Step | Hyp | Ref
| Expression |
1 | | df-ceil 13166 |
. 2
⊢ ⌈ =
(𝑥 ∈ ℝ ↦
-(⌊‘-𝑥)) |
2 | | zre 11988 |
. . . . . . 7
⊢ (𝑧 ∈ ℤ → 𝑧 ∈
ℝ) |
3 | | lenegcon2 11147 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 ≤ -𝑧 ↔ 𝑧 ≤ -𝑥)) |
4 | | peano2re 10815 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈
ℝ) |
5 | 4 | anim1ci 617 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑧 ∈ ℝ ∧ (𝑥 + 1) ∈
ℝ)) |
6 | | ltnegcon1 11143 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ ℝ ∧ (𝑥 + 1) ∈ ℝ) →
(-𝑧 < (𝑥 + 1) ↔ -(𝑥 + 1) < 𝑧)) |
7 | 5, 6 | syl 17 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-𝑧 < (𝑥 + 1) ↔ -(𝑥 + 1) < 𝑧)) |
8 | | recn 10629 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℂ) |
9 | | 1cnd 10638 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ → 1 ∈
ℂ) |
10 | 8, 9 | negdid 11012 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ → -(𝑥 + 1) = (-𝑥 + -1)) |
11 | 10 | adantr 483 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → -(𝑥 + 1) = (-𝑥 + -1)) |
12 | 11 | breq1d 5078 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-(𝑥 + 1) < 𝑧 ↔ (-𝑥 + -1) < 𝑧)) |
13 | | renegcl 10951 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ → -𝑥 ∈
ℝ) |
14 | 13 | adantr 483 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → -𝑥 ∈
ℝ) |
15 | | neg1rr 11755 |
. . . . . . . . . . . 12
⊢ -1 ∈
ℝ |
16 | 15 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → -1
∈ ℝ) |
17 | | simpr 487 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ∈
ℝ) |
18 | 14, 16, 17 | ltaddsubd 11242 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((-𝑥 + -1) < 𝑧 ↔ -𝑥 < (𝑧 − -1))) |
19 | | recn 10629 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ ℝ → 𝑧 ∈
ℂ) |
20 | | 1cnd 10638 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ ℝ → 1 ∈
ℂ) |
21 | 19, 20 | subnegd 11006 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ℝ → (𝑧 − -1) = (𝑧 + 1)) |
22 | 21 | adantl 484 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑧 − -1) = (𝑧 + 1)) |
23 | 22 | breq2d 5080 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-𝑥 < (𝑧 − -1) ↔ -𝑥 < (𝑧 + 1))) |
24 | 18, 23 | bitrd 281 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((-𝑥 + -1) < 𝑧 ↔ -𝑥 < (𝑧 + 1))) |
25 | 7, 12, 24 | 3bitrd 307 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-𝑧 < (𝑥 + 1) ↔ -𝑥 < (𝑧 + 1))) |
26 | 3, 25 | anbi12d 632 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑥 ≤ -𝑧 ∧ -𝑧 < (𝑥 + 1)) ↔ (𝑧 ≤ -𝑥 ∧ -𝑥 < (𝑧 + 1)))) |
27 | 2, 26 | sylan2 594 |
. . . . . 6
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℤ) → ((𝑥 ≤ -𝑧 ∧ -𝑧 < (𝑥 + 1)) ↔ (𝑧 ≤ -𝑥 ∧ -𝑥 < (𝑧 + 1)))) |
28 | 27 | riotabidva 7135 |
. . . . 5
⊢ (𝑥 ∈ ℝ →
(℩𝑧 ∈
ℤ (𝑥 ≤ -𝑧 ∧ -𝑧 < (𝑥 + 1))) = (℩𝑧 ∈ ℤ (𝑧 ≤ -𝑥 ∧ -𝑥 < (𝑧 + 1)))) |
29 | 28 | negeqd 10882 |
. . . 4
⊢ (𝑥 ∈ ℝ →
-(℩𝑧 ∈
ℤ (𝑥 ≤ -𝑧 ∧ -𝑧 < (𝑥 + 1))) = -(℩𝑧 ∈ ℤ (𝑧 ≤ -𝑥 ∧ -𝑥 < (𝑧 + 1)))) |
30 | | zbtwnre 12349 |
. . . . 5
⊢ (𝑥 ∈ ℝ →
∃!𝑦 ∈ ℤ
(𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))) |
31 | | breq2 5072 |
. . . . . . 7
⊢ (𝑦 = -𝑧 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ -𝑧)) |
32 | | breq1 5071 |
. . . . . . 7
⊢ (𝑦 = -𝑧 → (𝑦 < (𝑥 + 1) ↔ -𝑧 < (𝑥 + 1))) |
33 | 31, 32 | anbi12d 632 |
. . . . . 6
⊢ (𝑦 = -𝑧 → ((𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)) ↔ (𝑥 ≤ -𝑧 ∧ -𝑧 < (𝑥 + 1)))) |
34 | 33 | zriotaneg 12099 |
. . . . 5
⊢
(∃!𝑦 ∈
ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)) → (℩𝑦 ∈ ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))) = -(℩𝑧 ∈ ℤ (𝑥 ≤ -𝑧 ∧ -𝑧 < (𝑥 + 1)))) |
35 | 30, 34 | syl 17 |
. . . 4
⊢ (𝑥 ∈ ℝ →
(℩𝑦 ∈
ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))) = -(℩𝑧 ∈ ℤ (𝑥 ≤ -𝑧 ∧ -𝑧 < (𝑥 + 1)))) |
36 | | flval 13167 |
. . . . . 6
⊢ (-𝑥 ∈ ℝ →
(⌊‘-𝑥) =
(℩𝑧 ∈
ℤ (𝑧 ≤ -𝑥 ∧ -𝑥 < (𝑧 + 1)))) |
37 | 13, 36 | syl 17 |
. . . . 5
⊢ (𝑥 ∈ ℝ →
(⌊‘-𝑥) =
(℩𝑧 ∈
ℤ (𝑧 ≤ -𝑥 ∧ -𝑥 < (𝑧 + 1)))) |
38 | 37 | negeqd 10882 |
. . . 4
⊢ (𝑥 ∈ ℝ →
-(⌊‘-𝑥) =
-(℩𝑧 ∈
ℤ (𝑧 ≤ -𝑥 ∧ -𝑥 < (𝑧 + 1)))) |
39 | 29, 35, 38 | 3eqtr4rd 2869 |
. . 3
⊢ (𝑥 ∈ ℝ →
-(⌊‘-𝑥) =
(℩𝑦 ∈
ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)))) |
40 | 39 | mpteq2ia 5159 |
. 2
⊢ (𝑥 ∈ ℝ ↦
-(⌊‘-𝑥)) =
(𝑥 ∈ ℝ ↦
(℩𝑦 ∈
ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)))) |
41 | 1, 40 | eqtri 2846 |
1
⊢ ⌈ =
(𝑥 ∈ ℝ ↦
(℩𝑦 ∈
ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)))) |