Step | Hyp | Ref
| Expression |
1 | | simpl 473 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈
(mzPoly‘ℕ)) → 𝑁 ∈
ℕ0) |
2 | | simpr 477 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈
(mzPoly‘ℕ)) → 𝑃 ∈
(mzPoly‘ℕ)) |
3 | | eqidd 2623 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈
(mzPoly‘ℕ)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)}) |
4 | | fveq1 6190 |
. . . . . . . . . 10
⊢ (𝑝 = 𝑃 → (𝑝‘𝑏) = (𝑃‘𝑏)) |
5 | 4 | eqeq1d 2624 |
. . . . . . . . 9
⊢ (𝑝 = 𝑃 → ((𝑝‘𝑏) = 0 ↔ (𝑃‘𝑏) = 0)) |
6 | 5 | anbi2d 740 |
. . . . . . . 8
⊢ (𝑝 = 𝑃 → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0) ↔ (𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0))) |
7 | 6 | rexbidv 3052 |
. . . . . . 7
⊢ (𝑝 = 𝑃 → (∃𝑏 ∈ (ℕ0
↑𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0) ↔ ∃𝑏 ∈ (ℕ0
↑𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0))) |
8 | 7 | abbidv 2741 |
. . . . . 6
⊢ (𝑝 = 𝑃 → {𝑎 ∣ ∃𝑏 ∈ (ℕ0
↑𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0
↑𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0)}) |
9 | | eqeq1 2626 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁)))) |
10 | 9 | anbi1d 741 |
. . . . . . . . 9
⊢ (𝑎 = 𝑡 → ((𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0) ↔ (𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0))) |
11 | 10 | rexbidv 3052 |
. . . . . . . 8
⊢ (𝑎 = 𝑡 → (∃𝑏 ∈ (ℕ0
↑𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0) ↔ ∃𝑏 ∈ (ℕ0
↑𝑚 ℕ)(𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0))) |
12 | | reseq1 5390 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁))) |
13 | 12 | eqeq2d 2632 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁)))) |
14 | | fveq2 6191 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑢 → (𝑃‘𝑏) = (𝑃‘𝑢)) |
15 | 14 | eqeq1d 2624 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑢 → ((𝑃‘𝑏) = 0 ↔ (𝑃‘𝑢) = 0)) |
16 | 13, 15 | anbi12d 747 |
. . . . . . . . 9
⊢ (𝑏 = 𝑢 → ((𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0) ↔ (𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0))) |
17 | 16 | cbvrexv 3172 |
. . . . . . . 8
⊢
(∃𝑏 ∈
(ℕ0 ↑𝑚 ℕ)(𝑡 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0) ↔ ∃𝑢 ∈ (ℕ0
↑𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)) |
18 | 11, 17 | syl6bb 276 |
. . . . . . 7
⊢ (𝑎 = 𝑡 → (∃𝑏 ∈ (ℕ0
↑𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0) ↔ ∃𝑢 ∈ (ℕ0
↑𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0))) |
19 | 18 | cbvabv 2747 |
. . . . . 6
⊢ {𝑎 ∣ ∃𝑏 ∈ (ℕ0
↑𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑃‘𝑏) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} |
20 | 8, 19 | syl6eq 2672 |
. . . . 5
⊢ (𝑝 = 𝑃 → {𝑎 ∣ ∃𝑏 ∈ (ℕ0
↑𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)}) |
21 | 20 | eqeq2d 2632 |
. . . 4
⊢ (𝑝 = 𝑃 → ({𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0
↑𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0)} ↔ {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)})) |
22 | 21 | rspcev 3309 |
. . 3
⊢ ((𝑃 ∈ (mzPoly‘ℕ)
∧ {𝑡 ∣
∃𝑢 ∈
(ℕ0 ↑𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)}) → ∃𝑝 ∈ (mzPoly‘ℕ){𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0
↑𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0)}) |
23 | 2, 3, 22 | syl2anc 693 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈
(mzPoly‘ℕ)) → ∃𝑝 ∈ (mzPoly‘ℕ){𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0
↑𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0)}) |
24 | | eldioph3b 37328 |
. 2
⊢ ({𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧
∃𝑝 ∈
(mzPoly‘ℕ){𝑡
∣ ∃𝑢 ∈
(ℕ0 ↑𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0
↑𝑚 ℕ)(𝑎 = (𝑏 ↾ (1...𝑁)) ∧ (𝑝‘𝑏) = 0)})) |
25 | 1, 23, 24 | sylanbrc 698 |
1
⊢ ((𝑁 ∈ ℕ0
∧ 𝑃 ∈
(mzPoly‘ℕ)) → {𝑡 ∣ ∃𝑢 ∈ (ℕ0
↑𝑚 ℕ)(𝑡 = (𝑢 ↾ (1...𝑁)) ∧ (𝑃‘𝑢) = 0)} ∈ (Dioph‘𝑁)) |