Proof of Theorem dnibndlem13
| Step | Hyp | Ref
| Expression |
| 1 | | dnibndlem13.1 |
. . . 4
⊢ 𝑇 = (𝑥 ∈ ℝ ↦
(abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) |
| 2 | | dnibndlem13.2 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 3 | 2 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) ∧ ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2)))) → 𝐴 ∈
ℝ) |
| 4 | | dnibndlem13.3 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ ℝ) |
| 5 | 4 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) ∧ ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2)))) → 𝐵 ∈
ℝ) |
| 6 | | simpr 484 |
. . . 4
⊢ (((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) ∧ ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2)))) →
((⌊‘(𝐴 + (1 /
2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2)))) |
| 7 | 1, 3, 5, 6 | dnibndlem12 36490 |
. . 3
⊢ (((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) ∧ ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2)))) →
(abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |
| 8 | 2 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) ∧ ((⌊‘(𝐴 + (1 / 2))) + 1) = (⌊‘(𝐵 + (1 / 2)))) → 𝐴 ∈
ℝ) |
| 9 | 4 | ad2antrr 726 |
. . . 4
⊢ (((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) ∧ ((⌊‘(𝐴 + (1 / 2))) + 1) = (⌊‘(𝐵 + (1 / 2)))) → 𝐵 ∈
ℝ) |
| 10 | | simpr 484 |
. . . . 5
⊢ (((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) ∧ ((⌊‘(𝐴 + (1 / 2))) + 1) = (⌊‘(𝐵 + (1 / 2)))) →
((⌊‘(𝐴 + (1 /
2))) + 1) = (⌊‘(𝐵 + (1 / 2)))) |
| 11 | 10 | eqcomd 2743 |
. . . 4
⊢ (((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) ∧ ((⌊‘(𝐴 + (1 / 2))) + 1) = (⌊‘(𝐵 + (1 / 2)))) →
(⌊‘(𝐵 + (1 /
2))) = ((⌊‘(𝐴 +
(1 / 2))) + 1)) |
| 12 | 1, 8, 9, 11 | dnibndlem9 36487 |
. . 3
⊢ (((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) ∧ ((⌊‘(𝐴 + (1 / 2))) + 1) = (⌊‘(𝐵 + (1 / 2)))) →
(abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |
| 13 | | simpr 484 |
. . . . . 6
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) → (⌊‘(𝐴 + (1 / 2))) < (⌊‘(𝐵 + (1 / 2)))) |
| 14 | | halfre 12480 |
. . . . . . . . . . . 12
⊢ (1 / 2)
∈ ℝ |
| 15 | 14 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (1 / 2) ∈
ℝ) |
| 16 | 2, 15 | readdcld 11290 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴 + (1 / 2)) ∈ ℝ) |
| 17 | 16 | flcld 13838 |
. . . . . . . . 9
⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ∈
ℤ) |
| 18 | 4, 15 | readdcld 11290 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐵 + (1 / 2)) ∈ ℝ) |
| 19 | 18 | flcld 13838 |
. . . . . . . . 9
⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) ∈
ℤ) |
| 20 | 17, 19 | jca 511 |
. . . . . . . 8
⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) ∈ ℤ
∧ (⌊‘(𝐵 +
(1 / 2))) ∈ ℤ)) |
| 21 | 20 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) → ((⌊‘(𝐴 + (1 / 2))) ∈ ℤ ∧
(⌊‘(𝐵 + (1 /
2))) ∈ ℤ)) |
| 22 | | zltp1le 12667 |
. . . . . . 7
⊢
(((⌊‘(𝐴
+ (1 / 2))) ∈ ℤ ∧ (⌊‘(𝐵 + (1 / 2))) ∈ ℤ) →
((⌊‘(𝐴 + (1 /
2))) < (⌊‘(𝐵
+ (1 / 2))) ↔ ((⌊‘(𝐴 + (1 / 2))) + 1) ≤ (⌊‘(𝐵 + (1 / 2))))) |
| 23 | 21, 22 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) → ((⌊‘(𝐴 + (1 / 2))) < (⌊‘(𝐵 + (1 / 2))) ↔
((⌊‘(𝐴 + (1 /
2))) + 1) ≤ (⌊‘(𝐵 + (1 / 2))))) |
| 24 | 13, 23 | mpbid 232 |
. . . . 5
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) → ((⌊‘(𝐴 + (1 / 2))) + 1) ≤ (⌊‘(𝐵 + (1 / 2)))) |
| 25 | | reflcl 13836 |
. . . . . . . . 9
⊢ ((𝐴 + (1 / 2)) ∈ ℝ
→ (⌊‘(𝐴 +
(1 / 2))) ∈ ℝ) |
| 26 | 16, 25 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ∈
ℝ) |
| 27 | | peano2re 11434 |
. . . . . . . 8
⊢
((⌊‘(𝐴 +
(1 / 2))) ∈ ℝ → ((⌊‘(𝐴 + (1 / 2))) + 1) ∈
ℝ) |
| 28 | 26, 27 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 1) ∈
ℝ) |
| 29 | 28 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) → ((⌊‘(𝐴 + (1 / 2))) + 1) ∈
ℝ) |
| 30 | 19 | zred 12722 |
. . . . . . 7
⊢ (𝜑 → (⌊‘(𝐵 + (1 / 2))) ∈
ℝ) |
| 31 | 30 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) → (⌊‘(𝐵 + (1 / 2))) ∈
ℝ) |
| 32 | 29, 31 | leloed 11404 |
. . . . 5
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) → (((⌊‘(𝐴 + (1 / 2))) + 1) ≤ (⌊‘(𝐵 + (1 / 2))) ↔
(((⌊‘(𝐴 + (1 /
2))) + 1) < (⌊‘(𝐵 + (1 / 2))) ∨ ((⌊‘(𝐴 + (1 / 2))) + 1) =
(⌊‘(𝐵 + (1 /
2)))))) |
| 33 | 24, 32 | mpbid 232 |
. . . 4
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) → (((⌊‘(𝐴 + (1 / 2))) + 1) < (⌊‘(𝐵 + (1 / 2))) ∨
((⌊‘(𝐴 + (1 /
2))) + 1) = (⌊‘(𝐵 + (1 / 2))))) |
| 34 | 17 | peano2zd 12725 |
. . . . . . . . . 10
⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + 1) ∈
ℤ) |
| 35 | 34, 19 | jca 511 |
. . . . . . . . 9
⊢ (𝜑 → (((⌊‘(𝐴 + (1 / 2))) + 1) ∈ ℤ
∧ (⌊‘(𝐵 +
(1 / 2))) ∈ ℤ)) |
| 36 | | zltp1le 12667 |
. . . . . . . . 9
⊢
((((⌊‘(𝐴
+ (1 / 2))) + 1) ∈ ℤ ∧ (⌊‘(𝐵 + (1 / 2))) ∈ ℤ) →
(((⌊‘(𝐴 + (1 /
2))) + 1) < (⌊‘(𝐵 + (1 / 2))) ↔ (((⌊‘(𝐴 + (1 / 2))) + 1) + 1) ≤
(⌊‘(𝐵 + (1 /
2))))) |
| 37 | 35, 36 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (((⌊‘(𝐴 + (1 / 2))) + 1) <
(⌊‘(𝐵 + (1 /
2))) ↔ (((⌊‘(𝐴 + (1 / 2))) + 1) + 1) ≤
(⌊‘(𝐵 + (1 /
2))))) |
| 38 | 26 | recnd 11289 |
. . . . . . . . . . 11
⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ∈
ℂ) |
| 39 | | 1cnd 11256 |
. . . . . . . . . . 11
⊢ (𝜑 → 1 ∈
ℂ) |
| 40 | 38, 39, 39 | addassd 11283 |
. . . . . . . . . 10
⊢ (𝜑 → (((⌊‘(𝐴 + (1 / 2))) + 1) + 1) =
((⌊‘(𝐴 + (1 /
2))) + (1 + 1))) |
| 41 | | 1p1e2 12391 |
. . . . . . . . . . . 12
⊢ (1 + 1) =
2 |
| 42 | 41 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (1 + 1) =
2) |
| 43 | 42 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) + (1 + 1)) =
((⌊‘(𝐴 + (1 /
2))) + 2)) |
| 44 | 40, 43 | eqtrd 2777 |
. . . . . . . . 9
⊢ (𝜑 → (((⌊‘(𝐴 + (1 / 2))) + 1) + 1) =
((⌊‘(𝐴 + (1 /
2))) + 2)) |
| 45 | 44 | breq1d 5153 |
. . . . . . . 8
⊢ (𝜑 → ((((⌊‘(𝐴 + (1 / 2))) + 1) + 1) ≤
(⌊‘(𝐵 + (1 /
2))) ↔ ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2))))) |
| 46 | 37, 45 | bitrd 279 |
. . . . . . 7
⊢ (𝜑 → (((⌊‘(𝐴 + (1 / 2))) + 1) <
(⌊‘(𝐵 + (1 /
2))) ↔ ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2))))) |
| 47 | 46 | biimpd 229 |
. . . . . 6
⊢ (𝜑 → (((⌊‘(𝐴 + (1 / 2))) + 1) <
(⌊‘(𝐵 + (1 /
2))) → ((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2))))) |
| 48 | 47 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) → (((⌊‘(𝐴 + (1 / 2))) + 1) < (⌊‘(𝐵 + (1 / 2))) →
((⌊‘(𝐴 + (1 /
2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2))))) |
| 49 | 48 | orim1d 968 |
. . . 4
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) → ((((⌊‘(𝐴 + (1 / 2))) + 1) < (⌊‘(𝐵 + (1 / 2))) ∨
((⌊‘(𝐴 + (1 /
2))) + 1) = (⌊‘(𝐵 + (1 / 2)))) → (((⌊‘(𝐴 + (1 / 2))) + 2) ≤
(⌊‘(𝐵 + (1 /
2))) ∨ ((⌊‘(𝐴 + (1 / 2))) + 1) = (⌊‘(𝐵 + (1 / 2)))))) |
| 50 | 33, 49 | mpd 15 |
. . 3
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) → (((⌊‘(𝐴 + (1 / 2))) + 2) ≤ (⌊‘(𝐵 + (1 / 2))) ∨
((⌊‘(𝐴 + (1 /
2))) + 1) = (⌊‘(𝐵 + (1 / 2))))) |
| 51 | 7, 12, 50 | mpjaodan 961 |
. 2
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2)))) → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |
| 52 | 2 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) =
(⌊‘(𝐵 + (1 /
2)))) → 𝐴 ∈
ℝ) |
| 53 | 4 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) =
(⌊‘(𝐵 + (1 /
2)))) → 𝐵 ∈
ℝ) |
| 54 | | simpr 484 |
. . . 4
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) =
(⌊‘(𝐵 + (1 /
2)))) → (⌊‘(𝐴 + (1 / 2))) = (⌊‘(𝐵 + (1 / 2)))) |
| 55 | 54 | eqcomd 2743 |
. . 3
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) =
(⌊‘(𝐵 + (1 /
2)))) → (⌊‘(𝐵 + (1 / 2))) = (⌊‘(𝐴 + (1 / 2)))) |
| 56 | 1, 52, 53, 55 | dnibndlem2 36480 |
. 2
⊢ ((𝜑 ∧ (⌊‘(𝐴 + (1 / 2))) =
(⌊‘(𝐵 + (1 /
2)))) → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |
| 57 | | dnibndlem13.4 |
. . 3
⊢ (𝜑 → (⌊‘(𝐴 + (1 / 2))) ≤
(⌊‘(𝐵 + (1 /
2)))) |
| 58 | 26, 30 | leloed 11404 |
. . 3
⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) ≤
(⌊‘(𝐵 + (1 /
2))) ↔ ((⌊‘(𝐴 + (1 / 2))) < (⌊‘(𝐵 + (1 / 2))) ∨
(⌊‘(𝐴 + (1 /
2))) = (⌊‘(𝐵 +
(1 / 2)))))) |
| 59 | 57, 58 | mpbid 232 |
. 2
⊢ (𝜑 → ((⌊‘(𝐴 + (1 / 2))) <
(⌊‘(𝐵 + (1 /
2))) ∨ (⌊‘(𝐴
+ (1 / 2))) = (⌊‘(𝐵 + (1 / 2))))) |
| 60 | 51, 56, 59 | mpjaodan 961 |
1
⊢ (𝜑 → (abs‘((𝑇‘𝐵) − (𝑇‘𝐴))) ≤ (abs‘(𝐵 − 𝐴))) |