| Step | Hyp | Ref
| Expression |
| 1 | | vdw 17032 |
. . 3
⊢ ((𝑅 ∈ Fin ∧ 𝐾 ∈ ℕ0)
→ ∃𝑛 ∈
ℕ ∀𝑓 ∈
(𝑅 ↑m
(1...𝑛))∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝑓 “ {𝑐})) |
| 2 | 1 | 3adant2 1132 |
. 2
⊢ ((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅 ∧ 𝐾 ∈ ℕ0) →
∃𝑛 ∈ ℕ
∀𝑓 ∈ (𝑅 ↑m (1...𝑛))∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝑓 “ {𝑐})) |
| 3 | | simpl2 1193 |
. . . . . . 7
⊢ (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅 ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → 𝐹:ℕ⟶𝑅) |
| 4 | | fz1ssnn 13595 |
. . . . . . 7
⊢
(1...𝑛) ⊆
ℕ |
| 5 | | fssres 6774 |
. . . . . . 7
⊢ ((𝐹:ℕ⟶𝑅 ∧ (1...𝑛) ⊆ ℕ) → (𝐹 ↾ (1...𝑛)):(1...𝑛)⟶𝑅) |
| 6 | 3, 4, 5 | sylancl 586 |
. . . . . 6
⊢ (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅 ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → (𝐹 ↾ (1...𝑛)):(1...𝑛)⟶𝑅) |
| 7 | | simpl1 1192 |
. . . . . . 7
⊢ (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅 ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → 𝑅 ∈ Fin) |
| 8 | | ovex 7464 |
. . . . . . 7
⊢
(1...𝑛) ∈
V |
| 9 | | elmapg 8879 |
. . . . . . 7
⊢ ((𝑅 ∈ Fin ∧ (1...𝑛) ∈ V) → ((𝐹 ↾ (1...𝑛)) ∈ (𝑅 ↑m (1...𝑛)) ↔ (𝐹 ↾ (1...𝑛)):(1...𝑛)⟶𝑅)) |
| 10 | 7, 8, 9 | sylancl 586 |
. . . . . 6
⊢ (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅 ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → ((𝐹 ↾ (1...𝑛)) ∈ (𝑅 ↑m (1...𝑛)) ↔ (𝐹 ↾ (1...𝑛)):(1...𝑛)⟶𝑅)) |
| 11 | 6, 10 | mpbird 257 |
. . . . 5
⊢ (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅 ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) → (𝐹 ↾ (1...𝑛)) ∈ (𝑅 ↑m (1...𝑛))) |
| 12 | | cnveq 5884 |
. . . . . . . . . . 11
⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → ◡𝑓 = ◡(𝐹 ↾ (1...𝑛))) |
| 13 | 12 | imaeq1d 6077 |
. . . . . . . . . 10
⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → (◡𝑓 “ {𝑐}) = (◡(𝐹 ↾ (1...𝑛)) “ {𝑐})) |
| 14 | 13 | eleq2d 2827 |
. . . . . . . . 9
⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → ((𝑎 + (𝑚 · 𝑑)) ∈ (◡𝑓 “ {𝑐}) ↔ (𝑎 + (𝑚 · 𝑑)) ∈ (◡(𝐹 ↾ (1...𝑛)) “ {𝑐}))) |
| 15 | 14 | ralbidv 3178 |
. . . . . . . 8
⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → (∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝑓 “ {𝑐}) ↔ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡(𝐹 ↾ (1...𝑛)) “ {𝑐}))) |
| 16 | 15 | 2rexbidv 3222 |
. . . . . . 7
⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → (∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝑓 “ {𝑐}) ↔ ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡(𝐹 ↾ (1...𝑛)) “ {𝑐}))) |
| 17 | 16 | rexbidv 3179 |
. . . . . 6
⊢ (𝑓 = (𝐹 ↾ (1...𝑛)) → (∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝑓 “ {𝑐}) ↔ ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡(𝐹 ↾ (1...𝑛)) “ {𝑐}))) |
| 18 | 17 | rspcv 3618 |
. . . . 5
⊢ ((𝐹 ↾ (1...𝑛)) ∈ (𝑅 ↑m (1...𝑛)) → (∀𝑓 ∈ (𝑅 ↑m (1...𝑛))∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝑓 “ {𝑐}) → ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡(𝐹 ↾ (1...𝑛)) “ {𝑐}))) |
| 19 | 11, 18 | syl 17 |
. . . 4
⊢ (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅 ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) →
(∀𝑓 ∈ (𝑅 ↑m (1...𝑛))∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝑓 “ {𝑐}) → ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡(𝐹 ↾ (1...𝑛)) “ {𝑐}))) |
| 20 | | resss 6019 |
. . . . . . . . . 10
⊢ (𝐹 ↾ (1...𝑛)) ⊆ 𝐹 |
| 21 | | cnvss 5883 |
. . . . . . . . . 10
⊢ ((𝐹 ↾ (1...𝑛)) ⊆ 𝐹 → ◡(𝐹 ↾ (1...𝑛)) ⊆ ◡𝐹) |
| 22 | | imass1 6119 |
. . . . . . . . . 10
⊢ (◡(𝐹 ↾ (1...𝑛)) ⊆ ◡𝐹 → (◡(𝐹 ↾ (1...𝑛)) “ {𝑐}) ⊆ (◡𝐹 “ {𝑐})) |
| 23 | 20, 21, 22 | mp2b 10 |
. . . . . . . . 9
⊢ (◡(𝐹 ↾ (1...𝑛)) “ {𝑐}) ⊆ (◡𝐹 “ {𝑐}) |
| 24 | 23 | sseli 3979 |
. . . . . . . 8
⊢ ((𝑎 + (𝑚 · 𝑑)) ∈ (◡(𝐹 ↾ (1...𝑛)) “ {𝑐}) → (𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})) |
| 25 | 24 | ralimi 3083 |
. . . . . . 7
⊢
(∀𝑚 ∈
(0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡(𝐹 ↾ (1...𝑛)) “ {𝑐}) → ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})) |
| 26 | 25 | reximi 3084 |
. . . . . 6
⊢
(∃𝑑 ∈
ℕ ∀𝑚 ∈
(0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡(𝐹 ↾ (1...𝑛)) “ {𝑐}) → ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})) |
| 27 | 26 | reximi 3084 |
. . . . 5
⊢
(∃𝑎 ∈
ℕ ∃𝑑 ∈
ℕ ∀𝑚 ∈
(0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡(𝐹 ↾ (1...𝑛)) “ {𝑐}) → ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})) |
| 28 | 27 | reximi 3084 |
. . . 4
⊢
(∃𝑐 ∈
𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡(𝐹 ↾ (1...𝑛)) “ {𝑐}) → ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})) |
| 29 | 19, 28 | syl6 35 |
. . 3
⊢ (((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅 ∧ 𝐾 ∈ ℕ0) ∧ 𝑛 ∈ ℕ) →
(∀𝑓 ∈ (𝑅 ↑m (1...𝑛))∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝑓 “ {𝑐}) → ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}))) |
| 30 | 29 | rexlimdva 3155 |
. 2
⊢ ((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅 ∧ 𝐾 ∈ ℕ0) →
(∃𝑛 ∈ ℕ
∀𝑓 ∈ (𝑅 ↑m (1...𝑛))∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝑓 “ {𝑐}) → ∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐}))) |
| 31 | 2, 30 | mpd 15 |
1
⊢ ((𝑅 ∈ Fin ∧ 𝐹:ℕ⟶𝑅 ∧ 𝐾 ∈ ℕ0) →
∃𝑐 ∈ 𝑅 ∃𝑎 ∈ ℕ ∃𝑑 ∈ ℕ ∀𝑚 ∈ (0...(𝐾 − 1))(𝑎 + (𝑚 · 𝑑)) ∈ (◡𝐹 “ {𝑐})) |