Step | Hyp | Ref
| Expression |
1 | | eldiophb 40282 |
. . 3
⊢ (𝐴 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧
∃𝑎 ∈
(ℤ≥‘𝑁)∃𝑏 ∈ (mzPoly‘(1...𝑎))𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0
↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)})) |
2 | | simp-5r 786 |
. . . . . . . . 9
⊢
((((((𝑁 ∈
ℕ0 ∧ 𝑆
∈ V) ∧ (¬ 𝑆
∈ Fin ∧ (1...𝑁)
⊆ 𝑆)) ∧ (𝑎 ∈
(ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑆 ∈ V) |
3 | | simprr 773 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑏 ∈ (mzPoly‘(1...𝑎))) |
4 | 3 | ad2antrr 726 |
. . . . . . . . 9
⊢
((((((𝑁 ∈
ℕ0 ∧ 𝑆
∈ V) ∧ (¬ 𝑆
∈ Fin ∧ (1...𝑁)
⊆ 𝑆)) ∧ (𝑎 ∈
(ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑏 ∈ (mzPoly‘(1...𝑎))) |
5 | | simprl 771 |
. . . . . . . . . 10
⊢
((((((𝑁 ∈
ℕ0 ∧ 𝑆
∈ V) ∧ (¬ 𝑆
∈ Fin ∧ (1...𝑁)
⊆ 𝑆)) ∧ (𝑎 ∈
(ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑐:(1...𝑎)–1-1→𝑆) |
6 | | f1f 6615 |
. . . . . . . . . 10
⊢ (𝑐:(1...𝑎)–1-1→𝑆 → 𝑐:(1...𝑎)⟶𝑆) |
7 | 5, 6 | syl 17 |
. . . . . . . . 9
⊢
((((((𝑁 ∈
ℕ0 ∧ 𝑆
∈ V) ∧ (¬ 𝑆
∈ Fin ∧ (1...𝑁)
⊆ 𝑆)) ∧ (𝑎 ∈
(ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → 𝑐:(1...𝑎)⟶𝑆) |
8 | | mzprename 40274 |
. . . . . . . . 9
⊢ ((𝑆 ∈ V ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)) ∧ 𝑐:(1...𝑎)⟶𝑆) → (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) ∈ (mzPoly‘𝑆)) |
9 | 2, 4, 7, 8 | syl3anc 1373 |
. . . . . . . 8
⊢
((((((𝑁 ∈
ℕ0 ∧ 𝑆
∈ V) ∧ (¬ 𝑆
∈ Fin ∧ (1...𝑁)
⊆ 𝑆)) ∧ (𝑎 ∈
(ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) ∈ (mzPoly‘𝑆)) |
10 | | simprr 773 |
. . . . . . . . 9
⊢
((((((𝑁 ∈
ℕ0 ∧ 𝑆
∈ V) ∧ (¬ 𝑆
∈ Fin ∧ (1...𝑁)
⊆ 𝑆)) ∧ (𝑎 ∈
(ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) |
11 | | diophrw 40284 |
. . . . . . . . . 10
⊢ ((𝑆 ∈ V ∧ 𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0)} = {𝑡 ∣ ∃𝑑 ∈ (ℕ0
↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)}) |
12 | 11 | eqcomd 2743 |
. . . . . . . . 9
⊢ ((𝑆 ∈ V ∧ 𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) → {𝑡 ∣ ∃𝑑 ∈ (ℕ0
↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0)}) |
13 | 2, 5, 10, 12 | syl3anc 1373 |
. . . . . . . 8
⊢
((((((𝑁 ∈
ℕ0 ∧ 𝑆
∈ V) ∧ (¬ 𝑆
∈ Fin ∧ (1...𝑁)
⊆ 𝑆)) ∧ (𝑎 ∈
(ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → {𝑡 ∣ ∃𝑑 ∈ (ℕ0
↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0)}) |
14 | | fveq1 6716 |
. . . . . . . . . . . . 13
⊢ (𝑝 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → (𝑝‘𝑢) = ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢)) |
15 | 14 | eqeq1d 2739 |
. . . . . . . . . . . 12
⊢ (𝑝 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → ((𝑝‘𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0)) |
16 | 15 | anbi2d 632 |
. . . . . . . . . . 11
⊢ (𝑝 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0))) |
17 | 16 | rexbidv 3216 |
. . . . . . . . . 10
⊢ (𝑝 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → (∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0))) |
18 | 17 | abbidv 2807 |
. . . . . . . . 9
⊢ (𝑝 = (𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0)}) |
19 | 18 | rspceeqv 3552 |
. . . . . . . 8
⊢ (((𝑒 ∈ (ℤ
↑m 𝑆)
↦ (𝑏‘(𝑒 ∘ 𝑐))) ∈ (mzPoly‘𝑆) ∧ {𝑡 ∣ ∃𝑑 ∈ (ℕ0
↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ ((𝑒 ∈ (ℤ ↑m 𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0)}) → ∃𝑝 ∈ (mzPoly‘𝑆){𝑡 ∣ ∃𝑑 ∈ (ℕ0
↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) |
20 | 9, 13, 19 | syl2anc 587 |
. . . . . . 7
⊢
((((((𝑁 ∈
ℕ0 ∧ 𝑆
∈ V) ∧ (¬ 𝑆
∈ Fin ∧ (1...𝑁)
⊆ 𝑆)) ∧ (𝑎 ∈
(ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) ∧ 𝑐 ∈ V) ∧ (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) → ∃𝑝 ∈ (mzPoly‘𝑆){𝑡 ∣ ∃𝑑 ∈ (ℕ0
↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) |
21 | | simplll 775 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑁 ∈
ℕ0) |
22 | | simplrl 777 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ¬ 𝑆 ∈ Fin) |
23 | | simplrr 778 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → (1...𝑁) ⊆ 𝑆) |
24 | | simprl 771 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑎 ∈ (ℤ≥‘𝑁)) |
25 | | eldioph2lem2 40286 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ0
∧ ¬ 𝑆 ∈ Fin)
∧ ((1...𝑁) ⊆
𝑆 ∧ 𝑎 ∈ (ℤ≥‘𝑁))) → ∃𝑐(𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) |
26 | 21, 22, 23, 24, 25 | syl22anc 839 |
. . . . . . . 8
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ∃𝑐(𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) |
27 | | rexv 3433 |
. . . . . . . 8
⊢
(∃𝑐 ∈ V
(𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁))) ↔ ∃𝑐(𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) |
28 | 26, 27 | sylibr 237 |
. . . . . . 7
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ∃𝑐 ∈ V (𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) |
29 | 20, 28 | r19.29a 3208 |
. . . . . 6
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ∃𝑝 ∈ (mzPoly‘𝑆){𝑡 ∣ ∃𝑑 ∈ (ℕ0
↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) |
30 | | eqeq1 2741 |
. . . . . . 7
⊢ (𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0
↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} → (𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ↔ {𝑡 ∣ ∃𝑑 ∈ (ℕ0
↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) |
31 | 30 | rexbidv 3216 |
. . . . . 6
⊢ (𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0
↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} → (∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ↔ ∃𝑝 ∈ (mzPoly‘𝑆){𝑡 ∣ ∃𝑑 ∈ (ℕ0
↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) |
32 | 29, 31 | syl5ibrcom 250 |
. . . . 5
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → (𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0
↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} → ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) |
33 | 32 | rexlimdvva 3213 |
. . . 4
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) → (∃𝑎 ∈
(ℤ≥‘𝑁)∃𝑏 ∈ (mzPoly‘(1...𝑎))𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0
↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)} → ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) |
34 | 33 | adantld 494 |
. . 3
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) → ((𝑁 ∈ ℕ0 ∧
∃𝑎 ∈
(ℤ≥‘𝑁)∃𝑏 ∈ (mzPoly‘(1...𝑎))𝐴 = {𝑡 ∣ ∃𝑑 ∈ (ℕ0
↑m (1...𝑎))(𝑡 = (𝑑 ↾ (1...𝑁)) ∧ (𝑏‘𝑑) = 0)}) → ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) |
35 | 1, 34 | syl5bi 245 |
. 2
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) → (𝐴 ∈ (Dioph‘𝑁) → ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) |
36 | | simpr 488 |
. . . 4
⊢
(((((𝑁 ∈
ℕ0 ∧ 𝑆
∈ V) ∧ (¬ 𝑆
∈ Fin ∧ (1...𝑁)
⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) ∧ 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) → 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) |
37 | | simplll 775 |
. . . . . 6
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑁 ∈
ℕ0) |
38 | | simpllr 776 |
. . . . . 6
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑆 ∈ V) |
39 | | simplrr 778 |
. . . . . 6
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → (1...𝑁) ⊆ 𝑆) |
40 | | simpr 488 |
. . . . . 6
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑝 ∈ (mzPoly‘𝑆)) |
41 | | eldioph2 40287 |
. . . . . 6
⊢ ((𝑁 ∈ ℕ0
∧ (𝑆 ∈ V ∧
(1...𝑁) ⊆ 𝑆) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ∈ (Dioph‘𝑁)) |
42 | 37, 38, 39, 40, 41 | syl121anc 1377 |
. . . . 5
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ∈ (Dioph‘𝑁)) |
43 | 42 | adantr 484 |
. . . 4
⊢
(((((𝑁 ∈
ℕ0 ∧ 𝑆
∈ V) ∧ (¬ 𝑆
∈ Fin ∧ (1...𝑁)
⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) ∧ 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ∈ (Dioph‘𝑁)) |
44 | 36, 43 | eqeltrd 2838 |
. . 3
⊢
(((((𝑁 ∈
ℕ0 ∧ 𝑆
∈ V) ∧ (¬ 𝑆
∈ Fin ∧ (1...𝑁)
⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) ∧ 𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) → 𝐴 ∈ (Dioph‘𝑁)) |
45 | 44 | rexlimdva2 3206 |
. 2
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) → (∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} → 𝐴 ∈ (Dioph‘𝑁))) |
46 | 35, 45 | impbid 215 |
1
⊢ (((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) → (𝐴 ∈ (Dioph‘𝑁) ↔ ∃𝑝 ∈ (mzPoly‘𝑆)𝐴 = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑m 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) |