Step | Hyp | Ref
| Expression |
1 | | fzfid 13432 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) →
(1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin) |
2 | | simpr 488 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) |
3 | 2 | elin2d 4089 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ) |
4 | | prmnn 16115 |
. . . . . . . . 9
⊢ (𝑝 ∈ ℙ → 𝑝 ∈
ℕ) |
5 | 3, 4 | syl 17 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ) |
6 | 5 | nnrpd 12512 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ+) |
7 | 6 | relogcld 25366 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈
ℝ) |
8 | 7 | recnd 10747 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈
ℂ) |
9 | | fsumconst 15238 |
. . . . 5
⊢
(((1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ∈ Fin ∧ (log‘𝑝) ∈ ℂ) →
Σ𝑘 ∈
(1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝) =
((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝))) |
10 | 1, 8, 9 | syl2anc 587 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → Σ𝑘 ∈
(1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝) =
((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝))) |
11 | | simpl 486 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝐴 ∈ ℝ) |
12 | | 1red 10720 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 ∈
ℝ) |
13 | 5 | nnred 11731 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ) |
14 | | prmuz2 16137 |
. . . . . . . . . . . . 13
⊢ (𝑝 ∈ ℙ → 𝑝 ∈
(ℤ≥‘2)) |
15 | 3, 14 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈
(ℤ≥‘2)) |
16 | | eluz2gt1 12402 |
. . . . . . . . . . . 12
⊢ (𝑝 ∈
(ℤ≥‘2) → 1 < 𝑝) |
17 | 15, 16 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝑝) |
18 | 2 | elin1d 4088 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ (0[,]𝐴)) |
19 | | 0re 10721 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
ℝ |
20 | | elicc2 12886 |
. . . . . . . . . . . . . 14
⊢ ((0
∈ ℝ ∧ 𝐴
∈ ℝ) → (𝑝
∈ (0[,]𝐴) ↔
(𝑝 ∈ ℝ ∧ 0
≤ 𝑝 ∧ 𝑝 ≤ 𝐴))) |
21 | 19, 11, 20 | sylancr 590 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴))) |
22 | 18, 21 | mpbid 235 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ 𝐴)) |
23 | 22 | simp3d 1145 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ≤ 𝐴) |
24 | 12, 13, 11, 17, 23 | ltletrd 10878 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 1 < 𝐴) |
25 | 11, 24 | rplogcld 25372 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝐴) ∈
ℝ+) |
26 | 13, 17 | rplogcld 25372 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → (log‘𝑝) ∈
ℝ+) |
27 | 25, 26 | rpdivcld 12531 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈
ℝ+) |
28 | 27 | rpred 12514 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝐴) / (log‘𝑝)) ∈
ℝ) |
29 | 27 | rpge0d 12518 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ≤
((log‘𝐴) /
(log‘𝑝))) |
30 | | flge0nn0 13281 |
. . . . . . 7
⊢
((((log‘𝐴) /
(log‘𝑝)) ∈
ℝ ∧ 0 ≤ ((log‘𝐴) / (log‘𝑝))) → (⌊‘((log‘𝐴) / (log‘𝑝))) ∈
ℕ0) |
31 | 28, 29, 30 | syl2anc 587 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) →
(⌊‘((log‘𝐴) / (log‘𝑝))) ∈
ℕ0) |
32 | | hashfz1 13798 |
. . . . . 6
⊢
((⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℕ0 →
(♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) = (⌊‘((log‘𝐴) / (log‘𝑝)))) |
33 | 31, 32 | syl 17 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) →
(♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) = (⌊‘((log‘𝐴) / (log‘𝑝)))) |
34 | 33 | oveq1d 7185 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) →
((♯‘(1...(⌊‘((log‘𝐴) / (log‘𝑝))))) · (log‘𝑝)) = ((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝))) |
35 | 28 | flcld 13259 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) →
(⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ) |
36 | 35 | zcnd 12169 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) →
(⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℂ) |
37 | 36, 8 | mulcomd 10740 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) →
((⌊‘((log‘𝐴) / (log‘𝑝))) · (log‘𝑝)) = ((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝))))) |
38 | 10, 34, 37 | 3eqtrrd 2778 |
. . 3
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → ((log‘𝑝) ·
(⌊‘((log‘𝐴) / (log‘𝑝)))) = Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝)) |
39 | 38 | sumeq2dv 15153 |
. 2
⊢ (𝐴 ∈ ℝ →
Σ𝑝 ∈ ((0[,]𝐴) ∩
ℙ)((log‘𝑝)
· (⌊‘((log‘𝐴) / (log‘𝑝)))) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝)) |
40 | | chpval2 25954 |
. 2
⊢ (𝐴 ∈ ℝ →
(ψ‘𝐴) =
Σ𝑝 ∈ ((0[,]𝐴) ∩
ℙ)((log‘𝑝)
· (⌊‘((log‘𝐴) / (log‘𝑝))))) |
41 | | simpl 486 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈
(1...(⌊‘𝐴)))
→ 𝐴 ∈
ℝ) |
42 | | 0red 10722 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈
(1...(⌊‘𝐴)))
→ 0 ∈ ℝ) |
43 | | 1red 10720 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈
(1...(⌊‘𝐴)))
→ 1 ∈ ℝ) |
44 | | 0lt1 11240 |
. . . . . . . . 9
⊢ 0 <
1 |
45 | 44 | a1i 11 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈
(1...(⌊‘𝐴)))
→ 0 < 1) |
46 | | elfzuz2 13003 |
. . . . . . . . 9
⊢ (𝑘 ∈
(1...(⌊‘𝐴))
→ (⌊‘𝐴)
∈ (ℤ≥‘1)) |
47 | | eluzle 12337 |
. . . . . . . . . . 11
⊢
((⌊‘𝐴)
∈ (ℤ≥‘1) → 1 ≤ (⌊‘𝐴)) |
48 | 47 | adantl 485 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧
(⌊‘𝐴) ∈
(ℤ≥‘1)) → 1 ≤ (⌊‘𝐴)) |
49 | | simpl 486 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧
(⌊‘𝐴) ∈
(ℤ≥‘1)) → 𝐴 ∈ ℝ) |
50 | | 1z 12093 |
. . . . . . . . . . 11
⊢ 1 ∈
ℤ |
51 | | flge 13266 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 1 ∈
ℤ) → (1 ≤ 𝐴
↔ 1 ≤ (⌊‘𝐴))) |
52 | 49, 50, 51 | sylancl 589 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧
(⌊‘𝐴) ∈
(ℤ≥‘1)) → (1 ≤ 𝐴 ↔ 1 ≤ (⌊‘𝐴))) |
53 | 48, 52 | mpbird 260 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧
(⌊‘𝐴) ∈
(ℤ≥‘1)) → 1 ≤ 𝐴) |
54 | 46, 53 | sylan2 596 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈
(1...(⌊‘𝐴)))
→ 1 ≤ 𝐴) |
55 | 42, 43, 41, 45, 54 | ltletrd 10878 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈
(1...(⌊‘𝐴)))
→ 0 < 𝐴) |
56 | 42, 41, 55 | ltled 10866 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈
(1...(⌊‘𝐴)))
→ 0 ≤ 𝐴) |
57 | | elfznn 13027 |
. . . . . . . 8
⊢ (𝑘 ∈
(1...(⌊‘𝐴))
→ 𝑘 ∈
ℕ) |
58 | 57 | adantl 485 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈
(1...(⌊‘𝐴)))
→ 𝑘 ∈
ℕ) |
59 | 58 | nnrecred 11767 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈
(1...(⌊‘𝐴)))
→ (1 / 𝑘) ∈
ℝ) |
60 | 41, 56, 59 | recxpcld 25466 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈
(1...(⌊‘𝐴)))
→ (𝐴↑𝑐(1 / 𝑘)) ∈
ℝ) |
61 | | chtval 25847 |
. . . . 5
⊢ ((𝐴↑𝑐(1 /
𝑘)) ∈ ℝ →
(θ‘(𝐴↑𝑐(1 / 𝑘))) = Σ𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩
ℙ)(log‘𝑝)) |
62 | 60, 61 | syl 17 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈
(1...(⌊‘𝐴)))
→ (θ‘(𝐴↑𝑐(1 / 𝑘))) = Σ𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩
ℙ)(log‘𝑝)) |
63 | 62 | sumeq2dv 15153 |
. . 3
⊢ (𝐴 ∈ ℝ →
Σ𝑘 ∈
(1...(⌊‘𝐴))(θ‘(𝐴↑𝑐(1 / 𝑘))) = Σ𝑘 ∈ (1...(⌊‘𝐴))Σ𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩
ℙ)(log‘𝑝)) |
64 | | ppifi 25843 |
. . . 4
⊢ (𝐴 ∈ ℝ →
((0[,]𝐴) ∩ ℙ)
∈ Fin) |
65 | | fzfid 13432 |
. . . 4
⊢ (𝐴 ∈ ℝ →
(1...(⌊‘𝐴))
∈ Fin) |
66 | | elinel2 4086 |
. . . . . . . 8
⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 𝑝 ∈ ℙ) |
67 | | elfznn 13027 |
. . . . . . . 8
⊢ (𝑘 ∈
(1...(⌊‘((log‘𝐴) / (log‘𝑝)))) → 𝑘 ∈ ℕ) |
68 | 66, 67 | anim12i 616 |
. . . . . . 7
⊢ ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ)) |
69 | 68 | a1i 11 |
. . . . . 6
⊢ (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → (𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ))) |
70 | | 0red 10722 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 ∈
ℝ) |
71 | | inss2 4120 |
. . . . . . . . . . . . 13
⊢
((0[,]𝐴) ∩
ℙ) ⊆ ℙ |
72 | 71 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ℝ →
((0[,]𝐴) ∩ ℙ)
⊆ ℙ) |
73 | 72 | sselda 3877 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℙ) |
74 | 73, 4 | syl 17 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℕ) |
75 | 74 | nnred 11731 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 𝑝 ∈ ℝ) |
76 | 74 | nngt0d 11765 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝑝) |
77 | 70, 75, 11, 76, 23 | ltletrd 10878 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝑝 ∈ ((0[,]𝐴) ∩ ℙ)) → 0 < 𝐴) |
78 | 77 | ex 416 |
. . . . . . 7
⊢ (𝐴 ∈ ℝ → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) → 0 < 𝐴)) |
79 | 78 | adantrd 495 |
. . . . . 6
⊢ (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → 0 < 𝐴)) |
80 | 69, 79 | jcad 516 |
. . . . 5
⊢ (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) → ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴))) |
81 | | elinel2 4086 |
. . . . . . . 8
⊢ (𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ) → 𝑝 ∈
ℙ) |
82 | 57, 81 | anim12ci 617 |
. . . . . . 7
⊢ ((𝑘 ∈
(1...(⌊‘𝐴))
∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 /
𝑘))) ∩ ℙ)) →
(𝑝 ∈ ℙ ∧
𝑘 ∈
ℕ)) |
83 | 82 | a1i 11 |
. . . . . 6
⊢ (𝐴 ∈ ℝ → ((𝑘 ∈
(1...(⌊‘𝐴))
∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 /
𝑘))) ∩ ℙ)) →
(𝑝 ∈ ℙ ∧
𝑘 ∈
ℕ))) |
84 | 55 | ex 416 |
. . . . . . 7
⊢ (𝐴 ∈ ℝ → (𝑘 ∈
(1...(⌊‘𝐴))
→ 0 < 𝐴)) |
85 | 84 | adantrd 495 |
. . . . . 6
⊢ (𝐴 ∈ ℝ → ((𝑘 ∈
(1...(⌊‘𝐴))
∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 /
𝑘))) ∩ ℙ)) →
0 < 𝐴)) |
86 | 83, 85 | jcad 516 |
. . . . 5
⊢ (𝐴 ∈ ℝ → ((𝑘 ∈
(1...(⌊‘𝐴))
∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 /
𝑘))) ∩ ℙ)) →
((𝑝 ∈ ℙ ∧
𝑘 ∈ ℕ) ∧ 0
< 𝐴))) |
87 | | elin 3859 |
. . . . . . . . 9
⊢ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ)) |
88 | | simprll 779 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑝 ∈
ℙ) |
89 | 88 | biantrud 535 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ (𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ))) |
90 | | 0red 10722 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 0 ∈
ℝ) |
91 | | simpl 486 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝐴 ∈
ℝ) |
92 | 88, 4 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑝 ∈
ℕ) |
93 | 92 | nnred 11731 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑝 ∈
ℝ) |
94 | 92 | nnnn0d 12036 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑝 ∈
ℕ0) |
95 | 94 | nn0ge0d 12039 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 0 ≤ 𝑝) |
96 | | df-3an 1090 |
. . . . . . . . . . . . 13
⊢ ((𝑝 ∈ ℝ ∧ 0 ≤
𝑝 ∧ 𝑝 ≤ 𝐴) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ 𝐴)) |
97 | 20, 96 | bitrdi 290 |
. . . . . . . . . . . 12
⊢ ((0
∈ ℝ ∧ 𝐴
∈ ℝ) → (𝑝
∈ (0[,]𝐴) ↔
((𝑝 ∈ ℝ ∧ 0
≤ 𝑝) ∧ 𝑝 ≤ 𝐴))) |
98 | 97 | baibd 543 |
. . . . . . . . . . 11
⊢ (((0
∈ ℝ ∧ 𝐴
∈ ℝ) ∧ (𝑝
∈ ℝ ∧ 0 ≤ 𝑝)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝 ≤ 𝐴)) |
99 | 90, 91, 93, 95, 98 | syl22anc 838 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑝 ∈ (0[,]𝐴) ↔ 𝑝 ≤ 𝐴)) |
100 | 89, 99 | bitr3d 284 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑝 ∈ (0[,]𝐴) ∧ 𝑝 ∈ ℙ) ↔ 𝑝 ≤ 𝐴)) |
101 | 87, 100 | syl5bb 286 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ↔ 𝑝 ≤ 𝐴)) |
102 | | simprr 773 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 0 < 𝐴) |
103 | 91, 102 | elrpd 12511 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝐴 ∈
ℝ+) |
104 | 103 | relogcld 25366 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (log‘𝐴) ∈
ℝ) |
105 | 88, 14 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑝 ∈
(ℤ≥‘2)) |
106 | 105, 16 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 1 < 𝑝) |
107 | 93, 106 | rplogcld 25372 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (log‘𝑝) ∈
ℝ+) |
108 | 104, 107 | rerpdivcld 12545 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) →
((log‘𝐴) /
(log‘𝑝)) ∈
ℝ) |
109 | | simprlr 780 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑘 ∈
ℕ) |
110 | 109 | nnzd 12167 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑘 ∈
ℤ) |
111 | | flge 13266 |
. . . . . . . . . 10
⊢
((((log‘𝐴) /
(log‘𝑝)) ∈
ℝ ∧ 𝑘 ∈
ℤ) → (𝑘 ≤
((log‘𝐴) /
(log‘𝑝)) ↔ 𝑘 ≤
(⌊‘((log‘𝐴) / (log‘𝑝))))) |
112 | 108, 110,
111 | syl2anc 587 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑘 ≤ ((log‘𝐴) / (log‘𝑝)) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝))))) |
113 | 109 | nnnn0d 12036 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑘 ∈
ℕ0) |
114 | 92, 113 | nnexpcld 13698 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑝↑𝑘) ∈ ℕ) |
115 | 114 | nnrpd 12512 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑝↑𝑘) ∈
ℝ+) |
116 | 115, 103 | logled 25370 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (log‘(𝑝↑𝑘)) ≤ (log‘𝐴))) |
117 | 92 | nnrpd 12512 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑝 ∈
ℝ+) |
118 | | relogexp 25339 |
. . . . . . . . . . . 12
⊢ ((𝑝 ∈ ℝ+
∧ 𝑘 ∈ ℤ)
→ (log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝))) |
119 | 117, 110,
118 | syl2anc 587 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) →
(log‘(𝑝↑𝑘)) = (𝑘 · (log‘𝑝))) |
120 | 119 | breq1d 5040 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) →
((log‘(𝑝↑𝑘)) ≤ (log‘𝐴) ↔ (𝑘 · (log‘𝑝)) ≤ (log‘𝐴))) |
121 | 109 | nnred 11731 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑘 ∈
ℝ) |
122 | 121, 104,
107 | lemuldivd 12563 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑘 · (log‘𝑝)) ≤ (log‘𝐴) ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝)))) |
123 | 116, 120,
122 | 3bitrd 308 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ 𝑘 ≤ ((log‘𝐴) / (log‘𝑝)))) |
124 | | nnuz 12363 |
. . . . . . . . . . 11
⊢ ℕ =
(ℤ≥‘1) |
125 | 109, 124 | eleqtrdi 2843 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑘 ∈
(ℤ≥‘1)) |
126 | 108 | flcld 13259 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) →
(⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ) |
127 | | elfz5 12990 |
. . . . . . . . . 10
⊢ ((𝑘 ∈
(ℤ≥‘1) ∧ (⌊‘((log‘𝐴) / (log‘𝑝))) ∈ ℤ) →
(𝑘 ∈
(1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝))))) |
128 | 125, 126,
127 | syl2anc 587 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑘 ∈
(1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ 𝑘 ≤ (⌊‘((log‘𝐴) / (log‘𝑝))))) |
129 | 112, 123,
128 | 3bitr4rd 315 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑘 ∈
(1...(⌊‘((log‘𝐴) / (log‘𝑝)))) ↔ (𝑝↑𝑘) ≤ 𝐴)) |
130 | 101, 129 | anbi12d 634 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑝 ≤ 𝐴 ∧ (𝑝↑𝑘) ≤ 𝐴))) |
131 | 91 | flcld 13259 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) →
(⌊‘𝐴) ∈
ℤ) |
132 | | elfz5 12990 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈
(ℤ≥‘1) ∧ (⌊‘𝐴) ∈ ℤ) → (𝑘 ∈ (1...(⌊‘𝐴)) ↔ 𝑘 ≤ (⌊‘𝐴))) |
133 | 125, 131,
132 | syl2anc 587 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑘 ∈
(1...(⌊‘𝐴))
↔ 𝑘 ≤
(⌊‘𝐴))) |
134 | | flge 13266 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℤ) → (𝑘 ≤ 𝐴 ↔ 𝑘 ≤ (⌊‘𝐴))) |
135 | 91, 110, 134 | syl2anc 587 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑘 ≤ 𝐴 ↔ 𝑘 ≤ (⌊‘𝐴))) |
136 | 133, 135 | bitr4d 285 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑘 ∈
(1...(⌊‘𝐴))
↔ 𝑘 ≤ 𝐴)) |
137 | | elin 3859 |
. . . . . . . . . 10
⊢ (𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ)) |
138 | 88 | biantrud 535 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ↔ (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ))) |
139 | 103 | rpge0d 12518 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 0 ≤ 𝐴) |
140 | 109 | nnrecred 11767 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (1 / 𝑘) ∈
ℝ) |
141 | 91, 139, 140 | recxpcld 25466 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝐴↑𝑐(1 /
𝑘)) ∈
ℝ) |
142 | | elicc2 12886 |
. . . . . . . . . . . . . . 15
⊢ ((0
∈ ℝ ∧ (𝐴↑𝑐(1 / 𝑘)) ∈ ℝ) → (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ↔ (𝑝 ∈ ℝ ∧ 0 ≤ 𝑝 ∧ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))))) |
143 | | df-3an 1090 |
. . . . . . . . . . . . . . 15
⊢ ((𝑝 ∈ ℝ ∧ 0 ≤
𝑝 ∧ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘)))) |
144 | 142, 143 | bitrdi 290 |
. . . . . . . . . . . . . 14
⊢ ((0
∈ ℝ ∧ (𝐴↑𝑐(1 / 𝑘)) ∈ ℝ) → (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ↔ ((𝑝 ∈ ℝ ∧ 0 ≤ 𝑝) ∧ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘))))) |
145 | 144 | baibd 543 |
. . . . . . . . . . . . 13
⊢ (((0
∈ ℝ ∧ (𝐴↑𝑐(1 / 𝑘)) ∈ ℝ) ∧ (𝑝 ∈ ℝ ∧ 0 ≤
𝑝)) → (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ↔ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘)))) |
146 | 90, 141, 93, 95, 145 | syl22anc 838 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ↔ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘)))) |
147 | 138, 146 | bitr3d 284 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ) ↔ 𝑝 ≤ (𝐴↑𝑐(1 / 𝑘)))) |
148 | 91, 139, 140 | cxpge0d 25467 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 0 ≤ (𝐴↑𝑐(1 /
𝑘))) |
149 | 109 | nnrpd 12512 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑘 ∈
ℝ+) |
150 | 93, 95, 141, 148, 149 | cxple2d 25470 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑝 ≤ (𝐴↑𝑐(1 / 𝑘)) ↔ (𝑝↑𝑐𝑘) ≤ ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘))) |
151 | 92 | nncnd 11732 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑝 ∈
ℂ) |
152 | | cxpexp 25411 |
. . . . . . . . . . . . 13
⊢ ((𝑝 ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ (𝑝↑𝑐𝑘) = (𝑝↑𝑘)) |
153 | 151, 113,
152 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑝↑𝑐𝑘) = (𝑝↑𝑘)) |
154 | 109 | nncnd 11732 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑘 ∈
ℂ) |
155 | 109 | nnne0d 11766 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑘 ≠ 0) |
156 | 154, 155 | recid2d 11490 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((1 / 𝑘) · 𝑘) = 1) |
157 | 156 | oveq2d 7186 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝐴↑𝑐((1 /
𝑘) · 𝑘)) = (𝐴↑𝑐1)) |
158 | 103, 140,
154 | cxpmuld 25479 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝐴↑𝑐((1 /
𝑘) · 𝑘)) = ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘)) |
159 | 91 | recnd 10747 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝐴 ∈
ℂ) |
160 | 159 | cxp1d 25449 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝐴↑𝑐1) =
𝐴) |
161 | 157, 158,
160 | 3eqtr3d 2781 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝐴↑𝑐(1 /
𝑘))↑𝑐𝑘) = 𝐴) |
162 | 153, 161 | breq12d 5043 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑝↑𝑐𝑘) ≤ ((𝐴↑𝑐(1 / 𝑘))↑𝑐𝑘) ↔ (𝑝↑𝑘) ≤ 𝐴)) |
163 | 147, 150,
162 | 3bitrd 308 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑝 ∈ (0[,](𝐴↑𝑐(1 / 𝑘))) ∧ 𝑝 ∈ ℙ) ↔ (𝑝↑𝑘) ≤ 𝐴)) |
164 | 137, 163 | syl5bb 286 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩ ℙ) ↔ (𝑝↑𝑘) ≤ 𝐴)) |
165 | 136, 164 | anbi12d 634 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑘 ∈
(1...(⌊‘𝐴))
∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 /
𝑘))) ∩ ℙ)) ↔
(𝑘 ≤ 𝐴 ∧ (𝑝↑𝑘) ≤ 𝐴))) |
166 | 114 | nnred 11731 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑝↑𝑘) ∈ ℝ) |
167 | | bernneq3 13684 |
. . . . . . . . . . . 12
⊢ ((𝑝 ∈
(ℤ≥‘2) ∧ 𝑘 ∈ ℕ0) → 𝑘 < (𝑝↑𝑘)) |
168 | 105, 113,
167 | syl2anc 587 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑘 < (𝑝↑𝑘)) |
169 | 121, 166,
168 | ltled 10866 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑘 ≤ (𝑝↑𝑘)) |
170 | | letr 10812 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ ℝ ∧ (𝑝↑𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑘 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑘 ≤ 𝐴)) |
171 | 121, 166,
91, 170 | syl3anc 1372 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑘 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑘 ≤ 𝐴)) |
172 | 169, 171 | mpand 695 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 → 𝑘 ≤ 𝐴)) |
173 | 172 | pm4.71rd 566 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑘 ≤ 𝐴 ∧ (𝑝↑𝑘) ≤ 𝐴))) |
174 | 151 | exp1d 13597 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑝↑1) = 𝑝) |
175 | 92 | nnge1d 11764 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 1 ≤ 𝑝) |
176 | 93, 175, 125 | leexp2ad 13709 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → (𝑝↑1) ≤ (𝑝↑𝑘)) |
177 | 174, 176 | eqbrtrrd 5054 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → 𝑝 ≤ (𝑝↑𝑘)) |
178 | | letr 10812 |
. . . . . . . . . . 11
⊢ ((𝑝 ∈ ℝ ∧ (𝑝↑𝑘) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑝 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑝 ≤ 𝐴)) |
179 | 93, 166, 91, 178 | syl3anc 1372 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑝 ≤ (𝑝↑𝑘) ∧ (𝑝↑𝑘) ≤ 𝐴) → 𝑝 ≤ 𝐴)) |
180 | 177, 179 | mpand 695 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 → 𝑝 ≤ 𝐴)) |
181 | 180 | pm4.71rd 566 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑝↑𝑘) ≤ 𝐴 ↔ (𝑝 ≤ 𝐴 ∧ (𝑝↑𝑘) ≤ 𝐴))) |
182 | 165, 173,
181 | 3bitr2rd 311 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑝 ≤ 𝐴 ∧ (𝑝↑𝑘) ≤ 𝐴) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩
ℙ)))) |
183 | 130, 182 | bitrd 282 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ ((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴)) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩
ℙ)))) |
184 | 183 | ex 416 |
. . . . 5
⊢ (𝐴 ∈ ℝ → (((𝑝 ∈ ℙ ∧ 𝑘 ∈ ℕ) ∧ 0 <
𝐴) → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩
ℙ))))) |
185 | 80, 86, 184 | pm5.21ndd 384 |
. . . 4
⊢ (𝐴 ∈ ℝ → ((𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))) ↔ (𝑘 ∈ (1...(⌊‘𝐴)) ∧ 𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩
ℙ)))) |
186 | 8 | adantrr 717 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ (𝑝 ∈ ((0[,]𝐴) ∩ ℙ) ∧ 𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝)))))) → (log‘𝑝) ∈
ℂ) |
187 | 64, 65, 1, 185, 186 | fsumcom2 15222 |
. . 3
⊢ (𝐴 ∈ ℝ →
Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈
(1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝) = Σ𝑘 ∈ (1...(⌊‘𝐴))Σ𝑝 ∈ ((0[,](𝐴↑𝑐(1 / 𝑘))) ∩
ℙ)(log‘𝑝)) |
188 | 63, 187 | eqtr4d 2776 |
. 2
⊢ (𝐴 ∈ ℝ →
Σ𝑘 ∈
(1...(⌊‘𝐴))(θ‘(𝐴↑𝑐(1 / 𝑘))) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)Σ𝑘 ∈ (1...(⌊‘((log‘𝐴) / (log‘𝑝))))(log‘𝑝)) |
189 | 39, 40, 188 | 3eqtr4d 2783 |
1
⊢ (𝐴 ∈ ℝ →
(ψ‘𝐴) =
Σ𝑘 ∈
(1...(⌊‘𝐴))(θ‘(𝐴↑𝑐(1 / 𝑘)))) |