Proof of Theorem dignn0flhalflem1
Step | Hyp | Ref
| Expression |
1 | | zre 11258 |
. . . . . 6
⊢ (𝐴 ∈ ℤ → 𝐴 ∈
ℝ) |
2 | 1 | 3ad2ant1 1075 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ 𝐴 ∈
ℝ) |
3 | | 2rp 11713 |
. . . . . . . . 9
⊢ 2 ∈
ℝ+ |
4 | 3 | a1i 11 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ → 2 ∈
ℝ+) |
5 | | nnz 11276 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℤ) |
6 | 4, 5 | rpexpcld 12894 |
. . . . . . 7
⊢ (𝑁 ∈ ℕ →
(2↑𝑁) ∈
ℝ+) |
7 | 6 | rpred 11748 |
. . . . . 6
⊢ (𝑁 ∈ ℕ →
(2↑𝑁) ∈
ℝ) |
8 | 7 | 3ad2ant3 1077 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (2↑𝑁) ∈
ℝ) |
9 | 2, 8 | resubcld 10337 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (𝐴 −
(2↑𝑁)) ∈
ℝ) |
10 | 6 | 3ad2ant3 1077 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (2↑𝑁) ∈
ℝ+) |
11 | 9, 10 | modcld 12536 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((𝐴 −
(2↑𝑁)) mod
(2↑𝑁)) ∈
ℝ) |
12 | 9, 11 | resubcld 10337 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((𝐴 −
(2↑𝑁)) − ((𝐴 − (2↑𝑁)) mod (2↑𝑁))) ∈ ℝ) |
13 | | peano2zm 11297 |
. . . . . 6
⊢ (𝐴 ∈ ℤ → (𝐴 − 1) ∈
ℤ) |
14 | 13 | zred 11358 |
. . . . 5
⊢ (𝐴 ∈ ℤ → (𝐴 − 1) ∈
ℝ) |
15 | 14 | 3ad2ant1 1075 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (𝐴 − 1) ∈
ℝ) |
16 | 15, 10 | modcld 12536 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((𝐴 − 1) mod
(2↑𝑁)) ∈
ℝ) |
17 | 15, 16 | resubcld 10337 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((𝐴 − 1)
− ((𝐴 − 1) mod
(2↑𝑁))) ∈
ℝ) |
18 | | 1red 9934 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ 1 ∈ ℝ) |
19 | 18, 16 | readdcld 9948 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (1 + ((𝐴 − 1)
mod (2↑𝑁))) ∈
ℝ) |
20 | 8, 11 | readdcld 9948 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((2↑𝑁) +
((𝐴 − (2↑𝑁)) mod (2↑𝑁))) ∈ ℝ) |
21 | | 2nn 11062 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
ℕ |
22 | 21 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ → 2 ∈
ℕ) |
23 | | nnnn0 11176 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℕ0) |
24 | 22, 23 | nnexpcld 12892 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ →
(2↑𝑁) ∈
ℕ) |
25 | 24 | anim2i 591 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 ∈ ℤ ∧
(2↑𝑁) ∈
ℕ)) |
26 | 25 | 3adant2 1073 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (𝐴 ∈ ℤ
∧ (2↑𝑁) ∈
ℕ)) |
27 | | m1modmmod 42110 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℤ ∧
(2↑𝑁) ∈ ℕ)
→ (((𝐴 − 1) mod
(2↑𝑁)) − (𝐴 mod (2↑𝑁))) = if((𝐴 mod (2↑𝑁)) = 0, ((2↑𝑁) − 1), -1)) |
28 | 26, 27 | syl 17 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (((𝐴 − 1) mod
(2↑𝑁)) − (𝐴 mod (2↑𝑁))) = if((𝐴 mod (2↑𝑁)) = 0, ((2↑𝑁) − 1), -1)) |
29 | | nnz 11276 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 − 1) / 2) ∈ ℕ
→ ((𝐴 − 1) / 2)
∈ ℤ) |
30 | 29 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 − 1) / 2) ∈ ℕ
→ ((𝐴 − 1) / 2)
∈ ℤ)) |
31 | | zcn 11259 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℤ → 𝐴 ∈
ℂ) |
32 | | xp1d2m1eqxm1d2 11163 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴 ∈ ℂ → (((𝐴 + 1) / 2) − 1) = ((𝐴 − 1) /
2)) |
33 | 32 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ∈ ℂ → ((𝐴 − 1) / 2) = (((𝐴 + 1) / 2) −
1)) |
34 | 31, 33 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ ℤ → ((𝐴 − 1) / 2) = (((𝐴 + 1) / 2) −
1)) |
35 | 34 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 − 1) / 2) = (((𝐴 + 1) / 2) −
1)) |
36 | 35 | eleq1d 2672 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 − 1) / 2) ∈ ℤ
↔ (((𝐴 + 1) / 2)
− 1) ∈ ℤ)) |
37 | | peano2z 11295 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐴 + 1) / 2) − 1) ∈
ℤ → ((((𝐴 + 1) /
2) − 1) + 1) ∈ ℤ) |
38 | 31 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝐴 ∈
ℂ) |
39 | | 1cnd 9935 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 1 ∈
ℂ) |
40 | 38, 39 | addcld 9938 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 + 1) ∈
ℂ) |
41 | 40 | halfcld 11154 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 + 1) / 2) ∈
ℂ) |
42 | 41, 39 | npcand 10275 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) →
((((𝐴 + 1) / 2) − 1)
+ 1) = ((𝐴 + 1) /
2)) |
43 | 42 | eleq1d 2672 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) →
(((((𝐴 + 1) / 2) − 1)
+ 1) ∈ ℤ ↔ ((𝐴 + 1) / 2) ∈ ℤ)) |
44 | 37, 43 | syl5ib 233 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) →
((((𝐴 + 1) / 2) − 1)
∈ ℤ → ((𝐴 +
1) / 2) ∈ ℤ)) |
45 | 36, 44 | sylbid 229 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 − 1) / 2) ∈ ℤ
→ ((𝐴 + 1) / 2) ∈
ℤ)) |
46 | | mod0 12537 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ ℝ ∧
(2↑𝑁) ∈
ℝ+) → ((𝐴 mod (2↑𝑁)) = 0 ↔ (𝐴 / (2↑𝑁)) ∈ ℤ)) |
47 | 1, 6, 46 | syl2an 493 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 mod (2↑𝑁)) = 0 ↔ (𝐴 / (2↑𝑁)) ∈ ℤ)) |
48 | 22 | nnzd 11357 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈ ℕ → 2 ∈
ℤ) |
49 | | nnm1nn0 11211 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈
ℕ0) |
50 | | zexpcl 12737 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((2
∈ ℤ ∧ (𝑁
− 1) ∈ ℕ0) → (2↑(𝑁 − 1)) ∈
ℤ) |
51 | 48, 49, 50 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑁 ∈ ℕ →
(2↑(𝑁 − 1))
∈ ℤ) |
52 | 51 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) →
(2↑(𝑁 − 1))
∈ ℤ) |
53 | 52 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 / (2↑𝑁)) ∈ ℤ) → (2↑(𝑁 − 1)) ∈
ℤ) |
54 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 / (2↑𝑁)) ∈ ℤ) → (𝐴 / (2↑𝑁)) ∈ ℤ) |
55 | 53, 54 | zmulcld 11364 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) ∧ (𝐴 / (2↑𝑁)) ∈ ℤ) → ((2↑(𝑁 − 1)) · (𝐴 / (2↑𝑁))) ∈ ℤ) |
56 | 55 | ex 449 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 / (2↑𝑁)) ∈ ℤ → ((2↑(𝑁 − 1)) · (𝐴 / (2↑𝑁))) ∈ ℤ)) |
57 | 5 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈
ℤ) |
58 | 57 | zcnd 11359 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈
ℂ) |
59 | 58, 39 | negsubd 10277 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑁 + -1) = (𝑁 − 1)) |
60 | 59 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑁 − 1) = (𝑁 + -1)) |
61 | 60 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑁 − 1) − 𝑁) = ((𝑁 + -1) − 𝑁)) |
62 | 39 | negcld 10258 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → -1
∈ ℂ) |
63 | 58, 62 | pncan2d 10273 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑁 + -1) − 𝑁) = -1) |
64 | 61, 63 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑁 − 1) − 𝑁) = -1) |
65 | 64 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) →
(2↑((𝑁 − 1)
− 𝑁)) =
(2↑-1)) |
66 | | 2cnd 10970 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 2 ∈
ℂ) |
67 | | 2ne0 10990 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ 2 ≠
0 |
68 | 67 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 2 ≠
0) |
69 | | 1zzd 11285 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑁 ∈ ℕ → 1 ∈
ℤ) |
70 | 5, 69 | zsubcld 11363 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈
ℤ) |
71 | 70, 5 | jca 553 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) ∈ ℤ ∧
𝑁 ∈
ℤ)) |
72 | 71 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝑁 − 1) ∈ ℤ ∧
𝑁 ∈
ℤ)) |
73 | | expsub 12770 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((2
∈ ℂ ∧ 2 ≠ 0) ∧ ((𝑁 − 1) ∈ ℤ ∧ 𝑁 ∈ ℤ)) →
(2↑((𝑁 − 1)
− 𝑁)) =
((2↑(𝑁 − 1)) /
(2↑𝑁))) |
74 | 66, 68, 72, 73 | syl21anc 1317 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) →
(2↑((𝑁 − 1)
− 𝑁)) =
((2↑(𝑁 − 1)) /
(2↑𝑁))) |
75 | | expn1 12732 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (2 ∈
ℂ → (2↑-1) = (1 / 2)) |
76 | 66, 75 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) →
(2↑-1) = (1 / 2)) |
77 | 65, 74, 76 | 3eqtr3d 2652 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) →
((2↑(𝑁 − 1)) /
(2↑𝑁)) = (1 /
2)) |
78 | 77 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 · ((2↑(𝑁 − 1)) / (2↑𝑁))) = (𝐴 · (1 / 2))) |
79 | | 2cnd 10970 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑁 ∈ ℕ → 2 ∈
ℂ) |
80 | 79, 49 | expcld 12870 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑁 ∈ ℕ →
(2↑(𝑁 − 1))
∈ ℂ) |
81 | 80 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) →
(2↑(𝑁 − 1))
∈ ℂ) |
82 | 3 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → 2 ∈
ℝ+) |
83 | 82, 57 | rpexpcld 12894 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) →
(2↑𝑁) ∈
ℝ+) |
84 | 83 | rpcnne0d 11757 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) →
((2↑𝑁) ∈ ℂ
∧ (2↑𝑁) ≠
0)) |
85 | | div12 10586 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((2↑(𝑁 −
1)) ∈ ℂ ∧ 𝐴
∈ ℂ ∧ ((2↑𝑁) ∈ ℂ ∧ (2↑𝑁) ≠ 0)) →
((2↑(𝑁 − 1))
· (𝐴 / (2↑𝑁))) = (𝐴 · ((2↑(𝑁 − 1)) / (2↑𝑁)))) |
86 | 81, 38, 84, 85 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) →
((2↑(𝑁 − 1))
· (𝐴 / (2↑𝑁))) = (𝐴 · ((2↑(𝑁 − 1)) / (2↑𝑁)))) |
87 | 38, 66, 68 | divrecd 10683 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 / 2) = (𝐴 · (1 / 2))) |
88 | 78, 86, 87 | 3eqtr4d 2654 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) →
((2↑(𝑁 − 1))
· (𝐴 / (2↑𝑁))) = (𝐴 / 2)) |
89 | 88 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) →
(((2↑(𝑁 − 1))
· (𝐴 / (2↑𝑁))) ∈ ℤ ↔ (𝐴 / 2) ∈
ℤ)) |
90 | 56, 89 | sylibd 228 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 / (2↑𝑁)) ∈ ℤ → (𝐴 / 2) ∈ ℤ)) |
91 | 47, 90 | sylbid 229 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 mod (2↑𝑁)) = 0 → (𝐴 / 2) ∈ ℤ)) |
92 | | zeo2 11340 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∈ ℤ → ((𝐴 / 2) ∈ ℤ ↔
¬ ((𝐴 + 1) / 2) ∈
ℤ)) |
93 | 92 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 / 2) ∈ ℤ ↔
¬ ((𝐴 + 1) / 2) ∈
ℤ)) |
94 | 91, 93 | sylibd 228 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → ((𝐴 mod (2↑𝑁)) = 0 → ¬ ((𝐴 + 1) / 2) ∈ ℤ)) |
95 | 94 | necon2ad 2797 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 + 1) / 2) ∈ ℤ →
(𝐴 mod (2↑𝑁)) ≠ 0)) |
96 | 30, 45, 95 | 3syld 58 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝐴 − 1) / 2) ∈ ℕ
→ (𝐴 mod (2↑𝑁)) ≠ 0)) |
97 | 96 | ex 449 |
. . . . . . . . . . . . 13
⊢ (𝐴 ∈ ℤ → (𝑁 ∈ ℕ → (((𝐴 − 1) / 2) ∈ ℕ
→ (𝐴 mod (2↑𝑁)) ≠ 0))) |
98 | 97 | com23 84 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ℤ → (((𝐴 − 1) / 2) ∈ ℕ
→ (𝑁 ∈ ℕ
→ (𝐴 mod (2↑𝑁)) ≠ 0))) |
99 | 98 | 3imp 1249 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (𝐴 mod (2↑𝑁)) ≠ 0) |
100 | 99 | neneqd 2787 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ¬ (𝐴 mod
(2↑𝑁)) =
0) |
101 | 100 | iffalsed 4047 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ if((𝐴 mod
(2↑𝑁)) = 0,
((2↑𝑁) − 1), -1)
= -1) |
102 | 28, 101 | eqtrd 2644 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (((𝐴 − 1) mod
(2↑𝑁)) − (𝐴 mod (2↑𝑁))) = -1) |
103 | | neg1lt0 11004 |
. . . . . . . . . 10
⊢ -1 <
0 |
104 | | 2re 10967 |
. . . . . . . . . . . . 13
⊢ 2 ∈
ℝ |
105 | | 1lt2 11071 |
. . . . . . . . . . . . 13
⊢ 1 <
2 |
106 | | expgt1 12760 |
. . . . . . . . . . . . 13
⊢ ((2
∈ ℝ ∧ 𝑁
∈ ℕ ∧ 1 < 2) → 1 < (2↑𝑁)) |
107 | 104, 105,
106 | mp3an13 1407 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ → 1 <
(2↑𝑁)) |
108 | | 1red 9934 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ → 1 ∈
ℝ) |
109 | 108, 7 | posdifd 10493 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ → (1 <
(2↑𝑁) ↔ 0 <
((2↑𝑁) −
1))) |
110 | 107, 109 | mpbid 221 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ → 0 <
((2↑𝑁) −
1)) |
111 | 108 | renegcld 10336 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ → -1 ∈
ℝ) |
112 | | 0red 9920 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ → 0 ∈
ℝ) |
113 | 7, 108 | resubcld 10337 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℕ →
((2↑𝑁) − 1)
∈ ℝ) |
114 | | lttr 9993 |
. . . . . . . . . . . 12
⊢ ((-1
∈ ℝ ∧ 0 ∈ ℝ ∧ ((2↑𝑁) − 1) ∈ ℝ) → ((-1
< 0 ∧ 0 < ((2↑𝑁) − 1)) → -1 < ((2↑𝑁) − 1))) |
115 | 111, 112,
113, 114 | syl3anc 1318 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ → ((-1 <
0 ∧ 0 < ((2↑𝑁)
− 1)) → -1 < ((2↑𝑁) − 1))) |
116 | 110, 115 | mpan2d 706 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℕ → (-1 <
0 → -1 < ((2↑𝑁) − 1))) |
117 | 103, 116 | mpi 20 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ → -1 <
((2↑𝑁) −
1)) |
118 | 117 | 3ad2ant3 1077 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ -1 < ((2↑𝑁)
− 1)) |
119 | 102, 118 | eqbrtrd 4605 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (((𝐴 − 1) mod
(2↑𝑁)) − (𝐴 mod (2↑𝑁))) < ((2↑𝑁) − 1)) |
120 | 2, 10 | modcld 12536 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (𝐴 mod (2↑𝑁)) ∈
ℝ) |
121 | | ltsubadd2b 42100 |
. . . . . . . 8
⊢ (((1
∈ ℝ ∧ (2↑𝑁) ∈ ℝ) ∧ ((𝐴 mod (2↑𝑁)) ∈ ℝ ∧ ((𝐴 − 1) mod (2↑𝑁)) ∈ ℝ)) → ((((𝐴 − 1) mod (2↑𝑁)) − (𝐴 mod (2↑𝑁))) < ((2↑𝑁) − 1) ↔ (1 + ((𝐴 − 1) mod (2↑𝑁))) < ((2↑𝑁) + (𝐴 mod (2↑𝑁))))) |
122 | 18, 8, 120, 16, 121 | syl22anc 1319 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((((𝐴 − 1) mod
(2↑𝑁)) − (𝐴 mod (2↑𝑁))) < ((2↑𝑁) − 1) ↔ (1 + ((𝐴 − 1) mod (2↑𝑁))) < ((2↑𝑁) + (𝐴 mod (2↑𝑁))))) |
123 | 119, 122 | mpbid 221 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (1 + ((𝐴 − 1)
mod (2↑𝑁))) <
((2↑𝑁) + (𝐴 mod (2↑𝑁)))) |
124 | | modid0 12558 |
. . . . . . . . . . . 12
⊢
((2↑𝑁) ∈
ℝ+ → ((2↑𝑁) mod (2↑𝑁)) = 0) |
125 | 10, 124 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((2↑𝑁) mod
(2↑𝑁)) =
0) |
126 | 125 | oveq2d 6565 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((𝐴 mod
(2↑𝑁)) −
((2↑𝑁) mod
(2↑𝑁))) = ((𝐴 mod (2↑𝑁)) − 0)) |
127 | 120 | recnd 9947 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (𝐴 mod (2↑𝑁)) ∈
ℂ) |
128 | 127 | subid1d 10260 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((𝐴 mod
(2↑𝑁)) − 0) =
(𝐴 mod (2↑𝑁))) |
129 | 126, 128 | eqtrd 2644 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((𝐴 mod
(2↑𝑁)) −
((2↑𝑁) mod
(2↑𝑁))) = (𝐴 mod (2↑𝑁))) |
130 | 129 | oveq1d 6564 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (((𝐴 mod
(2↑𝑁)) −
((2↑𝑁) mod
(2↑𝑁))) mod
(2↑𝑁)) = ((𝐴 mod (2↑𝑁)) mod (2↑𝑁))) |
131 | | modsubmodmod 12591 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧
(2↑𝑁) ∈ ℝ
∧ (2↑𝑁) ∈
ℝ+) → (((𝐴 mod (2↑𝑁)) − ((2↑𝑁) mod (2↑𝑁))) mod (2↑𝑁)) = ((𝐴 − (2↑𝑁)) mod (2↑𝑁))) |
132 | 2, 8, 10, 131 | syl3anc 1318 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (((𝐴 mod
(2↑𝑁)) −
((2↑𝑁) mod
(2↑𝑁))) mod
(2↑𝑁)) = ((𝐴 − (2↑𝑁)) mod (2↑𝑁))) |
133 | | modabs2 12566 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧
(2↑𝑁) ∈
ℝ+) → ((𝐴 mod (2↑𝑁)) mod (2↑𝑁)) = (𝐴 mod (2↑𝑁))) |
134 | 2, 10, 133 | syl2anc 691 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((𝐴 mod
(2↑𝑁)) mod
(2↑𝑁)) = (𝐴 mod (2↑𝑁))) |
135 | 130, 132,
134 | 3eqtr3d 2652 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((𝐴 −
(2↑𝑁)) mod
(2↑𝑁)) = (𝐴 mod (2↑𝑁))) |
136 | 135 | oveq2d 6565 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((2↑𝑁) +
((𝐴 − (2↑𝑁)) mod (2↑𝑁))) = ((2↑𝑁) + (𝐴 mod (2↑𝑁)))) |
137 | 123, 136 | breqtrrd 4611 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (1 + ((𝐴 − 1)
mod (2↑𝑁))) <
((2↑𝑁) + ((𝐴 − (2↑𝑁)) mod (2↑𝑁)))) |
138 | 19, 20, 2, 137 | ltsub2dd 10519 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (𝐴 −
((2↑𝑁) + ((𝐴 − (2↑𝑁)) mod (2↑𝑁)))) < (𝐴 − (1 + ((𝐴 − 1) mod (2↑𝑁))))) |
139 | 31 | 3ad2ant1 1075 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ 𝐴 ∈
ℂ) |
140 | 8 | recnd 9947 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (2↑𝑁) ∈
ℂ) |
141 | 11 | recnd 9947 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((𝐴 −
(2↑𝑁)) mod
(2↑𝑁)) ∈
ℂ) |
142 | 139, 140,
141 | subsub4d 10302 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((𝐴 −
(2↑𝑁)) − ((𝐴 − (2↑𝑁)) mod (2↑𝑁))) = (𝐴 − ((2↑𝑁) + ((𝐴 − (2↑𝑁)) mod (2↑𝑁))))) |
143 | | 1cnd 9935 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ 1 ∈ ℂ) |
144 | 16 | recnd 9947 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((𝐴 − 1) mod
(2↑𝑁)) ∈
ℂ) |
145 | 139, 143,
144 | subsub4d 10302 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((𝐴 − 1)
− ((𝐴 − 1) mod
(2↑𝑁))) = (𝐴 − (1 + ((𝐴 − 1) mod (2↑𝑁))))) |
146 | 138, 142,
145 | 3brtr4d 4615 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ ((𝐴 −
(2↑𝑁)) − ((𝐴 − (2↑𝑁)) mod (2↑𝑁))) < ((𝐴 − 1) − ((𝐴 − 1) mod (2↑𝑁)))) |
147 | 12, 17, 10, 146 | ltdiv1dd 11805 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (((𝐴 −
(2↑𝑁)) − ((𝐴 − (2↑𝑁)) mod (2↑𝑁))) / (2↑𝑁)) < (((𝐴 − 1) − ((𝐴 − 1) mod (2↑𝑁))) / (2↑𝑁))) |
148 | 7 | recnd 9947 |
. . . . 5
⊢ (𝑁 ∈ ℕ →
(2↑𝑁) ∈
ℂ) |
149 | 148 | 3ad2ant3 1077 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (2↑𝑁) ∈
ℂ) |
150 | 67 | a1i 11 |
. . . . . 6
⊢ (𝑁 ∈ ℕ → 2 ≠
0) |
151 | 79, 150, 5 | expne0d 12876 |
. . . . 5
⊢ (𝑁 ∈ ℕ →
(2↑𝑁) ≠
0) |
152 | 151 | 3ad2ant3 1077 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (2↑𝑁) ≠
0) |
153 | | divsub1dir 42101 |
. . . . 5
⊢ ((𝐴 ∈ ℂ ∧
(2↑𝑁) ∈ ℂ
∧ (2↑𝑁) ≠ 0)
→ ((𝐴 / (2↑𝑁)) − 1) = ((𝐴 − (2↑𝑁)) / (2↑𝑁))) |
154 | 153 | fveq2d 6107 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧
(2↑𝑁) ∈ ℂ
∧ (2↑𝑁) ≠ 0)
→ (⌊‘((𝐴 /
(2↑𝑁)) − 1)) =
(⌊‘((𝐴 −
(2↑𝑁)) / (2↑𝑁)))) |
155 | 139, 149,
152, 154 | syl3anc 1318 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (⌊‘((𝐴 /
(2↑𝑁)) − 1)) =
(⌊‘((𝐴 −
(2↑𝑁)) / (2↑𝑁)))) |
156 | | fldivmod 42107 |
. . . 4
⊢ (((𝐴 − (2↑𝑁)) ∈ ℝ ∧
(2↑𝑁) ∈
ℝ+) → (⌊‘((𝐴 − (2↑𝑁)) / (2↑𝑁))) = (((𝐴 − (2↑𝑁)) − ((𝐴 − (2↑𝑁)) mod (2↑𝑁))) / (2↑𝑁))) |
157 | 9, 10, 156 | syl2anc 691 |
. . 3
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (⌊‘((𝐴
− (2↑𝑁)) /
(2↑𝑁))) = (((𝐴 − (2↑𝑁)) − ((𝐴 − (2↑𝑁)) mod (2↑𝑁))) / (2↑𝑁))) |
158 | 155, 157 | eqtrd 2644 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (⌊‘((𝐴 /
(2↑𝑁)) − 1)) =
(((𝐴 − (2↑𝑁)) − ((𝐴 − (2↑𝑁)) mod (2↑𝑁))) / (2↑𝑁))) |
159 | | fldivmod 42107 |
. . 3
⊢ (((𝐴 − 1) ∈ ℝ ∧
(2↑𝑁) ∈
ℝ+) → (⌊‘((𝐴 − 1) / (2↑𝑁))) = (((𝐴 − 1) − ((𝐴 − 1) mod (2↑𝑁))) / (2↑𝑁))) |
160 | 15, 10, 159 | syl2anc 691 |
. 2
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (⌊‘((𝐴
− 1) / (2↑𝑁))) =
(((𝐴 − 1) −
((𝐴 − 1) mod
(2↑𝑁))) /
(2↑𝑁))) |
161 | 147, 158,
160 | 3brtr4d 4615 |
1
⊢ ((𝐴 ∈ ℤ ∧ ((𝐴 − 1) / 2) ∈ ℕ
∧ 𝑁 ∈ ℕ)
→ (⌊‘((𝐴 /
(2↑𝑁)) − 1))
< (⌊‘((𝐴
− 1) / (2↑𝑁)))) |