Step | Hyp | Ref
| Expression |
1 | | mdegle0.f |
. . 3
⊢ (𝜑 → 𝐹 ∈ 𝐵) |
2 | | 0xr 10767 |
. . 3
⊢ 0 ∈
ℝ* |
3 | | mdegaddle.d |
. . . 4
⊢ 𝐷 = (𝐼 mDeg 𝑅) |
4 | | mdegaddle.y |
. . . 4
⊢ 𝑌 = (𝐼 mPoly 𝑅) |
5 | | mdegle0.b |
. . . 4
⊢ 𝐵 = (Base‘𝑌) |
6 | | eqid 2738 |
. . . 4
⊢
(0g‘𝑅) = (0g‘𝑅) |
7 | | eqid 2738 |
. . . 4
⊢ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} = {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈
Fin} |
8 | | eqid 2738 |
. . . 4
⊢ (𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏)) = (𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏)) |
9 | 3, 4, 5, 6, 7, 8 | mdegleb 24817 |
. . 3
⊢ ((𝐹 ∈ 𝐵 ∧ 0 ∈ ℝ*) →
((𝐷‘𝐹) ≤ 0 ↔ ∀𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} (0 < ((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) → (𝐹‘𝑥) = (0g‘𝑅)))) |
10 | 1, 2, 9 | sylancl 589 |
. 2
⊢ (𝜑 → ((𝐷‘𝐹) ≤ 0 ↔ ∀𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} (0 < ((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) → (𝐹‘𝑥) = (0g‘𝑅)))) |
11 | 7, 8 | tdeglem1 24808 |
. . . . . . . . . 10
⊢ (𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏)):{𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈
Fin}⟶ℕ0 |
12 | 11 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → (𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏)):{𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈
Fin}⟶ℕ0) |
13 | 12 | ffvelrnda 6862 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin}) → ((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) ∈
ℕ0) |
14 | | nn0re 11986 |
. . . . . . . . 9
⊢ (((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) ∈ ℕ0 → ((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) ∈ ℝ) |
15 | | nn0ge0 12002 |
. . . . . . . . 9
⊢ (((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) ∈ ℕ0 → 0 ≤
((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥)) |
16 | 14, 15 | jca 515 |
. . . . . . . 8
⊢ (((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) ∈ ℕ0 → (((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥))) |
17 | | ne0gt0 10824 |
. . . . . . . 8
⊢ ((((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) ∈ ℝ ∧ 0 ≤ ((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥)) → (((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) ≠ 0 ↔ 0 < ((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥))) |
18 | 13, 16, 17 | 3syl 18 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin}) →
(((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) ≠ 0 ↔ 0 < ((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥))) |
19 | 7, 8 | tdeglem4 24812 |
. . . . . . . . 9
⊢ (𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} → (((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) = 0 ↔ 𝑥 = (𝐼 × {0}))) |
20 | 19 | adantl 485 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin}) →
(((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) = 0 ↔ 𝑥 = (𝐼 × {0}))) |
21 | 20 | necon3abid 2970 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin}) →
(((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) ≠ 0 ↔ ¬ 𝑥 = (𝐼 × {0}))) |
22 | 18, 21 | bitr3d 284 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin}) → (0 <
((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) ↔ ¬ 𝑥 = (𝐼 × {0}))) |
23 | 22 | imbi1d 345 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin}) → ((0
< ((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) → (𝐹‘𝑥) = (0g‘𝑅)) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0g‘𝑅)))) |
24 | | eqeq2 2750 |
. . . . . . . 8
⊢ ((𝐹‘(𝐼 × {0})) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅)) → ((𝐹‘𝑥) = (𝐹‘(𝐼 × {0})) ↔ (𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅)))) |
25 | 24 | bibi1d 347 |
. . . . . . 7
⊢ ((𝐹‘(𝐼 × {0})) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅)) → (((𝐹‘𝑥) = (𝐹‘(𝐼 × {0})) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0g‘𝑅))) ↔ ((𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅)) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0g‘𝑅))))) |
26 | | eqeq2 2750 |
. . . . . . . 8
⊢
((0g‘𝑅) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅)) → ((𝐹‘𝑥) = (0g‘𝑅) ↔ (𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅)))) |
27 | 26 | bibi1d 347 |
. . . . . . 7
⊢
((0g‘𝑅) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅)) → (((𝐹‘𝑥) = (0g‘𝑅) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0g‘𝑅))) ↔ ((𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅)) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0g‘𝑅))))) |
28 | | fveq2 6675 |
. . . . . . . . 9
⊢ (𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (𝐹‘(𝐼 × {0}))) |
29 | | pm2.24 124 |
. . . . . . . . 9
⊢ (𝑥 = (𝐼 × {0}) → (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0g‘𝑅))) |
30 | 28, 29 | 2thd 268 |
. . . . . . . 8
⊢ (𝑥 = (𝐼 × {0}) → ((𝐹‘𝑥) = (𝐹‘(𝐼 × {0})) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0g‘𝑅)))) |
31 | 30 | adantl 485 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 = (𝐼 × {0})) → ((𝐹‘𝑥) = (𝐹‘(𝐼 × {0})) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0g‘𝑅)))) |
32 | | biimt 364 |
. . . . . . . 8
⊢ (¬
𝑥 = (𝐼 × {0}) → ((𝐹‘𝑥) = (0g‘𝑅) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0g‘𝑅)))) |
33 | 32 | adantl 485 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝑥 = (𝐼 × {0})) → ((𝐹‘𝑥) = (0g‘𝑅) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0g‘𝑅)))) |
34 | 25, 27, 31, 33 | ifbothda 4453 |
. . . . . 6
⊢ (𝜑 → ((𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅)) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0g‘𝑅)))) |
35 | 34 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin}) → ((𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅)) ↔ (¬ 𝑥 = (𝐼 × {0}) → (𝐹‘𝑥) = (0g‘𝑅)))) |
36 | 23, 35 | bitr4d 285 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin}) → ((0
< ((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) → (𝐹‘𝑥) = (0g‘𝑅)) ↔ (𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅)))) |
37 | 36 | ralbidva 3108 |
. . 3
⊢ (𝜑 → (∀𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} (0 < ((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) → (𝐹‘𝑥) = (0g‘𝑅)) ↔ ∀𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} (𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅)))) |
38 | | eqid 2738 |
. . . . . . 7
⊢
(Base‘𝑅) =
(Base‘𝑅) |
39 | 4, 38, 5, 7, 1 | mplelf 20815 |
. . . . . 6
⊢ (𝜑 → 𝐹:{𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈
Fin}⟶(Base‘𝑅)) |
40 | 39 | feqmptd 6738 |
. . . . 5
⊢ (𝜑 → 𝐹 = (𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (𝐹‘𝑥))) |
41 | | mdegle0.a |
. . . . . 6
⊢ 𝐴 = (algSc‘𝑌) |
42 | | mdegaddle.i |
. . . . . 6
⊢ (𝜑 → 𝐼 ∈ 𝑉) |
43 | | mdegaddle.r |
. . . . . 6
⊢ (𝜑 → 𝑅 ∈ Ring) |
44 | 7 | psrbag0 20875 |
. . . . . . . 8
⊢ (𝐼 ∈ 𝑉 → (𝐼 × {0}) ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈
Fin}) |
45 | 42, 44 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝐼 × {0}) ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈
Fin}) |
46 | 39, 45 | ffvelrnd 6863 |
. . . . . 6
⊢ (𝜑 → (𝐹‘(𝐼 × {0})) ∈ (Base‘𝑅)) |
47 | 4, 7, 6, 38, 41, 42, 43, 46 | mplascl 20877 |
. . . . 5
⊢ (𝜑 → (𝐴‘(𝐹‘(𝐼 × {0}))) = (𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅)))) |
48 | 40, 47 | eqeq12d 2754 |
. . . 4
⊢ (𝜑 → (𝐹 = (𝐴‘(𝐹‘(𝐼 × {0}))) ↔ (𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅))))) |
49 | | fvex 6688 |
. . . . . 6
⊢ (𝐹‘𝑥) ∈ V |
50 | 49 | rgenw 3065 |
. . . . 5
⊢
∀𝑥 ∈
{𝑎 ∈
(ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} (𝐹‘𝑥) ∈ V |
51 | | mpteqb 6795 |
. . . . 5
⊢
(∀𝑥 ∈
{𝑎 ∈
(ℕ0 ↑m 𝐼) ∣ (◡𝑎 “ ℕ) ∈ Fin} (𝐹‘𝑥) ∈ V → ((𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅))) ↔ ∀𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} (𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅)))) |
52 | 50, 51 | mp1i 13 |
. . . 4
⊢ (𝜑 → ((𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦ (𝐹‘𝑥)) = (𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅))) ↔ ∀𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} (𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅)))) |
53 | 48, 52 | bitrd 282 |
. . 3
⊢ (𝜑 → (𝐹 = (𝐴‘(𝐹‘(𝐼 × {0}))) ↔ ∀𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} (𝐹‘𝑥) = if(𝑥 = (𝐼 × {0}), (𝐹‘(𝐼 × {0})), (0g‘𝑅)))) |
54 | 37, 53 | bitr4d 285 |
. 2
⊢ (𝜑 → (∀𝑥 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} (0 < ((𝑏 ∈ {𝑎 ∈ (ℕ0
↑m 𝐼)
∣ (◡𝑎 “ ℕ) ∈ Fin} ↦
(ℂfld Σg 𝑏))‘𝑥) → (𝐹‘𝑥) = (0g‘𝑅)) ↔ 𝐹 = (𝐴‘(𝐹‘(𝐼 × {0}))))) |
55 | 10, 54 | bitrd 282 |
1
⊢ (𝜑 → ((𝐷‘𝐹) ≤ 0 ↔ 𝐹 = (𝐴‘(𝐹‘(𝐼 × {0}))))) |