Step | Hyp | Ref
| Expression |
1 | | df-ceil 13798 |
. 2
⊢ ⌈ =
(𝑥 ∈ ℝ ↦
-(⌊‘-𝑥)) |
2 | | zre 12600 |
. . . . . . 7
⊢ (𝑧 ∈ ℤ → 𝑧 ∈
ℝ) |
3 | | lenegcon2 11757 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 ≤ -𝑧 ↔ 𝑧 ≤ -𝑥)) |
4 | | peano2re 11425 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈
ℝ) |
5 | 4 | anim1ci 614 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑧 ∈ ℝ ∧ (𝑥 + 1) ∈
ℝ)) |
6 | | ltnegcon1 11753 |
. . . . . . . . . 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 11236 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℂ) |
9 | | 1cnd 11247 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ → 1 ∈
ℂ) |
10 | 8, 9 | negdid 11622 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ → -(𝑥 + 1) = (-𝑥 + -1)) |
11 | 10 | adantr 479 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → -(𝑥 + 1) = (-𝑥 + -1)) |
12 | 11 | breq1d 5162 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-(𝑥 + 1) < 𝑧 ↔ (-𝑥 + -1) < 𝑧)) |
13 | | renegcl 11561 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ → -𝑥 ∈
ℝ) |
14 | 13 | adantr 479 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → -𝑥 ∈
ℝ) |
15 | | neg1rr 12365 |
. . . . . . . . . . . 12
⊢ -1 ∈
ℝ |
16 | 15 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → -1
∈ ℝ) |
17 | | simpr 483 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ∈
ℝ) |
18 | 14, 16, 17 | ltaddsubd 11852 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((-𝑥 + -1) < 𝑧 ↔ -𝑥 < (𝑧 − -1))) |
19 | | recn 11236 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ ℝ → 𝑧 ∈
ℂ) |
20 | | 1cnd 11247 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ ℝ → 1 ∈
ℂ) |
21 | 19, 20 | subnegd 11616 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ℝ → (𝑧 − -1) = (𝑧 + 1)) |
22 | 21 | adantl 480 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑧 − -1) = (𝑧 + 1)) |
23 | 22 | breq2d 5164 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-𝑥 < (𝑧 − -1) ↔ -𝑥 < (𝑧 + 1))) |
24 | 18, 23 | bitrd 278 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((-𝑥 + -1) < 𝑧 ↔ -𝑥 < (𝑧 + 1))) |
25 | 7, 12, 24 | 3bitrd 304 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-𝑧 < (𝑥 + 1) ↔ -𝑥 < (𝑧 + 1))) |
26 | 3, 25 | anbi12d 630 |
. . . . . . 7
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((𝑥 ≤ -𝑧 ∧ -𝑧 < (𝑥 + 1)) ↔ (𝑧 ≤ -𝑥 ∧ -𝑥 < (𝑧 + 1)))) |
27 | 2, 26 | sylan2 591 |
. . . . . 6
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℤ) → ((𝑥 ≤ -𝑧 ∧ -𝑧 < (𝑥 + 1)) ↔ (𝑧 ≤ -𝑥 ∧ -𝑥 < (𝑧 + 1)))) |
28 | 27 | riotabidva 7402 |
. . . . 5
⊢ (𝑥 ∈ ℝ →
(℩𝑧 ∈
ℤ (𝑥 ≤ -𝑧 ∧ -𝑧 < (𝑥 + 1))) = (℩𝑧 ∈ ℤ (𝑧 ≤ -𝑥 ∧ -𝑥 < (𝑧 + 1)))) |
29 | 28 | negeqd 11492 |
. . . 4
⊢ (𝑥 ∈ ℝ →
-(℩𝑧 ∈
ℤ (𝑥 ≤ -𝑧 ∧ -𝑧 < (𝑥 + 1))) = -(℩𝑧 ∈ ℤ (𝑧 ≤ -𝑥 ∧ -𝑥 < (𝑧 + 1)))) |
30 | | zbtwnre 12968 |
. . . . 5
⊢ (𝑥 ∈ ℝ →
∃!𝑦 ∈ ℤ
(𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))) |
31 | | breq2 5156 |
. . . . . . 7
⊢ (𝑦 = -𝑧 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ -𝑧)) |
32 | | breq1 5155 |
. . . . . . 7
⊢ (𝑦 = -𝑧 → (𝑦 < (𝑥 + 1) ↔ -𝑧 < (𝑥 + 1))) |
33 | 31, 32 | anbi12d 630 |
. . . . . 6
⊢ (𝑦 = -𝑧 → ((𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)) ↔ (𝑥 ≤ -𝑧 ∧ -𝑧 < (𝑥 + 1)))) |
34 | 33 | zriotaneg 12713 |
. . . . 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 13799 |
. . . . . 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 11492 |
. . . 4
⊢ (𝑥 ∈ ℝ →
-(⌊‘-𝑥) =
-(℩𝑧 ∈
ℤ (𝑧 ≤ -𝑥 ∧ -𝑥 < (𝑧 + 1)))) |
39 | 29, 35, 38 | 3eqtr4rd 2779 |
. . 3
⊢ (𝑥 ∈ ℝ →
-(⌊‘-𝑥) =
(℩𝑦 ∈
ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)))) |
40 | 39 | mpteq2ia 5255 |
. 2
⊢ (𝑥 ∈ ℝ ↦
-(⌊‘-𝑥)) =
(𝑥 ∈ ℝ ↦
(℩𝑦 ∈
ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)))) |
41 | 1, 40 | eqtri 2756 |
1
⊢ ⌈ =
(𝑥 ∈ ℝ ↦
(℩𝑦 ∈
ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)))) |