| Step | Hyp | Ref
| Expression |
| 1 | | bitsval2 16462 |
. . . . . . 7
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (𝑚 ∈
(bits‘𝑁) ↔ ¬
2 ∥ (⌊‘(𝑁
/ (2↑𝑚))))) |
| 2 | | 2z 12649 |
. . . . . . . . . 10
⊢ 2 ∈
ℤ |
| 3 | 2 | a1i 11 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ 2 ∈ ℤ) |
| 4 | | simpl 482 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ 𝑁 ∈
ℤ) |
| 5 | 4 | zred 12722 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ 𝑁 ∈
ℝ) |
| 6 | | 2nn 12339 |
. . . . . . . . . . . . 13
⊢ 2 ∈
ℕ |
| 7 | 6 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ 2 ∈ ℕ) |
| 8 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ 𝑚 ∈
ℕ0) |
| 9 | 7, 8 | nnexpcld 14284 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (2↑𝑚) ∈
ℕ) |
| 10 | 5, 9 | nndivred 12320 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (𝑁 / (2↑𝑚)) ∈
ℝ) |
| 11 | 10 | flcld 13838 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (⌊‘(𝑁 /
(2↑𝑚))) ∈
ℤ) |
| 12 | | dvdsnegb 16311 |
. . . . . . . . 9
⊢ ((2
∈ ℤ ∧ (⌊‘(𝑁 / (2↑𝑚))) ∈ ℤ) → (2 ∥
(⌊‘(𝑁 /
(2↑𝑚))) ↔ 2
∥ -(⌊‘(𝑁
/ (2↑𝑚))))) |
| 13 | 3, 11, 12 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (2 ∥ (⌊‘(𝑁 / (2↑𝑚))) ↔ 2 ∥ -(⌊‘(𝑁 / (2↑𝑚))))) |
| 14 | 13 | notbid 318 |
. . . . . . 7
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))) ↔ ¬ 2 ∥
-(⌊‘(𝑁 /
(2↑𝑚))))) |
| 15 | 11 | znegcld 12724 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ -(⌊‘(𝑁 /
(2↑𝑚))) ∈
ℤ) |
| 16 | | oddm1even 16380 |
. . . . . . . . 9
⊢
(-(⌊‘(𝑁
/ (2↑𝑚))) ∈
ℤ → (¬ 2 ∥ -(⌊‘(𝑁 / (2↑𝑚))) ↔ 2 ∥ (-(⌊‘(𝑁 / (2↑𝑚))) − 1))) |
| 17 | 15, 16 | syl 17 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (¬ 2 ∥ -(⌊‘(𝑁 / (2↑𝑚))) ↔ 2 ∥ (-(⌊‘(𝑁 / (2↑𝑚))) − 1))) |
| 18 | | flltp1 13840 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 / (2↑𝑚)) ∈ ℝ → (𝑁 / (2↑𝑚)) < ((⌊‘(𝑁 / (2↑𝑚))) + 1)) |
| 19 | 10, 18 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (𝑁 / (2↑𝑚)) < ((⌊‘(𝑁 / (2↑𝑚))) + 1)) |
| 20 | 11 | zred 12722 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (⌊‘(𝑁 /
(2↑𝑚))) ∈
ℝ) |
| 21 | | 1red 11262 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ 1 ∈ ℝ) |
| 22 | 20, 21 | readdcld 11290 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ ((⌊‘(𝑁 /
(2↑𝑚))) + 1) ∈
ℝ) |
| 23 | 10, 22 | ltnegd 11841 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ ((𝑁 / (2↑𝑚)) < ((⌊‘(𝑁 / (2↑𝑚))) + 1) ↔ -((⌊‘(𝑁 / (2↑𝑚))) + 1) < -(𝑁 / (2↑𝑚)))) |
| 24 | 19, 23 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ -((⌊‘(𝑁
/ (2↑𝑚))) + 1) <
-(𝑁 / (2↑𝑚))) |
| 25 | 20 | recnd 11289 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (⌊‘(𝑁 /
(2↑𝑚))) ∈
ℂ) |
| 26 | 21 | recnd 11289 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ 1 ∈ ℂ) |
| 27 | 25, 26 | negdi2d 11634 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ -((⌊‘(𝑁
/ (2↑𝑚))) + 1) =
(-(⌊‘(𝑁 /
(2↑𝑚))) −
1)) |
| 28 | 5 | recnd 11289 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ 𝑁 ∈
ℂ) |
| 29 | 9 | nncnd 12282 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (2↑𝑚) ∈
ℂ) |
| 30 | 9 | nnne0d 12316 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (2↑𝑚) ≠
0) |
| 31 | 28, 29, 30 | divnegd 12056 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ -(𝑁 / (2↑𝑚)) = (-𝑁 / (2↑𝑚))) |
| 32 | 24, 27, 31 | 3brtr3d 5174 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (-(⌊‘(𝑁
/ (2↑𝑚))) − 1)
< (-𝑁 / (2↑𝑚))) |
| 33 | | 1zzd 12648 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ 1 ∈ ℤ) |
| 34 | 15, 33 | zsubcld 12727 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (-(⌊‘(𝑁
/ (2↑𝑚))) − 1)
∈ ℤ) |
| 35 | 34 | zred 12722 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (-(⌊‘(𝑁
/ (2↑𝑚))) − 1)
∈ ℝ) |
| 36 | 5 | renegcld 11690 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ -𝑁 ∈
ℝ) |
| 37 | 9 | nnrpd 13075 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (2↑𝑚) ∈
ℝ+) |
| 38 | 35, 36, 37 | ltmuldivd 13124 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (((-(⌊‘(𝑁 / (2↑𝑚))) − 1) · (2↑𝑚)) < -𝑁 ↔ (-(⌊‘(𝑁 / (2↑𝑚))) − 1) < (-𝑁 / (2↑𝑚)))) |
| 39 | 32, 38 | mpbird 257 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ ((-(⌊‘(𝑁
/ (2↑𝑚))) − 1)
· (2↑𝑚)) <
-𝑁) |
| 40 | 9 | nnzd 12640 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (2↑𝑚) ∈
ℤ) |
| 41 | 34, 40 | zmulcld 12728 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ ((-(⌊‘(𝑁
/ (2↑𝑚))) − 1)
· (2↑𝑚)) ∈
ℤ) |
| 42 | 4 | znegcld 12724 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ -𝑁 ∈
ℤ) |
| 43 | | zltlem1 12670 |
. . . . . . . . . . . . 13
⊢
((((-(⌊‘(𝑁 / (2↑𝑚))) − 1) · (2↑𝑚)) ∈ ℤ ∧ -𝑁 ∈ ℤ) →
(((-(⌊‘(𝑁 /
(2↑𝑚))) − 1)
· (2↑𝑚)) <
-𝑁 ↔
((-(⌊‘(𝑁 /
(2↑𝑚))) − 1)
· (2↑𝑚)) ≤
(-𝑁 −
1))) |
| 44 | 41, 42, 43 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (((-(⌊‘(𝑁 / (2↑𝑚))) − 1) · (2↑𝑚)) < -𝑁 ↔ ((-(⌊‘(𝑁 / (2↑𝑚))) − 1) · (2↑𝑚)) ≤ (-𝑁 − 1))) |
| 45 | 39, 44 | mpbid 232 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ ((-(⌊‘(𝑁
/ (2↑𝑚))) − 1)
· (2↑𝑚)) ≤
(-𝑁 −
1)) |
| 46 | 36, 21 | resubcld 11691 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (-𝑁 − 1)
∈ ℝ) |
| 47 | 35, 46, 37 | lemuldivd 13126 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (((-(⌊‘(𝑁 / (2↑𝑚))) − 1) · (2↑𝑚)) ≤ (-𝑁 − 1) ↔ (-(⌊‘(𝑁 / (2↑𝑚))) − 1) ≤ ((-𝑁 − 1) / (2↑𝑚)))) |
| 48 | 45, 47 | mpbid 232 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (-(⌊‘(𝑁
/ (2↑𝑚))) − 1)
≤ ((-𝑁 − 1) /
(2↑𝑚))) |
| 49 | | flle 13839 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑁 / (2↑𝑚)) ∈ ℝ →
(⌊‘(𝑁 /
(2↑𝑚))) ≤ (𝑁 / (2↑𝑚))) |
| 50 | 10, 49 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (⌊‘(𝑁 /
(2↑𝑚))) ≤ (𝑁 / (2↑𝑚))) |
| 51 | 20, 10 | lenegd 11842 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ ((⌊‘(𝑁 /
(2↑𝑚))) ≤ (𝑁 / (2↑𝑚)) ↔ -(𝑁 / (2↑𝑚)) ≤ -(⌊‘(𝑁 / (2↑𝑚))))) |
| 52 | 50, 51 | mpbid 232 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ -(𝑁 / (2↑𝑚)) ≤ -(⌊‘(𝑁 / (2↑𝑚)))) |
| 53 | 31, 52 | eqbrtrrd 5167 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (-𝑁 / (2↑𝑚)) ≤ -(⌊‘(𝑁 / (2↑𝑚)))) |
| 54 | 20 | renegcld 11690 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ -(⌊‘(𝑁 /
(2↑𝑚))) ∈
ℝ) |
| 55 | 36, 54, 37 | ledivmuld 13130 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ ((-𝑁 / (2↑𝑚)) ≤ -(⌊‘(𝑁 / (2↑𝑚))) ↔ -𝑁 ≤ ((2↑𝑚) · -(⌊‘(𝑁 / (2↑𝑚)))))) |
| 56 | 53, 55 | mpbid 232 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ -𝑁 ≤
((2↑𝑚) ·
-(⌊‘(𝑁 /
(2↑𝑚))))) |
| 57 | 40, 15 | zmulcld 12728 |
. . . . . . . . . . . . . 14
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ ((2↑𝑚) ·
-(⌊‘(𝑁 /
(2↑𝑚)))) ∈
ℤ) |
| 58 | | zlem1lt 12669 |
. . . . . . . . . . . . . 14
⊢ ((-𝑁 ∈ ℤ ∧
((2↑𝑚) ·
-(⌊‘(𝑁 /
(2↑𝑚)))) ∈
ℤ) → (-𝑁 ≤
((2↑𝑚) ·
-(⌊‘(𝑁 /
(2↑𝑚)))) ↔
(-𝑁 − 1) <
((2↑𝑚) ·
-(⌊‘(𝑁 /
(2↑𝑚)))))) |
| 59 | 42, 57, 58 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (-𝑁 ≤
((2↑𝑚) ·
-(⌊‘(𝑁 /
(2↑𝑚)))) ↔
(-𝑁 − 1) <
((2↑𝑚) ·
-(⌊‘(𝑁 /
(2↑𝑚)))))) |
| 60 | 56, 59 | mpbid 232 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (-𝑁 − 1) <
((2↑𝑚) ·
-(⌊‘(𝑁 /
(2↑𝑚))))) |
| 61 | 46, 54, 37 | ltdivmuld 13128 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (((-𝑁 − 1) /
(2↑𝑚)) <
-(⌊‘(𝑁 /
(2↑𝑚))) ↔ (-𝑁 − 1) < ((2↑𝑚) ·
-(⌊‘(𝑁 /
(2↑𝑚)))))) |
| 62 | 60, 61 | mpbird 257 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ ((-𝑁 − 1) /
(2↑𝑚)) <
-(⌊‘(𝑁 /
(2↑𝑚)))) |
| 63 | 25 | negcld 11607 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ -(⌊‘(𝑁 /
(2↑𝑚))) ∈
ℂ) |
| 64 | 63, 26 | npcand 11624 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ ((-(⌊‘(𝑁
/ (2↑𝑚))) − 1) +
1) = -(⌊‘(𝑁 /
(2↑𝑚)))) |
| 65 | 62, 64 | breqtrrd 5171 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ ((-𝑁 − 1) /
(2↑𝑚)) <
((-(⌊‘(𝑁 /
(2↑𝑚))) − 1) +
1)) |
| 66 | 46, 9 | nndivred 12320 |
. . . . . . . . . . 11
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ ((-𝑁 − 1) /
(2↑𝑚)) ∈
ℝ) |
| 67 | | flbi 13856 |
. . . . . . . . . . 11
⊢
((((-𝑁 − 1) /
(2↑𝑚)) ∈ ℝ
∧ (-(⌊‘(𝑁 /
(2↑𝑚))) − 1)
∈ ℤ) → ((⌊‘((-𝑁 − 1) / (2↑𝑚))) = (-(⌊‘(𝑁 / (2↑𝑚))) − 1) ↔
((-(⌊‘(𝑁 /
(2↑𝑚))) − 1)
≤ ((-𝑁 − 1) /
(2↑𝑚)) ∧ ((-𝑁 − 1) / (2↑𝑚)) < ((-(⌊‘(𝑁 / (2↑𝑚))) − 1) + 1)))) |
| 68 | 66, 34, 67 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ ((⌊‘((-𝑁
− 1) / (2↑𝑚))) =
(-(⌊‘(𝑁 /
(2↑𝑚))) − 1)
↔ ((-(⌊‘(𝑁
/ (2↑𝑚))) − 1)
≤ ((-𝑁 − 1) /
(2↑𝑚)) ∧ ((-𝑁 − 1) / (2↑𝑚)) < ((-(⌊‘(𝑁 / (2↑𝑚))) − 1) + 1)))) |
| 69 | 48, 65, 68 | mpbir2and 713 |
. . . . . . . . 9
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (⌊‘((-𝑁
− 1) / (2↑𝑚))) =
(-(⌊‘(𝑁 /
(2↑𝑚))) −
1)) |
| 70 | 69 | breq2d 5155 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (2 ∥ (⌊‘((-𝑁 − 1) / (2↑𝑚))) ↔ 2 ∥ (-(⌊‘(𝑁 / (2↑𝑚))) − 1))) |
| 71 | 17, 70 | bitr4d 282 |
. . . . . . 7
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (¬ 2 ∥ -(⌊‘(𝑁 / (2↑𝑚))) ↔ 2 ∥ (⌊‘((-𝑁 − 1) / (2↑𝑚))))) |
| 72 | 1, 14, 71 | 3bitrd 305 |
. . . . . 6
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (𝑚 ∈
(bits‘𝑁) ↔ 2
∥ (⌊‘((-𝑁
− 1) / (2↑𝑚))))) |
| 73 | 72 | notbid 318 |
. . . . 5
⊢ ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ0)
→ (¬ 𝑚 ∈
(bits‘𝑁) ↔ ¬
2 ∥ (⌊‘((-𝑁 − 1) / (2↑𝑚))))) |
| 74 | 73 | pm5.32da 579 |
. . . 4
⊢ (𝑁 ∈ ℤ → ((𝑚 ∈ ℕ0
∧ ¬ 𝑚 ∈
(bits‘𝑁)) ↔
(𝑚 ∈
ℕ0 ∧ ¬ 2 ∥ (⌊‘((-𝑁 − 1) / (2↑𝑚)))))) |
| 75 | | znegcl 12652 |
. . . . . 6
⊢ (𝑁 ∈ ℤ → -𝑁 ∈
ℤ) |
| 76 | | 1zzd 12648 |
. . . . . 6
⊢ (𝑁 ∈ ℤ → 1 ∈
ℤ) |
| 77 | 75, 76 | zsubcld 12727 |
. . . . 5
⊢ (𝑁 ∈ ℤ → (-𝑁 − 1) ∈
ℤ) |
| 78 | 77 | biantrurd 532 |
. . . 4
⊢ (𝑁 ∈ ℤ → ((𝑚 ∈ ℕ0
∧ ¬ 2 ∥ (⌊‘((-𝑁 − 1) / (2↑𝑚)))) ↔ ((-𝑁 − 1) ∈ ℤ ∧ (𝑚 ∈ ℕ0
∧ ¬ 2 ∥ (⌊‘((-𝑁 − 1) / (2↑𝑚))))))) |
| 79 | 74, 78 | bitrd 279 |
. . 3
⊢ (𝑁 ∈ ℤ → ((𝑚 ∈ ℕ0
∧ ¬ 𝑚 ∈
(bits‘𝑁)) ↔
((-𝑁 − 1) ∈
ℤ ∧ (𝑚 ∈
ℕ0 ∧ ¬ 2 ∥ (⌊‘((-𝑁 − 1) / (2↑𝑚))))))) |
| 80 | | eldif 3961 |
. . 3
⊢ (𝑚 ∈ (ℕ0
∖ (bits‘𝑁))
↔ (𝑚 ∈
ℕ0 ∧ ¬ 𝑚 ∈ (bits‘𝑁))) |
| 81 | | bitsval 16461 |
. . . 4
⊢ (𝑚 ∈ (bits‘(-𝑁 − 1)) ↔ ((-𝑁 − 1) ∈ ℤ ∧
𝑚 ∈
ℕ0 ∧ ¬ 2 ∥ (⌊‘((-𝑁 − 1) / (2↑𝑚))))) |
| 82 | | 3anass 1095 |
. . . 4
⊢ (((-𝑁 − 1) ∈ ℤ ∧
𝑚 ∈
ℕ0 ∧ ¬ 2 ∥ (⌊‘((-𝑁 − 1) / (2↑𝑚)))) ↔ ((-𝑁 − 1) ∈ ℤ ∧ (𝑚 ∈ ℕ0
∧ ¬ 2 ∥ (⌊‘((-𝑁 − 1) / (2↑𝑚)))))) |
| 83 | 81, 82 | bitri 275 |
. . 3
⊢ (𝑚 ∈ (bits‘(-𝑁 − 1)) ↔ ((-𝑁 − 1) ∈ ℤ ∧
(𝑚 ∈
ℕ0 ∧ ¬ 2 ∥ (⌊‘((-𝑁 − 1) / (2↑𝑚)))))) |
| 84 | 79, 80, 83 | 3bitr4g 314 |
. 2
⊢ (𝑁 ∈ ℤ → (𝑚 ∈ (ℕ0
∖ (bits‘𝑁))
↔ 𝑚 ∈
(bits‘(-𝑁 −
1)))) |
| 85 | 84 | eqrdv 2735 |
1
⊢ (𝑁 ∈ ℤ →
(ℕ0 ∖ (bits‘𝑁)) = (bits‘(-𝑁 − 1))) |