Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → 𝐴 ∈ ℤ ) |
2 |
|
2nn |
⊢ 2 ∈ ℕ |
3 |
2
|
a1i |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → 2 ∈ ℕ ) |
4 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → 𝑁 ∈ ℕ0 ) |
5 |
3 4
|
nnexpcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( 2 ↑ 𝑁 ) ∈ ℕ ) |
6 |
1 5
|
zmodcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ∈ ℕ0 ) |
7 |
6
|
nn0zd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ∈ ℤ ) |
8 |
7
|
znegcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ∈ ℤ ) |
9 |
|
sadadd |
⊢ ( ( - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ∈ ℤ ∧ 𝐴 ∈ ℤ ) → ( ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( bits ‘ 𝐴 ) ) = ( bits ‘ ( - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) + 𝐴 ) ) ) |
10 |
8 1 9
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( bits ‘ 𝐴 ) ) = ( bits ‘ ( - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) + 𝐴 ) ) ) |
11 |
|
sadadd |
⊢ ( ( - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ∈ ℤ ∧ ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ∈ ℤ ) → ( ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( bits ‘ ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ) = ( bits ‘ ( - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) + ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ) ) |
12 |
8 7 11
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( bits ‘ ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ) = ( bits ‘ ( - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) + ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ) ) |
13 |
8
|
zcnd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ∈ ℂ ) |
14 |
7
|
zcnd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ∈ ℂ ) |
15 |
13 14
|
addcomd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) + ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) = ( ( 𝐴 mod ( 2 ↑ 𝑁 ) ) + - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ) |
16 |
14
|
negidd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝐴 mod ( 2 ↑ 𝑁 ) ) + - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) = 0 ) |
17 |
15 16
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) + ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) = 0 ) |
18 |
17
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( bits ‘ ( - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) + ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ) = ( bits ‘ 0 ) ) |
19 |
|
0bits |
⊢ ( bits ‘ 0 ) = ∅ |
20 |
18 19
|
eqtrdi |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( bits ‘ ( - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) + ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ) = ∅ ) |
21 |
12 20
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( bits ‘ ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ) = ∅ ) |
22 |
21
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( bits ‘ ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ) sadd ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) = ( ∅ sadd ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) ) |
23 |
|
bitsss |
⊢ ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ⊆ ℕ0 |
24 |
|
bitsss |
⊢ ( bits ‘ ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ⊆ ℕ0 |
25 |
|
inss1 |
⊢ ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ⊆ ( bits ‘ 𝐴 ) |
26 |
|
bitsss |
⊢ ( bits ‘ 𝐴 ) ⊆ ℕ0 |
27 |
26
|
a1i |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( bits ‘ 𝐴 ) ⊆ ℕ0 ) |
28 |
25 27
|
sstrid |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ⊆ ℕ0 ) |
29 |
|
sadass |
⊢ ( ( ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ⊆ ℕ0 ∧ ( bits ‘ ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ⊆ ℕ0 ∧ ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ⊆ ℕ0 ) → ( ( ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( bits ‘ ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ) sadd ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) = ( ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( ( bits ‘ ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) ) ) |
30 |
23 24 28 29
|
mp3an12i |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( bits ‘ ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ) sadd ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) = ( ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( ( bits ‘ ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) ) ) |
31 |
|
bitsmod |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( bits ‘ ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) = ( ( bits ‘ 𝐴 ) ∩ ( 0 ..^ 𝑁 ) ) ) |
32 |
31
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( bits ‘ ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) = ( ( ( bits ‘ 𝐴 ) ∩ ( 0 ..^ 𝑁 ) ) sadd ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) ) |
33 |
|
inss1 |
⊢ ( ( bits ‘ 𝐴 ) ∩ ( 0 ..^ 𝑁 ) ) ⊆ ( bits ‘ 𝐴 ) |
34 |
33 27
|
sstrid |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( bits ‘ 𝐴 ) ∩ ( 0 ..^ 𝑁 ) ) ⊆ ℕ0 ) |
35 |
|
fzouzdisj |
⊢ ( ( 0 ..^ 𝑁 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) = ∅ |
36 |
35
|
ineq2i |
⊢ ( ( bits ‘ 𝐴 ) ∩ ( ( 0 ..^ 𝑁 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) = ( ( bits ‘ 𝐴 ) ∩ ∅ ) |
37 |
|
inindi |
⊢ ( ( bits ‘ 𝐴 ) ∩ ( ( 0 ..^ 𝑁 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) = ( ( ( bits ‘ 𝐴 ) ∩ ( 0 ..^ 𝑁 ) ) ∩ ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) |
38 |
|
in0 |
⊢ ( ( bits ‘ 𝐴 ) ∩ ∅ ) = ∅ |
39 |
36 37 38
|
3eqtr3i |
⊢ ( ( ( bits ‘ 𝐴 ) ∩ ( 0 ..^ 𝑁 ) ) ∩ ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) = ∅ |
40 |
39
|
a1i |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( ( bits ‘ 𝐴 ) ∩ ( 0 ..^ 𝑁 ) ) ∩ ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) = ∅ ) |
41 |
34 28 40
|
saddisj |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( ( bits ‘ 𝐴 ) ∩ ( 0 ..^ 𝑁 ) ) sadd ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) = ( ( ( bits ‘ 𝐴 ) ∩ ( 0 ..^ 𝑁 ) ) ∪ ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) ) |
42 |
|
indi |
⊢ ( ( bits ‘ 𝐴 ) ∩ ( ( 0 ..^ 𝑁 ) ∪ ( ℤ≥ ‘ 𝑁 ) ) ) = ( ( ( bits ‘ 𝐴 ) ∩ ( 0 ..^ 𝑁 ) ) ∪ ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) |
43 |
41 42
|
eqtr4di |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( ( bits ‘ 𝐴 ) ∩ ( 0 ..^ 𝑁 ) ) sadd ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) = ( ( bits ‘ 𝐴 ) ∩ ( ( 0 ..^ 𝑁 ) ∪ ( ℤ≥ ‘ 𝑁 ) ) ) ) |
44 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
45 |
4 44
|
eleqtrdi |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → 𝑁 ∈ ( ℤ≥ ‘ 0 ) ) |
46 |
|
fzouzsplit |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 0 ) → ( ℤ≥ ‘ 0 ) = ( ( 0 ..^ 𝑁 ) ∪ ( ℤ≥ ‘ 𝑁 ) ) ) |
47 |
45 46
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ℤ≥ ‘ 0 ) = ( ( 0 ..^ 𝑁 ) ∪ ( ℤ≥ ‘ 𝑁 ) ) ) |
48 |
44 47
|
eqtrid |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ℕ0 = ( ( 0 ..^ 𝑁 ) ∪ ( ℤ≥ ‘ 𝑁 ) ) ) |
49 |
26 48
|
sseqtrid |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( bits ‘ 𝐴 ) ⊆ ( ( 0 ..^ 𝑁 ) ∪ ( ℤ≥ ‘ 𝑁 ) ) ) |
50 |
|
df-ss |
⊢ ( ( bits ‘ 𝐴 ) ⊆ ( ( 0 ..^ 𝑁 ) ∪ ( ℤ≥ ‘ 𝑁 ) ) ↔ ( ( bits ‘ 𝐴 ) ∩ ( ( 0 ..^ 𝑁 ) ∪ ( ℤ≥ ‘ 𝑁 ) ) ) = ( bits ‘ 𝐴 ) ) |
51 |
49 50
|
sylib |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( bits ‘ 𝐴 ) ∩ ( ( 0 ..^ 𝑁 ) ∪ ( ℤ≥ ‘ 𝑁 ) ) ) = ( bits ‘ 𝐴 ) ) |
52 |
43 51
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( ( bits ‘ 𝐴 ) ∩ ( 0 ..^ 𝑁 ) ) sadd ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) = ( bits ‘ 𝐴 ) ) |
53 |
32 52
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( bits ‘ ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) = ( bits ‘ 𝐴 ) ) |
54 |
53
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( ( bits ‘ ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) ) = ( ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( bits ‘ 𝐴 ) ) ) |
55 |
30 54
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( bits ‘ ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ) sadd ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) = ( ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( bits ‘ 𝐴 ) ) ) |
56 |
|
sadid2 |
⊢ ( ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ⊆ ℕ0 → ( ∅ sadd ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) = ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) |
57 |
28 56
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ∅ sadd ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) = ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) |
58 |
22 55 57
|
3eqtr3d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( bits ‘ - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) sadd ( bits ‘ 𝐴 ) ) = ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) ) |
59 |
1
|
zcnd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → 𝐴 ∈ ℂ ) |
60 |
13 59
|
addcomd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) + 𝐴 ) = ( 𝐴 + - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ) |
61 |
59 14
|
negsubd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 + - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) = ( 𝐴 − ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ) |
62 |
59 14
|
subcld |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 − ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ∈ ℂ ) |
63 |
5
|
nncnd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( 2 ↑ 𝑁 ) ∈ ℂ ) |
64 |
5
|
nnne0d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( 2 ↑ 𝑁 ) ≠ 0 ) |
65 |
62 63 64
|
divcan1d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( ( 𝐴 − ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) / ( 2 ↑ 𝑁 ) ) · ( 2 ↑ 𝑁 ) ) = ( 𝐴 − ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) ) |
66 |
1
|
zred |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → 𝐴 ∈ ℝ ) |
67 |
5
|
nnrpd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( 2 ↑ 𝑁 ) ∈ ℝ+ ) |
68 |
|
moddiffl |
⊢ ( ( 𝐴 ∈ ℝ ∧ ( 2 ↑ 𝑁 ) ∈ ℝ+ ) → ( ( 𝐴 − ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) / ( 2 ↑ 𝑁 ) ) = ( ⌊ ‘ ( 𝐴 / ( 2 ↑ 𝑁 ) ) ) ) |
69 |
66 67 68
|
syl2anc |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝐴 − ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) / ( 2 ↑ 𝑁 ) ) = ( ⌊ ‘ ( 𝐴 / ( 2 ↑ 𝑁 ) ) ) ) |
70 |
69
|
oveq1d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( ( 𝐴 − ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) / ( 2 ↑ 𝑁 ) ) · ( 2 ↑ 𝑁 ) ) = ( ( ⌊ ‘ ( 𝐴 / ( 2 ↑ 𝑁 ) ) ) · ( 2 ↑ 𝑁 ) ) ) |
71 |
61 65 70
|
3eqtr2d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( 𝐴 + - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) ) = ( ( ⌊ ‘ ( 𝐴 / ( 2 ↑ 𝑁 ) ) ) · ( 2 ↑ 𝑁 ) ) ) |
72 |
60 71
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) + 𝐴 ) = ( ( ⌊ ‘ ( 𝐴 / ( 2 ↑ 𝑁 ) ) ) · ( 2 ↑ 𝑁 ) ) ) |
73 |
72
|
fveq2d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( bits ‘ ( - ( 𝐴 mod ( 2 ↑ 𝑁 ) ) + 𝐴 ) ) = ( bits ‘ ( ( ⌊ ‘ ( 𝐴 / ( 2 ↑ 𝑁 ) ) ) · ( 2 ↑ 𝑁 ) ) ) ) |
74 |
10 58 73
|
3eqtr3d |
⊢ ( ( 𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ0 ) → ( ( bits ‘ 𝐴 ) ∩ ( ℤ≥ ‘ 𝑁 ) ) = ( bits ‘ ( ( ⌊ ‘ ( 𝐴 / ( 2 ↑ 𝑁 ) ) ) · ( 2 ↑ 𝑁 ) ) ) ) |