Step |
Hyp |
Ref |
Expression |
1 |
|
deg1leb.d |
⊢ 𝐷 = ( deg1 ‘ 𝑅 ) |
2 |
|
deg1leb.p |
⊢ 𝑃 = ( Poly1 ‘ 𝑅 ) |
3 |
|
deg1leb.b |
⊢ 𝐵 = ( Base ‘ 𝑃 ) |
4 |
|
deg1leb.y |
⊢ 0 = ( 0g ‘ 𝑅 ) |
5 |
|
deg1leb.a |
⊢ 𝐴 = ( coe1 ‘ 𝐹 ) |
6 |
1
|
deg1fval |
⊢ 𝐷 = ( 1o mDeg 𝑅 ) |
7 |
|
eqid |
⊢ ( 1o mPoly 𝑅 ) = ( 1o mPoly 𝑅 ) |
8 |
|
eqid |
⊢ ( PwSer1 ‘ 𝑅 ) = ( PwSer1 ‘ 𝑅 ) |
9 |
2 8 3
|
ply1bas |
⊢ 𝐵 = ( Base ‘ ( 1o mPoly 𝑅 ) ) |
10 |
|
psr1baslem |
⊢ ( ℕ0 ↑m 1o ) = { 𝑎 ∈ ( ℕ0 ↑m 1o ) ∣ ( ◡ 𝑎 “ ℕ ) ∈ Fin } |
11 |
|
tdeglem2 |
⊢ ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) = ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( ℂfld Σg 𝑏 ) ) |
12 |
6 7 9 4 10 11
|
mdegleb |
⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ* ) → ( ( 𝐷 ‘ 𝐹 ) ≤ 𝐺 ↔ ∀ 𝑦 ∈ ( ℕ0 ↑m 1o ) ( 𝐺 < ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) → ( 𝐹 ‘ 𝑦 ) = 0 ) ) ) |
13 |
|
df1o2 |
⊢ 1o = { ∅ } |
14 |
|
nn0ex |
⊢ ℕ0 ∈ V |
15 |
|
0ex |
⊢ ∅ ∈ V |
16 |
|
eqid |
⊢ ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) = ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) |
17 |
13 14 15 16
|
mapsnf1o2 |
⊢ ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) : ( ℕ0 ↑m 1o ) –1-1-onto→ ℕ0 |
18 |
|
f1ofo |
⊢ ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) : ( ℕ0 ↑m 1o ) –1-1-onto→ ℕ0 → ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) : ( ℕ0 ↑m 1o ) –onto→ ℕ0 ) |
19 |
|
breq2 |
⊢ ( ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) = 𝑥 → ( 𝐺 < ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) ↔ 𝐺 < 𝑥 ) ) |
20 |
|
fveqeq2 |
⊢ ( ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) = 𝑥 → ( ( 𝐴 ‘ ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) ) = 0 ↔ ( 𝐴 ‘ 𝑥 ) = 0 ) ) |
21 |
19 20
|
imbi12d |
⊢ ( ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) = 𝑥 → ( ( 𝐺 < ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) → ( 𝐴 ‘ ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) ) = 0 ) ↔ ( 𝐺 < 𝑥 → ( 𝐴 ‘ 𝑥 ) = 0 ) ) ) |
22 |
21
|
cbvfo |
⊢ ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) : ( ℕ0 ↑m 1o ) –onto→ ℕ0 → ( ∀ 𝑦 ∈ ( ℕ0 ↑m 1o ) ( 𝐺 < ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) → ( 𝐴 ‘ ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) ) = 0 ) ↔ ∀ 𝑥 ∈ ℕ0 ( 𝐺 < 𝑥 → ( 𝐴 ‘ 𝑥 ) = 0 ) ) ) |
23 |
17 18 22
|
mp2b |
⊢ ( ∀ 𝑦 ∈ ( ℕ0 ↑m 1o ) ( 𝐺 < ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) → ( 𝐴 ‘ ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) ) = 0 ) ↔ ∀ 𝑥 ∈ ℕ0 ( 𝐺 < 𝑥 → ( 𝐴 ‘ 𝑥 ) = 0 ) ) |
24 |
|
fveq1 |
⊢ ( 𝑏 = 𝑦 → ( 𝑏 ‘ ∅ ) = ( 𝑦 ‘ ∅ ) ) |
25 |
|
fvex |
⊢ ( 𝑦 ‘ ∅ ) ∈ V |
26 |
24 16 25
|
fvmpt |
⊢ ( 𝑦 ∈ ( ℕ0 ↑m 1o ) → ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) = ( 𝑦 ‘ ∅ ) ) |
27 |
26
|
fveq2d |
⊢ ( 𝑦 ∈ ( ℕ0 ↑m 1o ) → ( 𝐴 ‘ ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) ) = ( 𝐴 ‘ ( 𝑦 ‘ ∅ ) ) ) |
28 |
27
|
adantl |
⊢ ( ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ* ) ∧ 𝑦 ∈ ( ℕ0 ↑m 1o ) ) → ( 𝐴 ‘ ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) ) = ( 𝐴 ‘ ( 𝑦 ‘ ∅ ) ) ) |
29 |
5
|
fvcoe1 |
⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝑦 ∈ ( ℕ0 ↑m 1o ) ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐴 ‘ ( 𝑦 ‘ ∅ ) ) ) |
30 |
29
|
adantlr |
⊢ ( ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ* ) ∧ 𝑦 ∈ ( ℕ0 ↑m 1o ) ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐴 ‘ ( 𝑦 ‘ ∅ ) ) ) |
31 |
28 30
|
eqtr4d |
⊢ ( ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ* ) ∧ 𝑦 ∈ ( ℕ0 ↑m 1o ) ) → ( 𝐴 ‘ ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) ) = ( 𝐹 ‘ 𝑦 ) ) |
32 |
31
|
eqeq1d |
⊢ ( ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ* ) ∧ 𝑦 ∈ ( ℕ0 ↑m 1o ) ) → ( ( 𝐴 ‘ ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) ) = 0 ↔ ( 𝐹 ‘ 𝑦 ) = 0 ) ) |
33 |
32
|
imbi2d |
⊢ ( ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ* ) ∧ 𝑦 ∈ ( ℕ0 ↑m 1o ) ) → ( ( 𝐺 < ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) → ( 𝐴 ‘ ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) ) = 0 ) ↔ ( 𝐺 < ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) → ( 𝐹 ‘ 𝑦 ) = 0 ) ) ) |
34 |
33
|
ralbidva |
⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ* ) → ( ∀ 𝑦 ∈ ( ℕ0 ↑m 1o ) ( 𝐺 < ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) → ( 𝐴 ‘ ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) ) = 0 ) ↔ ∀ 𝑦 ∈ ( ℕ0 ↑m 1o ) ( 𝐺 < ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) → ( 𝐹 ‘ 𝑦 ) = 0 ) ) ) |
35 |
23 34
|
bitr3id |
⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ* ) → ( ∀ 𝑥 ∈ ℕ0 ( 𝐺 < 𝑥 → ( 𝐴 ‘ 𝑥 ) = 0 ) ↔ ∀ 𝑦 ∈ ( ℕ0 ↑m 1o ) ( 𝐺 < ( ( 𝑏 ∈ ( ℕ0 ↑m 1o ) ↦ ( 𝑏 ‘ ∅ ) ) ‘ 𝑦 ) → ( 𝐹 ‘ 𝑦 ) = 0 ) ) ) |
36 |
12 35
|
bitr4d |
⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝐺 ∈ ℝ* ) → ( ( 𝐷 ‘ 𝐹 ) ≤ 𝐺 ↔ ∀ 𝑥 ∈ ℕ0 ( 𝐺 < 𝑥 → ( 𝐴 ‘ 𝑥 ) = 0 ) ) ) |