| Step | Hyp | Ref
| Expression |
| 1 | | df-ceil 13833 |
. 2
⊢ ⌈ =
(𝑥 ∈ ℝ ↦
-(⌊‘-𝑥)) |
| 2 | | zre 12617 |
. . . . . . 7
⊢ (𝑧 ∈ ℤ → 𝑧 ∈
ℝ) |
| 3 | | lenegcon2 11768 |
. . . . . . . 8
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑥 ≤ -𝑧 ↔ 𝑧 ≤ -𝑥)) |
| 4 | | peano2re 11434 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈
ℝ) |
| 5 | 4 | anim1ci 616 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑧 ∈ ℝ ∧ (𝑥 + 1) ∈
ℝ)) |
| 6 | | ltnegcon1 11764 |
. . . . . . . . . 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 11245 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ → 𝑥 ∈
ℂ) |
| 9 | | 1cnd 11256 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ → 1 ∈
ℂ) |
| 10 | 8, 9 | negdid 11633 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℝ → -(𝑥 + 1) = (-𝑥 + -1)) |
| 11 | 10 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → -(𝑥 + 1) = (-𝑥 + -1)) |
| 12 | 11 | breq1d 5153 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-(𝑥 + 1) < 𝑧 ↔ (-𝑥 + -1) < 𝑧)) |
| 13 | | renegcl 11572 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℝ → -𝑥 ∈
ℝ) |
| 14 | 13 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → -𝑥 ∈
ℝ) |
| 15 | | neg1rr 12381 |
. . . . . . . . . . . 12
⊢ -1 ∈
ℝ |
| 16 | 15 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → -1
∈ ℝ) |
| 17 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → 𝑧 ∈
ℝ) |
| 18 | 14, 16, 17 | ltaddsubd 11863 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((-𝑥 + -1) < 𝑧 ↔ -𝑥 < (𝑧 − -1))) |
| 19 | | recn 11245 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ ℝ → 𝑧 ∈
ℂ) |
| 20 | | 1cnd 11256 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ ℝ → 1 ∈
ℂ) |
| 21 | 19, 20 | subnegd 11627 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ℝ → (𝑧 − -1) = (𝑧 + 1)) |
| 22 | 21 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (𝑧 − -1) = (𝑧 + 1)) |
| 23 | 22 | breq2d 5155 |
. . . . . . . . . 10
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (-𝑥 < (𝑧 − -1) ↔ -𝑥 < (𝑧 + 1))) |
| 24 | 18, 23 | bitrd 279 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℝ) → ((-𝑥 + -1) < 𝑧 ↔ -𝑥 < (𝑧 + 1))) |
| 25 | 7, 12, 24 | 3bitrd 305 |
. . . . . . . 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 593 |
. . . . . 6
⊢ ((𝑥 ∈ ℝ ∧ 𝑧 ∈ ℤ) → ((𝑥 ≤ -𝑧 ∧ -𝑧 < (𝑥 + 1)) ↔ (𝑧 ≤ -𝑥 ∧ -𝑥 < (𝑧 + 1)))) |
| 28 | 27 | riotabidva 7407 |
. . . . 5
⊢ (𝑥 ∈ ℝ →
(℩𝑧 ∈
ℤ (𝑥 ≤ -𝑧 ∧ -𝑧 < (𝑥 + 1))) = (℩𝑧 ∈ ℤ (𝑧 ≤ -𝑥 ∧ -𝑥 < (𝑧 + 1)))) |
| 29 | 28 | negeqd 11502 |
. . . 4
⊢ (𝑥 ∈ ℝ →
-(℩𝑧 ∈
ℤ (𝑥 ≤ -𝑧 ∧ -𝑧 < (𝑥 + 1))) = -(℩𝑧 ∈ ℤ (𝑧 ≤ -𝑥 ∧ -𝑥 < (𝑧 + 1)))) |
| 30 | | zbtwnre 12988 |
. . . . 5
⊢ (𝑥 ∈ ℝ →
∃!𝑦 ∈ ℤ
(𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))) |
| 31 | | breq2 5147 |
. . . . . . 7
⊢ (𝑦 = -𝑧 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ -𝑧)) |
| 32 | | breq1 5146 |
. . . . . . 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 12731 |
. . . . 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 13834 |
. . . . . 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 11502 |
. . . 4
⊢ (𝑥 ∈ ℝ →
-(⌊‘-𝑥) =
-(℩𝑧 ∈
ℤ (𝑧 ≤ -𝑥 ∧ -𝑥 < (𝑧 + 1)))) |
| 39 | 29, 35, 38 | 3eqtr4rd 2788 |
. . 3
⊢ (𝑥 ∈ ℝ →
-(⌊‘-𝑥) =
(℩𝑦 ∈
ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)))) |
| 40 | 39 | mpteq2ia 5245 |
. 2
⊢ (𝑥 ∈ ℝ ↦
-(⌊‘-𝑥)) =
(𝑥 ∈ ℝ ↦
(℩𝑦 ∈
ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)))) |
| 41 | 1, 40 | eqtri 2765 |
1
⊢ ⌈ =
(𝑥 ∈ ℝ ↦
(℩𝑦 ∈
ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)))) |