Step |
Hyp |
Ref |
Expression |
1 |
|
prmrec.1 |
⊢ 𝐹 = ( 𝑛 ∈ ℕ ↦ if ( 𝑛 ∈ ℙ , ( 1 / 𝑛 ) , 0 ) ) |
2 |
|
prmrec.2 |
⊢ ( 𝜑 → 𝐾 ∈ ℕ ) |
3 |
|
prmrec.3 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
4 |
|
prmrec.4 |
⊢ 𝑀 = { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ∀ 𝑝 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑝 ∥ 𝑛 } |
5 |
|
prmrec.5 |
⊢ ( 𝜑 → seq 1 ( + , 𝐹 ) ∈ dom ⇝ ) |
6 |
|
prmrec.6 |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐾 + 1 ) ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) < ( 1 / 2 ) ) |
7 |
|
prmrec.7 |
⊢ 𝑊 = ( 𝑝 ∈ ℕ ↦ { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛 ) } ) |
8 |
3
|
nnred |
⊢ ( 𝜑 → 𝑁 ∈ ℝ ) |
9 |
8
|
rehalfcld |
⊢ ( 𝜑 → ( 𝑁 / 2 ) ∈ ℝ ) |
10 |
|
fzfi |
⊢ ( 1 ... 𝑁 ) ∈ Fin |
11 |
4
|
ssrab3 |
⊢ 𝑀 ⊆ ( 1 ... 𝑁 ) |
12 |
|
ssfi |
⊢ ( ( ( 1 ... 𝑁 ) ∈ Fin ∧ 𝑀 ⊆ ( 1 ... 𝑁 ) ) → 𝑀 ∈ Fin ) |
13 |
10 11 12
|
mp2an |
⊢ 𝑀 ∈ Fin |
14 |
|
hashcl |
⊢ ( 𝑀 ∈ Fin → ( ♯ ‘ 𝑀 ) ∈ ℕ0 ) |
15 |
13 14
|
ax-mp |
⊢ ( ♯ ‘ 𝑀 ) ∈ ℕ0 |
16 |
15
|
nn0rei |
⊢ ( ♯ ‘ 𝑀 ) ∈ ℝ |
17 |
16
|
a1i |
⊢ ( 𝜑 → ( ♯ ‘ 𝑀 ) ∈ ℝ ) |
18 |
|
2nn |
⊢ 2 ∈ ℕ |
19 |
2
|
nnnn0d |
⊢ ( 𝜑 → 𝐾 ∈ ℕ0 ) |
20 |
|
nnexpcl |
⊢ ( ( 2 ∈ ℕ ∧ 𝐾 ∈ ℕ0 ) → ( 2 ↑ 𝐾 ) ∈ ℕ ) |
21 |
18 19 20
|
sylancr |
⊢ ( 𝜑 → ( 2 ↑ 𝐾 ) ∈ ℕ ) |
22 |
21
|
nnred |
⊢ ( 𝜑 → ( 2 ↑ 𝐾 ) ∈ ℝ ) |
23 |
3
|
nnrpd |
⊢ ( 𝜑 → 𝑁 ∈ ℝ+ ) |
24 |
23
|
rpsqrtcld |
⊢ ( 𝜑 → ( √ ‘ 𝑁 ) ∈ ℝ+ ) |
25 |
24
|
rpred |
⊢ ( 𝜑 → ( √ ‘ 𝑁 ) ∈ ℝ ) |
26 |
22 25
|
remulcld |
⊢ ( 𝜑 → ( ( 2 ↑ 𝐾 ) · ( √ ‘ 𝑁 ) ) ∈ ℝ ) |
27 |
8
|
recnd |
⊢ ( 𝜑 → 𝑁 ∈ ℂ ) |
28 |
27
|
2halvesd |
⊢ ( 𝜑 → ( ( 𝑁 / 2 ) + ( 𝑁 / 2 ) ) = 𝑁 ) |
29 |
11
|
a1i |
⊢ ( 𝜑 → 𝑀 ⊆ ( 1 ... 𝑁 ) ) |
30 |
2
|
peano2nnd |
⊢ ( 𝜑 → ( 𝐾 + 1 ) ∈ ℕ ) |
31 |
|
elfzuz |
⊢ ( 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) → 𝑘 ∈ ( ℤ≥ ‘ ( 𝐾 + 1 ) ) ) |
32 |
|
eluznn |
⊢ ( ( ( 𝐾 + 1 ) ∈ ℕ ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐾 + 1 ) ) ) → 𝑘 ∈ ℕ ) |
33 |
30 31 32
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → 𝑘 ∈ ℕ ) |
34 |
|
eleq1w |
⊢ ( 𝑝 = 𝑘 → ( 𝑝 ∈ ℙ ↔ 𝑘 ∈ ℙ ) ) |
35 |
|
breq1 |
⊢ ( 𝑝 = 𝑘 → ( 𝑝 ∥ 𝑛 ↔ 𝑘 ∥ 𝑛 ) ) |
36 |
34 35
|
anbi12d |
⊢ ( 𝑝 = 𝑘 → ( ( 𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛 ) ↔ ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛 ) ) ) |
37 |
36
|
rabbidv |
⊢ ( 𝑝 = 𝑘 → { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛 ) } = { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛 ) } ) |
38 |
|
ovex |
⊢ ( 1 ... 𝑁 ) ∈ V |
39 |
38
|
rabex |
⊢ { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛 ) } ∈ V |
40 |
37 7 39
|
fvmpt |
⊢ ( 𝑘 ∈ ℕ → ( 𝑊 ‘ 𝑘 ) = { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛 ) } ) |
41 |
40
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑊 ‘ 𝑘 ) = { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛 ) } ) |
42 |
|
ssrab2 |
⊢ { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛 ) } ⊆ ( 1 ... 𝑁 ) |
43 |
41 42
|
eqsstrdi |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑊 ‘ 𝑘 ) ⊆ ( 1 ... 𝑁 ) ) |
44 |
33 43
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → ( 𝑊 ‘ 𝑘 ) ⊆ ( 1 ... 𝑁 ) ) |
45 |
44
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ⊆ ( 1 ... 𝑁 ) ) |
46 |
|
iunss |
⊢ ( ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ⊆ ( 1 ... 𝑁 ) ↔ ∀ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ⊆ ( 1 ... 𝑁 ) ) |
47 |
45 46
|
sylibr |
⊢ ( 𝜑 → ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ⊆ ( 1 ... 𝑁 ) ) |
48 |
29 47
|
unssd |
⊢ ( 𝜑 → ( 𝑀 ∪ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ⊆ ( 1 ... 𝑁 ) ) |
49 |
|
breq1 |
⊢ ( 𝑝 = 𝑞 → ( 𝑝 ∥ 𝑛 ↔ 𝑞 ∥ 𝑛 ) ) |
50 |
49
|
notbid |
⊢ ( 𝑝 = 𝑞 → ( ¬ 𝑝 ∥ 𝑛 ↔ ¬ 𝑞 ∥ 𝑛 ) ) |
51 |
50
|
cbvralvw |
⊢ ( ∀ 𝑝 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑝 ∥ 𝑛 ↔ ∀ 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑞 ∥ 𝑛 ) |
52 |
|
breq2 |
⊢ ( 𝑛 = 𝑥 → ( 𝑞 ∥ 𝑛 ↔ 𝑞 ∥ 𝑥 ) ) |
53 |
52
|
notbid |
⊢ ( 𝑛 = 𝑥 → ( ¬ 𝑞 ∥ 𝑛 ↔ ¬ 𝑞 ∥ 𝑥 ) ) |
54 |
53
|
ralbidv |
⊢ ( 𝑛 = 𝑥 → ( ∀ 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑞 ∥ 𝑛 ↔ ∀ 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑞 ∥ 𝑥 ) ) |
55 |
51 54
|
syl5bb |
⊢ ( 𝑛 = 𝑥 → ( ∀ 𝑝 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑝 ∥ 𝑛 ↔ ∀ 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑞 ∥ 𝑥 ) ) |
56 |
55 4
|
elrab2 |
⊢ ( 𝑥 ∈ 𝑀 ↔ ( 𝑥 ∈ ( 1 ... 𝑁 ) ∧ ∀ 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑞 ∥ 𝑥 ) ) |
57 |
|
elun1 |
⊢ ( 𝑥 ∈ 𝑀 → 𝑥 ∈ ( 𝑀 ∪ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) |
58 |
56 57
|
sylbir |
⊢ ( ( 𝑥 ∈ ( 1 ... 𝑁 ) ∧ ∀ 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑞 ∥ 𝑥 ) → 𝑥 ∈ ( 𝑀 ∪ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) |
59 |
58
|
ex |
⊢ ( 𝑥 ∈ ( 1 ... 𝑁 ) → ( ∀ 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑞 ∥ 𝑥 → 𝑥 ∈ ( 𝑀 ∪ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) ) |
60 |
59
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) → ( ∀ 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑞 ∥ 𝑥 → 𝑥 ∈ ( 𝑀 ∪ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) ) |
61 |
|
dfrex2 |
⊢ ( ∃ 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) 𝑞 ∥ 𝑥 ↔ ¬ ∀ 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑞 ∥ 𝑥 ) |
62 |
|
eldifn |
⊢ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) → ¬ 𝑞 ∈ ( 1 ... 𝐾 ) ) |
63 |
62
|
ad2antrl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → ¬ 𝑞 ∈ ( 1 ... 𝐾 ) ) |
64 |
|
eldifi |
⊢ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) → 𝑞 ∈ ℙ ) |
65 |
64
|
ad2antrl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑞 ∈ ℙ ) |
66 |
|
prmnn |
⊢ ( 𝑞 ∈ ℙ → 𝑞 ∈ ℕ ) |
67 |
65 66
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑞 ∈ ℕ ) |
68 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
69 |
67 68
|
eleqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑞 ∈ ( ℤ≥ ‘ 1 ) ) |
70 |
2
|
nnzd |
⊢ ( 𝜑 → 𝐾 ∈ ℤ ) |
71 |
70
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝐾 ∈ ℤ ) |
72 |
|
elfz5 |
⊢ ( ( 𝑞 ∈ ( ℤ≥ ‘ 1 ) ∧ 𝐾 ∈ ℤ ) → ( 𝑞 ∈ ( 1 ... 𝐾 ) ↔ 𝑞 ≤ 𝐾 ) ) |
73 |
69 71 72
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → ( 𝑞 ∈ ( 1 ... 𝐾 ) ↔ 𝑞 ≤ 𝐾 ) ) |
74 |
63 73
|
mtbid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → ¬ 𝑞 ≤ 𝐾 ) |
75 |
2
|
nnred |
⊢ ( 𝜑 → 𝐾 ∈ ℝ ) |
76 |
75
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝐾 ∈ ℝ ) |
77 |
67
|
nnred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑞 ∈ ℝ ) |
78 |
76 77
|
ltnled |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → ( 𝐾 < 𝑞 ↔ ¬ 𝑞 ≤ 𝐾 ) ) |
79 |
74 78
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝐾 < 𝑞 ) |
80 |
|
prmz |
⊢ ( 𝑞 ∈ ℙ → 𝑞 ∈ ℤ ) |
81 |
65 80
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑞 ∈ ℤ ) |
82 |
|
zltp1le |
⊢ ( ( 𝐾 ∈ ℤ ∧ 𝑞 ∈ ℤ ) → ( 𝐾 < 𝑞 ↔ ( 𝐾 + 1 ) ≤ 𝑞 ) ) |
83 |
71 81 82
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → ( 𝐾 < 𝑞 ↔ ( 𝐾 + 1 ) ≤ 𝑞 ) ) |
84 |
79 83
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → ( 𝐾 + 1 ) ≤ 𝑞 ) |
85 |
|
elfznn |
⊢ ( 𝑥 ∈ ( 1 ... 𝑁 ) → 𝑥 ∈ ℕ ) |
86 |
85
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑥 ∈ ℕ ) |
87 |
86
|
nnred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑥 ∈ ℝ ) |
88 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑁 ∈ ℝ ) |
89 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑞 ∥ 𝑥 ) |
90 |
|
dvdsle |
⊢ ( ( 𝑞 ∈ ℤ ∧ 𝑥 ∈ ℕ ) → ( 𝑞 ∥ 𝑥 → 𝑞 ≤ 𝑥 ) ) |
91 |
81 86 90
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → ( 𝑞 ∥ 𝑥 → 𝑞 ≤ 𝑥 ) ) |
92 |
89 91
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑞 ≤ 𝑥 ) |
93 |
|
elfzle2 |
⊢ ( 𝑥 ∈ ( 1 ... 𝑁 ) → 𝑥 ≤ 𝑁 ) |
94 |
93
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑥 ≤ 𝑁 ) |
95 |
77 87 88 92 94
|
letrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑞 ≤ 𝑁 ) |
96 |
70
|
peano2zd |
⊢ ( 𝜑 → ( 𝐾 + 1 ) ∈ ℤ ) |
97 |
96
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → ( 𝐾 + 1 ) ∈ ℤ ) |
98 |
3
|
nnzd |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
99 |
98
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑁 ∈ ℤ ) |
100 |
|
elfz |
⊢ ( ( 𝑞 ∈ ℤ ∧ ( 𝐾 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑞 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ↔ ( ( 𝐾 + 1 ) ≤ 𝑞 ∧ 𝑞 ≤ 𝑁 ) ) ) |
101 |
81 97 99 100
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → ( 𝑞 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ↔ ( ( 𝐾 + 1 ) ≤ 𝑞 ∧ 𝑞 ≤ 𝑁 ) ) ) |
102 |
84 95 101
|
mpbir2and |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑞 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) |
103 |
52
|
anbi2d |
⊢ ( 𝑛 = 𝑥 → ( ( 𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑛 ) ↔ ( 𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑥 ) ) ) |
104 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑥 ∈ ( 1 ... 𝑁 ) ) |
105 |
65 89
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → ( 𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑥 ) ) |
106 |
103 104 105
|
elrabd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑥 ∈ { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑛 ) } ) |
107 |
|
eleq1w |
⊢ ( 𝑝 = 𝑞 → ( 𝑝 ∈ ℙ ↔ 𝑞 ∈ ℙ ) ) |
108 |
107 49
|
anbi12d |
⊢ ( 𝑝 = 𝑞 → ( ( 𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛 ) ↔ ( 𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑛 ) ) ) |
109 |
108
|
rabbidv |
⊢ ( 𝑝 = 𝑞 → { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑝 ∈ ℙ ∧ 𝑝 ∥ 𝑛 ) } = { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑛 ) } ) |
110 |
38
|
rabex |
⊢ { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑛 ) } ∈ V |
111 |
109 7 110
|
fvmpt |
⊢ ( 𝑞 ∈ ℕ → ( 𝑊 ‘ 𝑞 ) = { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑛 ) } ) |
112 |
67 111
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → ( 𝑊 ‘ 𝑞 ) = { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑞 ∈ ℙ ∧ 𝑞 ∥ 𝑛 ) } ) |
113 |
106 112
|
eleqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑥 ∈ ( 𝑊 ‘ 𝑞 ) ) |
114 |
|
fveq2 |
⊢ ( 𝑘 = 𝑞 → ( 𝑊 ‘ 𝑘 ) = ( 𝑊 ‘ 𝑞 ) ) |
115 |
114
|
eliuni |
⊢ ( ( 𝑞 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ∧ 𝑥 ∈ ( 𝑊 ‘ 𝑞 ) ) → 𝑥 ∈ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) |
116 |
102 113 115
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑥 ∈ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) |
117 |
|
elun2 |
⊢ ( 𝑥 ∈ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) → 𝑥 ∈ ( 𝑀 ∪ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) |
118 |
116 117
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) ∧ ( 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ∧ 𝑞 ∥ 𝑥 ) ) → 𝑥 ∈ ( 𝑀 ∪ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) |
119 |
118
|
rexlimdvaa |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) → ( ∃ 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) 𝑞 ∥ 𝑥 → 𝑥 ∈ ( 𝑀 ∪ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) ) |
120 |
61 119
|
syl5bir |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) → ( ¬ ∀ 𝑞 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑞 ∥ 𝑥 → 𝑥 ∈ ( 𝑀 ∪ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) ) |
121 |
60 120
|
pm2.61d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 1 ... 𝑁 ) ) → 𝑥 ∈ ( 𝑀 ∪ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) |
122 |
48 121
|
eqelssd |
⊢ ( 𝜑 → ( 𝑀 ∪ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) = ( 1 ... 𝑁 ) ) |
123 |
122
|
fveq2d |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝑀 ∪ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) = ( ♯ ‘ ( 1 ... 𝑁 ) ) ) |
124 |
3
|
nnnn0d |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
125 |
|
hashfz1 |
⊢ ( 𝑁 ∈ ℕ0 → ( ♯ ‘ ( 1 ... 𝑁 ) ) = 𝑁 ) |
126 |
124 125
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ ( 1 ... 𝑁 ) ) = 𝑁 ) |
127 |
123 126
|
eqtr2d |
⊢ ( 𝜑 → 𝑁 = ( ♯ ‘ ( 𝑀 ∪ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) ) |
128 |
13
|
a1i |
⊢ ( 𝜑 → 𝑀 ∈ Fin ) |
129 |
|
ssfi |
⊢ ( ( ( 1 ... 𝑁 ) ∈ Fin ∧ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ⊆ ( 1 ... 𝑁 ) ) → ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ∈ Fin ) |
130 |
10 47 129
|
sylancr |
⊢ ( 𝜑 → ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ∈ Fin ) |
131 |
|
breq1 |
⊢ ( 𝑝 = 𝑘 → ( 𝑝 ∥ 𝑥 ↔ 𝑘 ∥ 𝑥 ) ) |
132 |
131
|
notbid |
⊢ ( 𝑝 = 𝑘 → ( ¬ 𝑝 ∥ 𝑥 ↔ ¬ 𝑘 ∥ 𝑥 ) ) |
133 |
|
breq2 |
⊢ ( 𝑛 = 𝑥 → ( 𝑝 ∥ 𝑛 ↔ 𝑝 ∥ 𝑥 ) ) |
134 |
133
|
notbid |
⊢ ( 𝑛 = 𝑥 → ( ¬ 𝑝 ∥ 𝑛 ↔ ¬ 𝑝 ∥ 𝑥 ) ) |
135 |
134
|
ralbidv |
⊢ ( 𝑛 = 𝑥 → ( ∀ 𝑝 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑝 ∥ 𝑛 ↔ ∀ 𝑝 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑝 ∥ 𝑥 ) ) |
136 |
135 4
|
elrab2 |
⊢ ( 𝑥 ∈ 𝑀 ↔ ( 𝑥 ∈ ( 1 ... 𝑁 ) ∧ ∀ 𝑝 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑝 ∥ 𝑥 ) ) |
137 |
136
|
simprbi |
⊢ ( 𝑥 ∈ 𝑀 → ∀ 𝑝 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑝 ∥ 𝑥 ) |
138 |
137
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ ( 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ∧ 𝑘 ∈ ℙ ) ) → ∀ 𝑝 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ¬ 𝑝 ∥ 𝑥 ) |
139 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ ( 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ∧ 𝑘 ∈ ℙ ) ) → 𝑘 ∈ ℙ ) |
140 |
|
noel |
⊢ ¬ 𝑘 ∈ ∅ |
141 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ ( 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ∧ 𝑘 ∈ ℙ ) ) → 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) |
142 |
141
|
biantrud |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ ( 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ∧ 𝑘 ∈ ℙ ) ) → ( 𝑘 ∈ ( 1 ... 𝐾 ) ↔ ( 𝑘 ∈ ( 1 ... 𝐾 ) ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) ) ) |
143 |
|
elin |
⊢ ( 𝑘 ∈ ( ( 1 ... 𝐾 ) ∩ ( ( 𝐾 + 1 ) ... 𝑁 ) ) ↔ ( 𝑘 ∈ ( 1 ... 𝐾 ) ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) ) |
144 |
142 143
|
bitr4di |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ ( 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ∧ 𝑘 ∈ ℙ ) ) → ( 𝑘 ∈ ( 1 ... 𝐾 ) ↔ 𝑘 ∈ ( ( 1 ... 𝐾 ) ∩ ( ( 𝐾 + 1 ) ... 𝑁 ) ) ) ) |
145 |
75
|
ltp1d |
⊢ ( 𝜑 → 𝐾 < ( 𝐾 + 1 ) ) |
146 |
|
fzdisj |
⊢ ( 𝐾 < ( 𝐾 + 1 ) → ( ( 1 ... 𝐾 ) ∩ ( ( 𝐾 + 1 ) ... 𝑁 ) ) = ∅ ) |
147 |
145 146
|
syl |
⊢ ( 𝜑 → ( ( 1 ... 𝐾 ) ∩ ( ( 𝐾 + 1 ) ... 𝑁 ) ) = ∅ ) |
148 |
147
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ ( 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ∧ 𝑘 ∈ ℙ ) ) → ( ( 1 ... 𝐾 ) ∩ ( ( 𝐾 + 1 ) ... 𝑁 ) ) = ∅ ) |
149 |
148
|
eleq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ ( 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ∧ 𝑘 ∈ ℙ ) ) → ( 𝑘 ∈ ( ( 1 ... 𝐾 ) ∩ ( ( 𝐾 + 1 ) ... 𝑁 ) ) ↔ 𝑘 ∈ ∅ ) ) |
150 |
144 149
|
bitrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ ( 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ∧ 𝑘 ∈ ℙ ) ) → ( 𝑘 ∈ ( 1 ... 𝐾 ) ↔ 𝑘 ∈ ∅ ) ) |
151 |
140 150
|
mtbiri |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ ( 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ∧ 𝑘 ∈ ℙ ) ) → ¬ 𝑘 ∈ ( 1 ... 𝐾 ) ) |
152 |
139 151
|
eldifd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ ( 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ∧ 𝑘 ∈ ℙ ) ) → 𝑘 ∈ ( ℙ ∖ ( 1 ... 𝐾 ) ) ) |
153 |
132 138 152
|
rspcdva |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ ( 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ∧ 𝑘 ∈ ℙ ) ) → ¬ 𝑘 ∥ 𝑥 ) |
154 |
153
|
expr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → ( 𝑘 ∈ ℙ → ¬ 𝑘 ∥ 𝑥 ) ) |
155 |
|
imnan |
⊢ ( ( 𝑘 ∈ ℙ → ¬ 𝑘 ∥ 𝑥 ) ↔ ¬ ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑥 ) ) |
156 |
154 155
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → ¬ ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑥 ) ) |
157 |
33
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → 𝑘 ∈ ℕ ) |
158 |
157 40
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → ( 𝑊 ‘ 𝑘 ) = { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛 ) } ) |
159 |
158
|
eleq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → ( 𝑥 ∈ ( 𝑊 ‘ 𝑘 ) ↔ 𝑥 ∈ { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛 ) } ) ) |
160 |
|
breq2 |
⊢ ( 𝑛 = 𝑥 → ( 𝑘 ∥ 𝑛 ↔ 𝑘 ∥ 𝑥 ) ) |
161 |
160
|
anbi2d |
⊢ ( 𝑛 = 𝑥 → ( ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛 ) ↔ ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑥 ) ) ) |
162 |
161
|
elrab |
⊢ ( 𝑥 ∈ { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛 ) } ↔ ( 𝑥 ∈ ( 1 ... 𝑁 ) ∧ ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑥 ) ) ) |
163 |
162
|
simprbi |
⊢ ( 𝑥 ∈ { 𝑛 ∈ ( 1 ... 𝑁 ) ∣ ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑛 ) } → ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑥 ) ) |
164 |
159 163
|
syl6bi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → ( 𝑥 ∈ ( 𝑊 ‘ 𝑘 ) → ( 𝑘 ∈ ℙ ∧ 𝑘 ∥ 𝑥 ) ) ) |
165 |
156 164
|
mtod |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → ¬ 𝑥 ∈ ( 𝑊 ‘ 𝑘 ) ) |
166 |
165
|
nrexdv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → ¬ ∃ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) 𝑥 ∈ ( 𝑊 ‘ 𝑘 ) ) |
167 |
|
eliun |
⊢ ( 𝑥 ∈ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ↔ ∃ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) 𝑥 ∈ ( 𝑊 ‘ 𝑘 ) ) |
168 |
166 167
|
sylnibr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑀 ) → ¬ 𝑥 ∈ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) |
169 |
168
|
ex |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑀 → ¬ 𝑥 ∈ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) |
170 |
|
imnan |
⊢ ( ( 𝑥 ∈ 𝑀 → ¬ 𝑥 ∈ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ↔ ¬ ( 𝑥 ∈ 𝑀 ∧ 𝑥 ∈ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) |
171 |
169 170
|
sylib |
⊢ ( 𝜑 → ¬ ( 𝑥 ∈ 𝑀 ∧ 𝑥 ∈ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) |
172 |
|
elin |
⊢ ( 𝑥 ∈ ( 𝑀 ∩ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ↔ ( 𝑥 ∈ 𝑀 ∧ 𝑥 ∈ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) |
173 |
171 172
|
sylnibr |
⊢ ( 𝜑 → ¬ 𝑥 ∈ ( 𝑀 ∩ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) |
174 |
173
|
eq0rdv |
⊢ ( 𝜑 → ( 𝑀 ∩ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) = ∅ ) |
175 |
|
hashun |
⊢ ( ( 𝑀 ∈ Fin ∧ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ∈ Fin ∧ ( 𝑀 ∩ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) = ∅ ) → ( ♯ ‘ ( 𝑀 ∪ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) = ( ( ♯ ‘ 𝑀 ) + ( ♯ ‘ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) ) |
176 |
128 130 174 175
|
syl3anc |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝑀 ∪ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) = ( ( ♯ ‘ 𝑀 ) + ( ♯ ‘ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) ) |
177 |
28 127 176
|
3eqtrd |
⊢ ( 𝜑 → ( ( 𝑁 / 2 ) + ( 𝑁 / 2 ) ) = ( ( ♯ ‘ 𝑀 ) + ( ♯ ‘ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) ) |
178 |
|
hashcl |
⊢ ( ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ∈ Fin → ( ♯ ‘ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ∈ ℕ0 ) |
179 |
130 178
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ∈ ℕ0 ) |
180 |
179
|
nn0red |
⊢ ( 𝜑 → ( ♯ ‘ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ∈ ℝ ) |
181 |
|
fzfid |
⊢ ( 𝜑 → ( ( 𝐾 + 1 ) ... 𝑁 ) ∈ Fin ) |
182 |
30 32
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐾 + 1 ) ) ) → 𝑘 ∈ ℕ ) |
183 |
|
nnrecre |
⊢ ( 𝑘 ∈ ℕ → ( 1 / 𝑘 ) ∈ ℝ ) |
184 |
|
0re |
⊢ 0 ∈ ℝ |
185 |
|
ifcl |
⊢ ( ( ( 1 / 𝑘 ) ∈ ℝ ∧ 0 ∈ ℝ ) → if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ∈ ℝ ) |
186 |
183 184 185
|
sylancl |
⊢ ( 𝑘 ∈ ℕ → if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ∈ ℝ ) |
187 |
182 186
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐾 + 1 ) ) ) → if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ∈ ℝ ) |
188 |
31 187
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ) → if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ∈ ℝ ) |
189 |
181 188
|
fsumrecl |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ∈ ℝ ) |
190 |
8 189
|
remulcld |
⊢ ( 𝜑 → ( 𝑁 · Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) ∈ ℝ ) |
191 |
1 2 3 4 5 6 7
|
prmreclem4 |
⊢ ( 𝜑 → ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) → ( ♯ ‘ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ≤ ( 𝑁 · Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) ) ) |
192 |
|
eluz |
⊢ ( ( 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℤ ) → ( 𝐾 ∈ ( ℤ≥ ‘ 𝑁 ) ↔ 𝑁 ≤ 𝐾 ) ) |
193 |
98 70 192
|
syl2anc |
⊢ ( 𝜑 → ( 𝐾 ∈ ( ℤ≥ ‘ 𝑁 ) ↔ 𝑁 ≤ 𝐾 ) ) |
194 |
|
nnleltp1 |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐾 ∈ ℕ ) → ( 𝑁 ≤ 𝐾 ↔ 𝑁 < ( 𝐾 + 1 ) ) ) |
195 |
3 2 194
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ≤ 𝐾 ↔ 𝑁 < ( 𝐾 + 1 ) ) ) |
196 |
|
fzn |
⊢ ( ( ( 𝐾 + 1 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 < ( 𝐾 + 1 ) ↔ ( ( 𝐾 + 1 ) ... 𝑁 ) = ∅ ) ) |
197 |
96 98 196
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 < ( 𝐾 + 1 ) ↔ ( ( 𝐾 + 1 ) ... 𝑁 ) = ∅ ) ) |
198 |
193 195 197
|
3bitrd |
⊢ ( 𝜑 → ( 𝐾 ∈ ( ℤ≥ ‘ 𝑁 ) ↔ ( ( 𝐾 + 1 ) ... 𝑁 ) = ∅ ) ) |
199 |
|
0le0 |
⊢ 0 ≤ 0 |
200 |
27
|
mul01d |
⊢ ( 𝜑 → ( 𝑁 · 0 ) = 0 ) |
201 |
199 200
|
breqtrrid |
⊢ ( 𝜑 → 0 ≤ ( 𝑁 · 0 ) ) |
202 |
|
iuneq1 |
⊢ ( ( ( 𝐾 + 1 ) ... 𝑁 ) = ∅ → ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) = ∪ 𝑘 ∈ ∅ ( 𝑊 ‘ 𝑘 ) ) |
203 |
|
0iun |
⊢ ∪ 𝑘 ∈ ∅ ( 𝑊 ‘ 𝑘 ) = ∅ |
204 |
202 203
|
eqtrdi |
⊢ ( ( ( 𝐾 + 1 ) ... 𝑁 ) = ∅ → ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) = ∅ ) |
205 |
204
|
fveq2d |
⊢ ( ( ( 𝐾 + 1 ) ... 𝑁 ) = ∅ → ( ♯ ‘ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) = ( ♯ ‘ ∅ ) ) |
206 |
|
hash0 |
⊢ ( ♯ ‘ ∅ ) = 0 |
207 |
205 206
|
eqtrdi |
⊢ ( ( ( 𝐾 + 1 ) ... 𝑁 ) = ∅ → ( ♯ ‘ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) = 0 ) |
208 |
|
sumeq1 |
⊢ ( ( ( 𝐾 + 1 ) ... 𝑁 ) = ∅ → Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) = Σ 𝑘 ∈ ∅ if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) |
209 |
|
sum0 |
⊢ Σ 𝑘 ∈ ∅ if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) = 0 |
210 |
208 209
|
eqtrdi |
⊢ ( ( ( 𝐾 + 1 ) ... 𝑁 ) = ∅ → Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) = 0 ) |
211 |
210
|
oveq2d |
⊢ ( ( ( 𝐾 + 1 ) ... 𝑁 ) = ∅ → ( 𝑁 · Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) = ( 𝑁 · 0 ) ) |
212 |
207 211
|
breq12d |
⊢ ( ( ( 𝐾 + 1 ) ... 𝑁 ) = ∅ → ( ( ♯ ‘ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ≤ ( 𝑁 · Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) ↔ 0 ≤ ( 𝑁 · 0 ) ) ) |
213 |
201 212
|
syl5ibrcom |
⊢ ( 𝜑 → ( ( ( 𝐾 + 1 ) ... 𝑁 ) = ∅ → ( ♯ ‘ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ≤ ( 𝑁 · Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) ) ) |
214 |
198 213
|
sylbid |
⊢ ( 𝜑 → ( 𝐾 ∈ ( ℤ≥ ‘ 𝑁 ) → ( ♯ ‘ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ≤ ( 𝑁 · Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) ) ) |
215 |
|
uztric |
⊢ ( ( 𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ∨ 𝐾 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) |
216 |
70 98 215
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 ∈ ( ℤ≥ ‘ 𝐾 ) ∨ 𝐾 ∈ ( ℤ≥ ‘ 𝑁 ) ) ) |
217 |
191 214 216
|
mpjaod |
⊢ ( 𝜑 → ( ♯ ‘ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ≤ ( 𝑁 · Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) ) |
218 |
|
eqid |
⊢ ( ℤ≥ ‘ ( 𝐾 + 1 ) ) = ( ℤ≥ ‘ ( 𝐾 + 1 ) ) |
219 |
|
eleq1w |
⊢ ( 𝑛 = 𝑘 → ( 𝑛 ∈ ℙ ↔ 𝑘 ∈ ℙ ) ) |
220 |
|
oveq2 |
⊢ ( 𝑛 = 𝑘 → ( 1 / 𝑛 ) = ( 1 / 𝑘 ) ) |
221 |
219 220
|
ifbieq1d |
⊢ ( 𝑛 = 𝑘 → if ( 𝑛 ∈ ℙ , ( 1 / 𝑛 ) , 0 ) = if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) |
222 |
|
ovex |
⊢ ( 1 / 𝑘 ) ∈ V |
223 |
|
c0ex |
⊢ 0 ∈ V |
224 |
222 223
|
ifex |
⊢ if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ∈ V |
225 |
221 1 224
|
fvmpt |
⊢ ( 𝑘 ∈ ℕ → ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) |
226 |
182 225
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐾 + 1 ) ) ) → ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) |
227 |
186
|
recnd |
⊢ ( 𝑘 ∈ ℕ → if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ∈ ℂ ) |
228 |
225 227
|
eqeltrd |
⊢ ( 𝑘 ∈ ℕ → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
229 |
228
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
230 |
68 30 229
|
iserex |
⊢ ( 𝜑 → ( seq 1 ( + , 𝐹 ) ∈ dom ⇝ ↔ seq ( 𝐾 + 1 ) ( + , 𝐹 ) ∈ dom ⇝ ) ) |
231 |
5 230
|
mpbid |
⊢ ( 𝜑 → seq ( 𝐾 + 1 ) ( + , 𝐹 ) ∈ dom ⇝ ) |
232 |
218 96 226 187 231
|
isumrecl |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐾 + 1 ) ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ∈ ℝ ) |
233 |
|
halfre |
⊢ ( 1 / 2 ) ∈ ℝ |
234 |
233
|
a1i |
⊢ ( 𝜑 → ( 1 / 2 ) ∈ ℝ ) |
235 |
|
fzssuz |
⊢ ( ( 𝐾 + 1 ) ... 𝑁 ) ⊆ ( ℤ≥ ‘ ( 𝐾 + 1 ) ) |
236 |
235
|
a1i |
⊢ ( 𝜑 → ( ( 𝐾 + 1 ) ... 𝑁 ) ⊆ ( ℤ≥ ‘ ( 𝐾 + 1 ) ) ) |
237 |
|
nnrp |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℝ+ ) |
238 |
237
|
rpreccld |
⊢ ( 𝑘 ∈ ℕ → ( 1 / 𝑘 ) ∈ ℝ+ ) |
239 |
238
|
rpge0d |
⊢ ( 𝑘 ∈ ℕ → 0 ≤ ( 1 / 𝑘 ) ) |
240 |
|
breq2 |
⊢ ( ( 1 / 𝑘 ) = if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) → ( 0 ≤ ( 1 / 𝑘 ) ↔ 0 ≤ if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) ) |
241 |
|
breq2 |
⊢ ( 0 = if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) → ( 0 ≤ 0 ↔ 0 ≤ if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) ) |
242 |
240 241
|
ifboth |
⊢ ( ( 0 ≤ ( 1 / 𝑘 ) ∧ 0 ≤ 0 ) → 0 ≤ if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) |
243 |
239 199 242
|
sylancl |
⊢ ( 𝑘 ∈ ℕ → 0 ≤ if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) |
244 |
182 243
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐾 + 1 ) ) ) → 0 ≤ if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) |
245 |
218 96 181 236 226 187 244 231
|
isumless |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ≤ Σ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐾 + 1 ) ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) |
246 |
189 232 234 245 6
|
lelttrd |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) < ( 1 / 2 ) ) |
247 |
3
|
nngt0d |
⊢ ( 𝜑 → 0 < 𝑁 ) |
248 |
|
ltmul2 |
⊢ ( ( Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ∈ ℝ ∧ ( 1 / 2 ) ∈ ℝ ∧ ( 𝑁 ∈ ℝ ∧ 0 < 𝑁 ) ) → ( Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) < ( 1 / 2 ) ↔ ( 𝑁 · Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) < ( 𝑁 · ( 1 / 2 ) ) ) ) |
249 |
189 234 8 247 248
|
syl112anc |
⊢ ( 𝜑 → ( Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) < ( 1 / 2 ) ↔ ( 𝑁 · Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) < ( 𝑁 · ( 1 / 2 ) ) ) ) |
250 |
246 249
|
mpbid |
⊢ ( 𝜑 → ( 𝑁 · Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) < ( 𝑁 · ( 1 / 2 ) ) ) |
251 |
|
2cn |
⊢ 2 ∈ ℂ |
252 |
|
2ne0 |
⊢ 2 ≠ 0 |
253 |
|
divrec |
⊢ ( ( 𝑁 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0 ) → ( 𝑁 / 2 ) = ( 𝑁 · ( 1 / 2 ) ) ) |
254 |
251 252 253
|
mp3an23 |
⊢ ( 𝑁 ∈ ℂ → ( 𝑁 / 2 ) = ( 𝑁 · ( 1 / 2 ) ) ) |
255 |
27 254
|
syl |
⊢ ( 𝜑 → ( 𝑁 / 2 ) = ( 𝑁 · ( 1 / 2 ) ) ) |
256 |
250 255
|
breqtrrd |
⊢ ( 𝜑 → ( 𝑁 · Σ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) if ( 𝑘 ∈ ℙ , ( 1 / 𝑘 ) , 0 ) ) < ( 𝑁 / 2 ) ) |
257 |
180 190 9 217 256
|
lelttrd |
⊢ ( 𝜑 → ( ♯ ‘ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) < ( 𝑁 / 2 ) ) |
258 |
180 9 17 257
|
ltadd2dd |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝑀 ) + ( ♯ ‘ ∪ 𝑘 ∈ ( ( 𝐾 + 1 ) ... 𝑁 ) ( 𝑊 ‘ 𝑘 ) ) ) < ( ( ♯ ‘ 𝑀 ) + ( 𝑁 / 2 ) ) ) |
259 |
177 258
|
eqbrtrd |
⊢ ( 𝜑 → ( ( 𝑁 / 2 ) + ( 𝑁 / 2 ) ) < ( ( ♯ ‘ 𝑀 ) + ( 𝑁 / 2 ) ) ) |
260 |
9 17 9
|
ltadd1d |
⊢ ( 𝜑 → ( ( 𝑁 / 2 ) < ( ♯ ‘ 𝑀 ) ↔ ( ( 𝑁 / 2 ) + ( 𝑁 / 2 ) ) < ( ( ♯ ‘ 𝑀 ) + ( 𝑁 / 2 ) ) ) ) |
261 |
259 260
|
mpbird |
⊢ ( 𝜑 → ( 𝑁 / 2 ) < ( ♯ ‘ 𝑀 ) ) |
262 |
|
oveq1 |
⊢ ( 𝑘 = 𝑟 → ( 𝑘 ↑ 2 ) = ( 𝑟 ↑ 2 ) ) |
263 |
262
|
breq1d |
⊢ ( 𝑘 = 𝑟 → ( ( 𝑘 ↑ 2 ) ∥ 𝑥 ↔ ( 𝑟 ↑ 2 ) ∥ 𝑥 ) ) |
264 |
263
|
cbvrabv |
⊢ { 𝑘 ∈ ℕ ∣ ( 𝑘 ↑ 2 ) ∥ 𝑥 } = { 𝑟 ∈ ℕ ∣ ( 𝑟 ↑ 2 ) ∥ 𝑥 } |
265 |
|
breq2 |
⊢ ( 𝑥 = 𝑛 → ( ( 𝑟 ↑ 2 ) ∥ 𝑥 ↔ ( 𝑟 ↑ 2 ) ∥ 𝑛 ) ) |
266 |
265
|
rabbidv |
⊢ ( 𝑥 = 𝑛 → { 𝑟 ∈ ℕ ∣ ( 𝑟 ↑ 2 ) ∥ 𝑥 } = { 𝑟 ∈ ℕ ∣ ( 𝑟 ↑ 2 ) ∥ 𝑛 } ) |
267 |
264 266
|
syl5eq |
⊢ ( 𝑥 = 𝑛 → { 𝑘 ∈ ℕ ∣ ( 𝑘 ↑ 2 ) ∥ 𝑥 } = { 𝑟 ∈ ℕ ∣ ( 𝑟 ↑ 2 ) ∥ 𝑛 } ) |
268 |
267
|
supeq1d |
⊢ ( 𝑥 = 𝑛 → sup ( { 𝑘 ∈ ℕ ∣ ( 𝑘 ↑ 2 ) ∥ 𝑥 } , ℝ , < ) = sup ( { 𝑟 ∈ ℕ ∣ ( 𝑟 ↑ 2 ) ∥ 𝑛 } , ℝ , < ) ) |
269 |
268
|
cbvmptv |
⊢ ( 𝑥 ∈ ℕ ↦ sup ( { 𝑘 ∈ ℕ ∣ ( 𝑘 ↑ 2 ) ∥ 𝑥 } , ℝ , < ) ) = ( 𝑛 ∈ ℕ ↦ sup ( { 𝑟 ∈ ℕ ∣ ( 𝑟 ↑ 2 ) ∥ 𝑛 } , ℝ , < ) ) |
270 |
1 2 3 4 269
|
prmreclem3 |
⊢ ( 𝜑 → ( ♯ ‘ 𝑀 ) ≤ ( ( 2 ↑ 𝐾 ) · ( √ ‘ 𝑁 ) ) ) |
271 |
9 17 26 261 270
|
ltletrd |
⊢ ( 𝜑 → ( 𝑁 / 2 ) < ( ( 2 ↑ 𝐾 ) · ( √ ‘ 𝑁 ) ) ) |