Step |
Hyp |
Ref |
Expression |
1 |
|
zre |
⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℝ ) |
2 |
|
chtval |
⊢ ( 𝑁 ∈ ℝ → ( θ ‘ 𝑁 ) = Σ 𝑛 ∈ ( ( 0 [,] 𝑁 ) ∩ ℙ ) ( log ‘ 𝑛 ) ) |
3 |
1 2
|
syl |
⊢ ( 𝑁 ∈ ℤ → ( θ ‘ 𝑁 ) = Σ 𝑛 ∈ ( ( 0 [,] 𝑁 ) ∩ ℙ ) ( log ‘ 𝑛 ) ) |
4 |
|
nnz |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ ) |
5 |
|
ppisval |
⊢ ( 𝑁 ∈ ℝ → ( ( 0 [,] 𝑁 ) ∩ ℙ ) = ( ( 2 ... ( ⌊ ‘ 𝑁 ) ) ∩ ℙ ) ) |
6 |
1 5
|
syl |
⊢ ( 𝑁 ∈ ℤ → ( ( 0 [,] 𝑁 ) ∩ ℙ ) = ( ( 2 ... ( ⌊ ‘ 𝑁 ) ) ∩ ℙ ) ) |
7 |
|
flid |
⊢ ( 𝑁 ∈ ℤ → ( ⌊ ‘ 𝑁 ) = 𝑁 ) |
8 |
7
|
oveq2d |
⊢ ( 𝑁 ∈ ℤ → ( 2 ... ( ⌊ ‘ 𝑁 ) ) = ( 2 ... 𝑁 ) ) |
9 |
8
|
ineq1d |
⊢ ( 𝑁 ∈ ℤ → ( ( 2 ... ( ⌊ ‘ 𝑁 ) ) ∩ ℙ ) = ( ( 2 ... 𝑁 ) ∩ ℙ ) ) |
10 |
6 9
|
eqtrd |
⊢ ( 𝑁 ∈ ℤ → ( ( 0 [,] 𝑁 ) ∩ ℙ ) = ( ( 2 ... 𝑁 ) ∩ ℙ ) ) |
11 |
4 10
|
syl |
⊢ ( 𝑁 ∈ ℕ → ( ( 0 [,] 𝑁 ) ∩ ℙ ) = ( ( 2 ... 𝑁 ) ∩ ℙ ) ) |
12 |
|
2nn |
⊢ 2 ∈ ℕ |
13 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
14 |
12 13
|
eleqtri |
⊢ 2 ∈ ( ℤ≥ ‘ 1 ) |
15 |
|
fzss1 |
⊢ ( 2 ∈ ( ℤ≥ ‘ 1 ) → ( 2 ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) ) |
16 |
14 15
|
ax-mp |
⊢ ( 2 ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) |
17 |
|
ssdif0 |
⊢ ( ( 2 ... 𝑁 ) ⊆ ( 1 ... 𝑁 ) ↔ ( ( 2 ... 𝑁 ) ∖ ( 1 ... 𝑁 ) ) = ∅ ) |
18 |
16 17
|
mpbi |
⊢ ( ( 2 ... 𝑁 ) ∖ ( 1 ... 𝑁 ) ) = ∅ |
19 |
18
|
ineq1i |
⊢ ( ( ( 2 ... 𝑁 ) ∖ ( 1 ... 𝑁 ) ) ∩ ℙ ) = ( ∅ ∩ ℙ ) |
20 |
|
0in |
⊢ ( ∅ ∩ ℙ ) = ∅ |
21 |
19 20
|
eqtri |
⊢ ( ( ( 2 ... 𝑁 ) ∖ ( 1 ... 𝑁 ) ) ∩ ℙ ) = ∅ |
22 |
21
|
a1i |
⊢ ( 𝑁 ∈ ℕ → ( ( ( 2 ... 𝑁 ) ∖ ( 1 ... 𝑁 ) ) ∩ ℙ ) = ∅ ) |
23 |
13
|
eleq2i |
⊢ ( 𝑁 ∈ ℕ ↔ 𝑁 ∈ ( ℤ≥ ‘ 1 ) ) |
24 |
|
fzpred |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 1 ) → ( 1 ... 𝑁 ) = ( { 1 } ∪ ( ( 1 + 1 ) ... 𝑁 ) ) ) |
25 |
23 24
|
sylbi |
⊢ ( 𝑁 ∈ ℕ → ( 1 ... 𝑁 ) = ( { 1 } ∪ ( ( 1 + 1 ) ... 𝑁 ) ) ) |
26 |
25
|
eqcomd |
⊢ ( 𝑁 ∈ ℕ → ( { 1 } ∪ ( ( 1 + 1 ) ... 𝑁 ) ) = ( 1 ... 𝑁 ) ) |
27 |
|
1p1e2 |
⊢ ( 1 + 1 ) = 2 |
28 |
27
|
oveq1i |
⊢ ( ( 1 + 1 ) ... 𝑁 ) = ( 2 ... 𝑁 ) |
29 |
28
|
a1i |
⊢ ( 𝑁 ∈ ℕ → ( ( 1 + 1 ) ... 𝑁 ) = ( 2 ... 𝑁 ) ) |
30 |
26 29
|
difeq12d |
⊢ ( 𝑁 ∈ ℕ → ( ( { 1 } ∪ ( ( 1 + 1 ) ... 𝑁 ) ) ∖ ( ( 1 + 1 ) ... 𝑁 ) ) = ( ( 1 ... 𝑁 ) ∖ ( 2 ... 𝑁 ) ) ) |
31 |
|
difun2 |
⊢ ( ( { 1 } ∪ ( ( 1 + 1 ) ... 𝑁 ) ) ∖ ( ( 1 + 1 ) ... 𝑁 ) ) = ( { 1 } ∖ ( ( 1 + 1 ) ... 𝑁 ) ) |
32 |
|
fzpreddisj |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 1 ) → ( { 1 } ∩ ( ( 1 + 1 ) ... 𝑁 ) ) = ∅ ) |
33 |
23 32
|
sylbi |
⊢ ( 𝑁 ∈ ℕ → ( { 1 } ∩ ( ( 1 + 1 ) ... 𝑁 ) ) = ∅ ) |
34 |
|
disjdif2 |
⊢ ( ( { 1 } ∩ ( ( 1 + 1 ) ... 𝑁 ) ) = ∅ → ( { 1 } ∖ ( ( 1 + 1 ) ... 𝑁 ) ) = { 1 } ) |
35 |
33 34
|
syl |
⊢ ( 𝑁 ∈ ℕ → ( { 1 } ∖ ( ( 1 + 1 ) ... 𝑁 ) ) = { 1 } ) |
36 |
31 35
|
syl5eq |
⊢ ( 𝑁 ∈ ℕ → ( ( { 1 } ∪ ( ( 1 + 1 ) ... 𝑁 ) ) ∖ ( ( 1 + 1 ) ... 𝑁 ) ) = { 1 } ) |
37 |
30 36
|
eqtr3d |
⊢ ( 𝑁 ∈ ℕ → ( ( 1 ... 𝑁 ) ∖ ( 2 ... 𝑁 ) ) = { 1 } ) |
38 |
37
|
ineq1d |
⊢ ( 𝑁 ∈ ℕ → ( ( ( 1 ... 𝑁 ) ∖ ( 2 ... 𝑁 ) ) ∩ ℙ ) = ( { 1 } ∩ ℙ ) ) |
39 |
|
incom |
⊢ ( ℙ ∩ { 1 } ) = ( { 1 } ∩ ℙ ) |
40 |
|
1nprm |
⊢ ¬ 1 ∈ ℙ |
41 |
|
disjsn |
⊢ ( ( ℙ ∩ { 1 } ) = ∅ ↔ ¬ 1 ∈ ℙ ) |
42 |
40 41
|
mpbir |
⊢ ( ℙ ∩ { 1 } ) = ∅ |
43 |
39 42
|
eqtr3i |
⊢ ( { 1 } ∩ ℙ ) = ∅ |
44 |
38 43
|
eqtrdi |
⊢ ( 𝑁 ∈ ℕ → ( ( ( 1 ... 𝑁 ) ∖ ( 2 ... 𝑁 ) ) ∩ ℙ ) = ∅ ) |
45 |
|
difininv |
⊢ ( ( ( ( ( 2 ... 𝑁 ) ∖ ( 1 ... 𝑁 ) ) ∩ ℙ ) = ∅ ∧ ( ( ( 1 ... 𝑁 ) ∖ ( 2 ... 𝑁 ) ) ∩ ℙ ) = ∅ ) → ( ( 2 ... 𝑁 ) ∩ ℙ ) = ( ( 1 ... 𝑁 ) ∩ ℙ ) ) |
46 |
22 44 45
|
syl2anc |
⊢ ( 𝑁 ∈ ℕ → ( ( 2 ... 𝑁 ) ∩ ℙ ) = ( ( 1 ... 𝑁 ) ∩ ℙ ) ) |
47 |
11 46
|
eqtrd |
⊢ ( 𝑁 ∈ ℕ → ( ( 0 [,] 𝑁 ) ∩ ℙ ) = ( ( 1 ... 𝑁 ) ∩ ℙ ) ) |
48 |
47
|
adantl |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 ∈ ℕ ) → ( ( 0 [,] 𝑁 ) ∩ ℙ ) = ( ( 1 ... 𝑁 ) ∩ ℙ ) ) |
49 |
|
znnnlt1 |
⊢ ( 𝑁 ∈ ℤ → ( ¬ 𝑁 ∈ ℕ ↔ 𝑁 < 1 ) ) |
50 |
49
|
biimpa |
⊢ ( ( 𝑁 ∈ ℤ ∧ ¬ 𝑁 ∈ ℕ ) → 𝑁 < 1 ) |
51 |
|
incom |
⊢ ( ( 0 [,] 𝑁 ) ∩ ℙ ) = ( ℙ ∩ ( 0 [,] 𝑁 ) ) |
52 |
|
isprm3 |
⊢ ( 𝑛 ∈ ℙ ↔ ( 𝑛 ∈ ( ℤ≥ ‘ 2 ) ∧ ∀ 𝑖 ∈ ( 2 ... ( 𝑛 − 1 ) ) ¬ 𝑖 ∥ 𝑛 ) ) |
53 |
52
|
simplbi |
⊢ ( 𝑛 ∈ ℙ → 𝑛 ∈ ( ℤ≥ ‘ 2 ) ) |
54 |
53
|
ssriv |
⊢ ℙ ⊆ ( ℤ≥ ‘ 2 ) |
55 |
12
|
nnzi |
⊢ 2 ∈ ℤ |
56 |
|
uzssico |
⊢ ( 2 ∈ ℤ → ( ℤ≥ ‘ 2 ) ⊆ ( 2 [,) +∞ ) ) |
57 |
55 56
|
ax-mp |
⊢ ( ℤ≥ ‘ 2 ) ⊆ ( 2 [,) +∞ ) |
58 |
54 57
|
sstri |
⊢ ℙ ⊆ ( 2 [,) +∞ ) |
59 |
|
incom |
⊢ ( ( 0 [,] 𝑁 ) ∩ ( 2 [,) +∞ ) ) = ( ( 2 [,) +∞ ) ∩ ( 0 [,] 𝑁 ) ) |
60 |
|
0xr |
⊢ 0 ∈ ℝ* |
61 |
60
|
a1i |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → 0 ∈ ℝ* ) |
62 |
12
|
nnrei |
⊢ 2 ∈ ℝ |
63 |
62
|
rexri |
⊢ 2 ∈ ℝ* |
64 |
63
|
a1i |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → 2 ∈ ℝ* ) |
65 |
|
0le0 |
⊢ 0 ≤ 0 |
66 |
65
|
a1i |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → 0 ≤ 0 ) |
67 |
1
|
adantr |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → 𝑁 ∈ ℝ ) |
68 |
|
1red |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → 1 ∈ ℝ ) |
69 |
62
|
a1i |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → 2 ∈ ℝ ) |
70 |
|
simpr |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → 𝑁 < 1 ) |
71 |
|
1lt2 |
⊢ 1 < 2 |
72 |
71
|
a1i |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → 1 < 2 ) |
73 |
67 68 69 70 72
|
lttrd |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → 𝑁 < 2 ) |
74 |
|
iccssico |
⊢ ( ( ( 0 ∈ ℝ* ∧ 2 ∈ ℝ* ) ∧ ( 0 ≤ 0 ∧ 𝑁 < 2 ) ) → ( 0 [,] 𝑁 ) ⊆ ( 0 [,) 2 ) ) |
75 |
61 64 66 73 74
|
syl22anc |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → ( 0 [,] 𝑁 ) ⊆ ( 0 [,) 2 ) ) |
76 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
77 |
|
icodisj |
⊢ ( ( 0 ∈ ℝ* ∧ 2 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( ( 0 [,) 2 ) ∩ ( 2 [,) +∞ ) ) = ∅ ) |
78 |
60 63 76 77
|
mp3an |
⊢ ( ( 0 [,) 2 ) ∩ ( 2 [,) +∞ ) ) = ∅ |
79 |
|
ssdisj |
⊢ ( ( ( 0 [,] 𝑁 ) ⊆ ( 0 [,) 2 ) ∧ ( ( 0 [,) 2 ) ∩ ( 2 [,) +∞ ) ) = ∅ ) → ( ( 0 [,] 𝑁 ) ∩ ( 2 [,) +∞ ) ) = ∅ ) |
80 |
75 78 79
|
sylancl |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → ( ( 0 [,] 𝑁 ) ∩ ( 2 [,) +∞ ) ) = ∅ ) |
81 |
59 80
|
eqtr3id |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → ( ( 2 [,) +∞ ) ∩ ( 0 [,] 𝑁 ) ) = ∅ ) |
82 |
|
ssdisj |
⊢ ( ( ℙ ⊆ ( 2 [,) +∞ ) ∧ ( ( 2 [,) +∞ ) ∩ ( 0 [,] 𝑁 ) ) = ∅ ) → ( ℙ ∩ ( 0 [,] 𝑁 ) ) = ∅ ) |
83 |
58 81 82
|
sylancr |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → ( ℙ ∩ ( 0 [,] 𝑁 ) ) = ∅ ) |
84 |
51 83
|
syl5eq |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → ( ( 0 [,] 𝑁 ) ∩ ℙ ) = ∅ ) |
85 |
|
1zzd |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → 1 ∈ ℤ ) |
86 |
|
simpl |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → 𝑁 ∈ ℤ ) |
87 |
|
fzn |
⊢ ( ( 1 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 < 1 ↔ ( 1 ... 𝑁 ) = ∅ ) ) |
88 |
87
|
biimpa |
⊢ ( ( ( 1 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ 𝑁 < 1 ) → ( 1 ... 𝑁 ) = ∅ ) |
89 |
85 86 70 88
|
syl21anc |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → ( 1 ... 𝑁 ) = ∅ ) |
90 |
89
|
ineq1d |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → ( ( 1 ... 𝑁 ) ∩ ℙ ) = ( ∅ ∩ ℙ ) ) |
91 |
90 20
|
eqtrdi |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → ( ( 1 ... 𝑁 ) ∩ ℙ ) = ∅ ) |
92 |
84 91
|
eqtr4d |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑁 < 1 ) → ( ( 0 [,] 𝑁 ) ∩ ℙ ) = ( ( 1 ... 𝑁 ) ∩ ℙ ) ) |
93 |
50 92
|
syldan |
⊢ ( ( 𝑁 ∈ ℤ ∧ ¬ 𝑁 ∈ ℕ ) → ( ( 0 [,] 𝑁 ) ∩ ℙ ) = ( ( 1 ... 𝑁 ) ∩ ℙ ) ) |
94 |
|
exmidd |
⊢ ( 𝑁 ∈ ℤ → ( 𝑁 ∈ ℕ ∨ ¬ 𝑁 ∈ ℕ ) ) |
95 |
48 93 94
|
mpjaodan |
⊢ ( 𝑁 ∈ ℤ → ( ( 0 [,] 𝑁 ) ∩ ℙ ) = ( ( 1 ... 𝑁 ) ∩ ℙ ) ) |
96 |
95
|
sumeq1d |
⊢ ( 𝑁 ∈ ℤ → Σ 𝑛 ∈ ( ( 0 [,] 𝑁 ) ∩ ℙ ) ( log ‘ 𝑛 ) = Σ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∩ ℙ ) ( log ‘ 𝑛 ) ) |
97 |
3 96
|
eqtrd |
⊢ ( 𝑁 ∈ ℤ → ( θ ‘ 𝑁 ) = Σ 𝑛 ∈ ( ( 1 ... 𝑁 ) ∩ ℙ ) ( log ‘ 𝑛 ) ) |