Step | Hyp | Ref
| Expression |
1 | | eldiophb 36338 |
. . 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 805 |
. . . . . . . . 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 792 |
. . . . . . . . . 10
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑏 ∈ (mzPoly‘(1...𝑎))) |
4 | 3 | ad2antrr 758 |
. . . . . . . . 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 790 |
. . . . . . . . . 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 6014 |
. . . . . . . . . 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 36330 |
. . . . . . . . 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 1318 |
. . . . . . . 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 792 |
. . . . . . . . 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 36340 |
. . . . . . . . . 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 2616 |
. . . . . . . . 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 1318 |
. . . . . . . 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 6102 |
. . . . . . . . . . . . . 14
⊢ (𝑝 = (𝑒 ∈ (ℤ ↑𝑚
𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → (𝑝‘𝑢) = ((𝑒 ∈ (ℤ ↑𝑚
𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢)) |
15 | 14 | eqeq1d 2612 |
. . . . . . . . . . . . 13
⊢ (𝑝 = (𝑒 ∈ (ℤ ↑𝑚
𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → ((𝑝‘𝑢) = 0 ↔ ((𝑒 ∈ (ℤ ↑𝑚
𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐)))‘𝑢) = 0)) |
16 | 15 | anbi2d 736 |
. . . . . . . . . . . 12
⊢ (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 3034 |
. . . . . . . . . . 11
⊢ (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 2728 |
. . . . . . . . . 10
⊢ (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 | eqeq2d 2620 |
. . . . . . . . 9
⊢ (𝑝 = (𝑒 ∈ (ℤ ↑𝑚
𝑆) ↦ (𝑏‘(𝑒 ∘ 𝑐))) → ({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)} ↔ {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)})) |
20 | 19 | rspcev 3282 |
. . . . . . . 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)}) |
21 | 9, 13, 20 | syl2anc 691 |
. . . . . . 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)}) |
22 | | simplll 794 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑁 ∈
ℕ0) |
23 | | simplrl 796 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → ¬ 𝑆 ∈ Fin) |
24 | | simplrr 797 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → (1...𝑁) ⊆ 𝑆) |
25 | | simprl 790 |
. . . . . . . . 9
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ (𝑎 ∈ (ℤ≥‘𝑁) ∧ 𝑏 ∈ (mzPoly‘(1...𝑎)))) → 𝑎 ∈ (ℤ≥‘𝑁)) |
26 | | eldioph2lem2 36342 |
. . . . . . . . 9
⊢ (((𝑁 ∈ ℕ0
∧ ¬ 𝑆 ∈ Fin)
∧ ((1...𝑁) ⊆
𝑆 ∧ 𝑎 ∈ (ℤ≥‘𝑁))) → ∃𝑐(𝑐:(1...𝑎)–1-1→𝑆 ∧ (𝑐 ↾ (1...𝑁)) = ( I ↾ (1...𝑁)))) |
27 | 22, 23, 24, 25, 26 | syl22anc 1319 |
. . . . . . . 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)))) |
28 | | rexv 3193 |
. . . . . . . 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)))) |
29 | 27, 28 | sylibr 223 |
. . . . . . 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)))) |
30 | 21, 29 | r19.29a 3060 |
. . . . . 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)}) |
31 | | eqeq1 2614 |
. . . . . . 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)})) |
32 | 31 | rexbidv 3034 |
. . . . . 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)})) |
33 | 30, 32 | syl5ibrcom 236 |
. . . . 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)})) |
34 | 33 | rexlimdvva 3020 |
. . . 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)})) |
35 | 34 | adantld 482 |
. . 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)})) |
36 | 1, 35 | syl5bi 231 |
. 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)})) |
37 | | simpr 476 |
. . . . 5
⊢
(((((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)}) |
38 | | simplll 794 |
. . . . . . 7
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑁 ∈
ℕ0) |
39 | | simpllr 795 |
. . . . . . 7
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑆 ∈ V) |
40 | | simplrr 797 |
. . . . . . 7
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → (1...𝑁) ⊆ 𝑆) |
41 | | simpr 476 |
. . . . . . 7
⊢ ((((𝑁 ∈ ℕ0
∧ 𝑆 ∈ V) ∧
(¬ 𝑆 ∈ Fin ∧
(1...𝑁) ⊆ 𝑆)) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → 𝑝 ∈ (mzPoly‘𝑆)) |
42 | | eldioph2 36343 |
. . . . . . 7
⊢ ((𝑁 ∈ ℕ0
∧ (𝑆 ∈ V ∧
(1...𝑁) ⊆ 𝑆) ∧ 𝑝 ∈ (mzPoly‘𝑆)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 𝑆)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ∈ (Dioph‘𝑁)) |
43 | 38, 39, 40, 41, 42 | syl121anc 1323 |
. . . . . 6
⊢ ((((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)) |
44 | 43 | adantr 480 |
. . . . 5
⊢
(((((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)) |
45 | 37, 44 | eqeltrd 2688 |
. . . 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 ∈ (Dioph‘𝑁)) |
46 | 45 | ex 449 |
. . 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))) |
47 | 46 | rexlimdva 3013 |
. 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))) |
48 | 36, 47 | impbid 201 |
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)})) |