Step | Hyp | Ref
| Expression |
1 | | simp1 1061 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈
(ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → 𝑁 ∈
ℕ0) |
2 | | simp2 1062 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈
(ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → 𝐾 ∈ (ℤ≥‘𝑁)) |
3 | | simp3 1063 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈
(ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → 𝑃 ∈ (mzPoly‘(1...𝐾))) |
4 | | eqidd 2623 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈
(ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)}) |
5 | | fveq1 6190 |
. . . . . . . . . 10
⊢ (𝑝 = 𝑃 → (𝑝‘𝑢) = (𝑃‘𝑢)) |
6 | 5 | eqeq1d 2624 |
. . . . . . . . 9
⊢ (𝑝 = 𝑃 → ((𝑝‘𝑢) = 0 ↔ (𝑃‘𝑢) = 0)) |
7 | 6 | anbi2d 740 |
. . . . . . . 8
⊢ (𝑝 = 𝑃 → ((𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0))) |
8 | 7 | rexbidv 3052 |
. . . . . . 7
⊢ (𝑝 = 𝑃 → (∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0))) |
9 | 8 | abbidv 2741 |
. . . . . 6
⊢ (𝑝 = 𝑃 → {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)}) |
10 | 9 | eqeq2d 2632 |
. . . . 5
⊢ (𝑝 = 𝑃 → ({𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ↔ {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)})) |
11 | 10 | rspcev 3309 |
. . . 4
⊢ ((𝑃 ∈ (mzPoly‘(1...𝐾)) ∧ {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)}) → ∃𝑝 ∈ (mzPoly‘(1...𝐾)){𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) |
12 | 3, 4, 11 | syl2anc 693 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈
(ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → ∃𝑝 ∈ (mzPoly‘(1...𝐾)){𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) |
13 | | oveq2 6658 |
. . . . . 6
⊢ (𝑘 = 𝐾 → (1...𝑘) = (1...𝐾)) |
14 | 13 | fveq2d 6195 |
. . . . 5
⊢ (𝑘 = 𝐾 → (mzPoly‘(1...𝑘)) = (mzPoly‘(1...𝐾))) |
15 | 13 | oveq2d 6666 |
. . . . . . . 8
⊢ (𝑘 = 𝐾 → (ℕ0
↑𝑚 (1...𝑘)) = (ℕ0
↑𝑚 (1...𝐾))) |
16 | 15 | rexeqdv 3145 |
. . . . . . 7
⊢ (𝑘 = 𝐾 → (∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0) ↔ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0))) |
17 | 16 | abbidv 2741 |
. . . . . 6
⊢ (𝑘 = 𝐾 → {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) |
18 | 17 | eqeq2d 2632 |
. . . . 5
⊢ (𝑘 = 𝐾 → ({𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ↔ {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) |
19 | 14, 18 | rexeqbidv 3153 |
. . . 4
⊢ (𝑘 = 𝐾 → (∃𝑝 ∈ (mzPoly‘(1...𝑘)){𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)} ↔ ∃𝑝 ∈ (mzPoly‘(1...𝐾)){𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) |
20 | 19 | rspcev 3309 |
. . 3
⊢ ((𝐾 ∈
(ℤ≥‘𝑁) ∧ ∃𝑝 ∈ (mzPoly‘(1...𝐾)){𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) → ∃𝑘 ∈ (ℤ≥‘𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘)){𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) |
21 | 2, 12, 20 | syl2anc 693 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈
(ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → ∃𝑘 ∈ (ℤ≥‘𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘)){𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)}) |
22 | | eldiophb 37320 |
. 2
⊢ ({𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧
∃𝑘 ∈
(ℤ≥‘𝑁)∃𝑝 ∈ (mzPoly‘(1...𝑘)){𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝑘))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑝‘𝑢) = 0)})) |
23 | 1, 21, 22 | sylanbrc 698 |
1
⊢ ((𝑁 ∈ ℕ0
∧ 𝐾 ∈
(ℤ≥‘𝑁) ∧ 𝑃 ∈ (mzPoly‘(1...𝐾))) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 (1...𝐾))(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁)) |