31
|
supeq1d |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → sup ( ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) “ ( 0 supp ( 0g ‘ 𝑅 ) ) ) , ℝ* , < ) = sup ( ∅ , ℝ* , < ) ) |
33 |
|
xrsup0 |
⊢ sup ( ∅ , ℝ* , < ) = -∞ |
34 |
32 33
|
eqtrdi |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → sup ( ( ( 𝑦 ∈ { 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑥 “ ℕ ) ∈ Fin } ↦ ( ℂfld Σg 𝑦 ) ) “ ( 0 supp ( 0g ‘ 𝑅 ) ) ) , ℝ* , < ) = -∞ ) |
35 |
13 34
|
eqtrd |
⊢ ( ( 𝐼 ∈ 𝑉 ∧ 𝑅 ∈ Ring ) → ( 𝐷 ‘ 0 ) = -∞ ) |