Step | Hyp | Ref
| Expression |
1 | | ostth2.4 |
. . . . 5
⊢ 𝑅 = ((log‘(𝐹‘𝑁)) / (log‘𝑁)) |
2 | | ostth.1 |
. . . . . . . 8
⊢ (𝜑 → 𝐹 ∈ 𝐴) |
3 | | ostth2.2 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘2)) |
4 | | eluz2b2 12396 |
. . . . . . . . . . 11
⊢ (𝑁 ∈
(ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁)) |
5 | 3, 4 | sylib 221 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 1 < 𝑁)) |
6 | 5 | simpld 498 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ ℕ) |
7 | | nnq 12437 |
. . . . . . . . 9
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℚ) |
8 | 6, 7 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℚ) |
9 | | qabsabv.a |
. . . . . . . . 9
⊢ 𝐴 = (AbsVal‘𝑄) |
10 | | qrng.q |
. . . . . . . . . 10
⊢ 𝑄 = (ℂfld
↾s ℚ) |
11 | 10 | qrngbas 26347 |
. . . . . . . . 9
⊢ ℚ =
(Base‘𝑄) |
12 | 9, 11 | abvcl 19707 |
. . . . . . . 8
⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℚ) → (𝐹‘𝑁) ∈ ℝ) |
13 | 2, 8, 12 | syl2anc 587 |
. . . . . . 7
⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ) |
14 | | ostth2.3 |
. . . . . . 7
⊢ (𝜑 → 1 < (𝐹‘𝑁)) |
15 | 13, 14 | rplogcld 25364 |
. . . . . 6
⊢ (𝜑 → (log‘(𝐹‘𝑁)) ∈
ℝ+) |
16 | 6 | nnred 11724 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℝ) |
17 | 5 | simprd 499 |
. . . . . . 7
⊢ (𝜑 → 1 < 𝑁) |
18 | 16, 17 | rplogcld 25364 |
. . . . . 6
⊢ (𝜑 → (log‘𝑁) ∈
ℝ+) |
19 | 15, 18 | rpdivcld 12524 |
. . . . 5
⊢ (𝜑 → ((log‘(𝐹‘𝑁)) / (log‘𝑁)) ∈
ℝ+) |
20 | 1, 19 | eqeltrid 2837 |
. . . 4
⊢ (𝜑 → 𝑅 ∈
ℝ+) |
21 | 20 | rpred 12507 |
. . 3
⊢ (𝜑 → 𝑅 ∈ ℝ) |
22 | 20 | rpgt0d 12510 |
. . 3
⊢ (𝜑 → 0 < 𝑅) |
23 | 6 | nnnn0d 12029 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
24 | 10, 9 | qabvle 26353 |
. . . . . . . . 9
⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝐹‘𝑁) ≤ 𝑁) |
25 | 2, 23, 24 | syl2anc 587 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘𝑁) ≤ 𝑁) |
26 | 6 | nnne0d 11759 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ≠ 0) |
27 | 10 | qrng0 26349 |
. . . . . . . . . . . 12
⊢ 0 =
(0g‘𝑄) |
28 | 9, 11, 27 | abvgt0 19711 |
. . . . . . . . . . 11
⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℚ ∧ 𝑁 ≠ 0) → 0 < (𝐹‘𝑁)) |
29 | 2, 8, 26, 28 | syl3anc 1372 |
. . . . . . . . . 10
⊢ (𝜑 → 0 < (𝐹‘𝑁)) |
30 | 13, 29 | elrpd 12504 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘𝑁) ∈
ℝ+) |
31 | 30 | reeflogd 25359 |
. . . . . . . 8
⊢ (𝜑 →
(exp‘(log‘(𝐹‘𝑁))) = (𝐹‘𝑁)) |
32 | 6 | nnrpd 12505 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈
ℝ+) |
33 | 32 | reeflogd 25359 |
. . . . . . . 8
⊢ (𝜑 →
(exp‘(log‘𝑁)) =
𝑁) |
34 | 25, 31, 33 | 3brtr4d 5059 |
. . . . . . 7
⊢ (𝜑 →
(exp‘(log‘(𝐹‘𝑁))) ≤ (exp‘(log‘𝑁))) |
35 | 15 | rpred 12507 |
. . . . . . . 8
⊢ (𝜑 → (log‘(𝐹‘𝑁)) ∈ ℝ) |
36 | 32 | relogcld 25358 |
. . . . . . . 8
⊢ (𝜑 → (log‘𝑁) ∈
ℝ) |
37 | | efle 15556 |
. . . . . . . 8
⊢
(((log‘(𝐹‘𝑁)) ∈ ℝ ∧ (log‘𝑁) ∈ ℝ) →
((log‘(𝐹‘𝑁)) ≤ (log‘𝑁) ↔
(exp‘(log‘(𝐹‘𝑁))) ≤ (exp‘(log‘𝑁)))) |
38 | 35, 36, 37 | syl2anc 587 |
. . . . . . 7
⊢ (𝜑 → ((log‘(𝐹‘𝑁)) ≤ (log‘𝑁) ↔ (exp‘(log‘(𝐹‘𝑁))) ≤ (exp‘(log‘𝑁)))) |
39 | 34, 38 | mpbird 260 |
. . . . . 6
⊢ (𝜑 → (log‘(𝐹‘𝑁)) ≤ (log‘𝑁)) |
40 | 18 | rpcnd 12509 |
. . . . . . 7
⊢ (𝜑 → (log‘𝑁) ∈
ℂ) |
41 | 40 | mulid1d 10729 |
. . . . . 6
⊢ (𝜑 → ((log‘𝑁) · 1) = (log‘𝑁)) |
42 | 39, 41 | breqtrrd 5055 |
. . . . 5
⊢ (𝜑 → (log‘(𝐹‘𝑁)) ≤ ((log‘𝑁) · 1)) |
43 | | 1red 10713 |
. . . . . 6
⊢ (𝜑 → 1 ∈
ℝ) |
44 | 35, 43, 18 | ledivmuld 12560 |
. . . . 5
⊢ (𝜑 → (((log‘(𝐹‘𝑁)) / (log‘𝑁)) ≤ 1 ↔ (log‘(𝐹‘𝑁)) ≤ ((log‘𝑁) · 1))) |
45 | 42, 44 | mpbird 260 |
. . . 4
⊢ (𝜑 → ((log‘(𝐹‘𝑁)) / (log‘𝑁)) ≤ 1) |
46 | 1, 45 | eqbrtrid 5062 |
. . 3
⊢ (𝜑 → 𝑅 ≤ 1) |
47 | | 0xr 10759 |
. . . 4
⊢ 0 ∈
ℝ* |
48 | | 1re 10712 |
. . . 4
⊢ 1 ∈
ℝ |
49 | | elioc2 12877 |
. . . 4
⊢ ((0
∈ ℝ* ∧ 1 ∈ ℝ) → (𝑅 ∈ (0(,]1) ↔ (𝑅 ∈ ℝ ∧ 0 < 𝑅 ∧ 𝑅 ≤ 1))) |
50 | 47, 48, 49 | mp2an 692 |
. . 3
⊢ (𝑅 ∈ (0(,]1) ↔ (𝑅 ∈ ℝ ∧ 0 <
𝑅 ∧ 𝑅 ≤ 1)) |
51 | 21, 22, 46, 50 | syl3anbrc 1344 |
. 2
⊢ (𝜑 → 𝑅 ∈ (0(,]1)) |
52 | 10, 9 | qabsabv 26357 |
. . . 4
⊢ (abs
↾ ℚ) ∈ 𝐴 |
53 | | fvres 6687 |
. . . . . . . 8
⊢ (𝑦 ∈ ℚ → ((abs
↾ ℚ)‘𝑦) =
(abs‘𝑦)) |
54 | 53 | oveq1d 7179 |
. . . . . . 7
⊢ (𝑦 ∈ ℚ → (((abs
↾ ℚ)‘𝑦)↑𝑐𝑅) = ((abs‘𝑦)↑𝑐𝑅)) |
55 | 54 | mpteq2ia 5118 |
. . . . . 6
⊢ (𝑦 ∈ ℚ ↦ (((abs
↾ ℚ)‘𝑦)↑𝑐𝑅)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑𝑐𝑅)) |
56 | 55 | eqcomi 2747 |
. . . . 5
⊢ (𝑦 ∈ ℚ ↦
((abs‘𝑦)↑𝑐𝑅)) = (𝑦 ∈ ℚ ↦ (((abs ↾
ℚ)‘𝑦)↑𝑐𝑅)) |
57 | 9, 11, 56 | abvcxp 26343 |
. . . 4
⊢ (((abs
↾ ℚ) ∈ 𝐴
∧ 𝑅 ∈ (0(,]1))
→ (𝑦 ∈ ℚ
↦ ((abs‘𝑦)↑𝑐𝑅)) ∈ 𝐴) |
58 | 52, 51, 57 | sylancr 590 |
. . 3
⊢ (𝜑 → (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑𝑐𝑅)) ∈ 𝐴) |
59 | | eluzelz 12327 |
. . . . . 6
⊢ (𝑧 ∈
(ℤ≥‘2) → 𝑧 ∈ ℤ) |
60 | | zq 12429 |
. . . . . 6
⊢ (𝑧 ∈ ℤ → 𝑧 ∈
ℚ) |
61 | | fveq2 6668 |
. . . . . . . 8
⊢ (𝑦 = 𝑧 → (abs‘𝑦) = (abs‘𝑧)) |
62 | 61 | oveq1d 7179 |
. . . . . . 7
⊢ (𝑦 = 𝑧 → ((abs‘𝑦)↑𝑐𝑅) = ((abs‘𝑧)↑𝑐𝑅)) |
63 | | eqid 2738 |
. . . . . . 7
⊢ (𝑦 ∈ ℚ ↦
((abs‘𝑦)↑𝑐𝑅)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑𝑐𝑅)) |
64 | | ovex 7197 |
. . . . . . 7
⊢
((abs‘𝑧)↑𝑐𝑅) ∈ V |
65 | 62, 63, 64 | fvmpt 6769 |
. . . . . 6
⊢ (𝑧 ∈ ℚ → ((𝑦 ∈ ℚ ↦
((abs‘𝑦)↑𝑐𝑅))‘𝑧) = ((abs‘𝑧)↑𝑐𝑅)) |
66 | 59, 60, 65 | 3syl 18 |
. . . . 5
⊢ (𝑧 ∈
(ℤ≥‘2) → ((𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑𝑐𝑅))‘𝑧) = ((abs‘𝑧)↑𝑐𝑅)) |
67 | 66 | adantl 485 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ ((𝑦 ∈ ℚ
↦ ((abs‘𝑦)↑𝑐𝑅))‘𝑧) = ((abs‘𝑧)↑𝑐𝑅)) |
68 | | simpr 488 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 𝑧 ∈
(ℤ≥‘2)) |
69 | | eluz2b2 12396 |
. . . . . . . . 9
⊢ (𝑧 ∈
(ℤ≥‘2) ↔ (𝑧 ∈ ℕ ∧ 1 < 𝑧)) |
70 | 68, 69 | sylib 221 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (𝑧 ∈ ℕ
∧ 1 < 𝑧)) |
71 | 70 | simpld 498 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 𝑧 ∈
ℕ) |
72 | 71 | nnred 11724 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 𝑧 ∈
ℝ) |
73 | 71 | nnnn0d 12029 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 𝑧 ∈
ℕ0) |
74 | 73 | nn0ge0d 12032 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 0 ≤ 𝑧) |
75 | 72, 74 | absidd 14865 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (abs‘𝑧) =
𝑧) |
76 | 75 | oveq1d 7179 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ ((abs‘𝑧)↑𝑐𝑅) = (𝑧↑𝑐𝑅)) |
77 | 72 | recnd 10740 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 𝑧 ∈
ℂ) |
78 | 71 | nnne0d 11759 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 𝑧 ≠
0) |
79 | 20 | rpcnd 12509 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ ℂ) |
80 | 79 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 𝑅 ∈
ℂ) |
81 | 77, 78, 80 | cxpefd 25447 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (𝑧↑𝑐𝑅) = (exp‘(𝑅 · (log‘𝑧)))) |
82 | | padic.j |
. . . . . . . . . . 11
⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥))))) |
83 | | ostth.k |
. . . . . . . . . . 11
⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1)) |
84 | 2 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 𝐹 ∈ 𝐴) |
85 | 3 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 𝑁 ∈
(ℤ≥‘2)) |
86 | 14 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 1 < (𝐹‘𝑁)) |
87 | | eqid 2738 |
. . . . . . . . . . 11
⊢
((log‘(𝐹‘𝑧)) / (log‘𝑧)) = ((log‘(𝐹‘𝑧)) / (log‘𝑧)) |
88 | | eqid 2738 |
. . . . . . . . . . 11
⊢ if((𝐹‘𝑧) ≤ 1, 1, (𝐹‘𝑧)) = if((𝐹‘𝑧) ≤ 1, 1, (𝐹‘𝑧)) |
89 | | eqid 2738 |
. . . . . . . . . . 11
⊢
((log‘𝑁) /
(log‘𝑧)) =
((log‘𝑁) /
(log‘𝑧)) |
90 | 10, 9, 82, 83, 84, 85, 86, 1, 68, 87, 88, 89 | ostth2lem4 26364 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (1 < (𝐹‘𝑧) ∧ 𝑅 ≤ ((log‘(𝐹‘𝑧)) / (log‘𝑧)))) |
91 | 90 | simprd 499 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 𝑅 ≤
((log‘(𝐹‘𝑧)) / (log‘𝑧))) |
92 | 90 | simpld 498 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 1 < (𝐹‘𝑧)) |
93 | | eqid 2738 |
. . . . . . . . . . 11
⊢ if((𝐹‘𝑁) ≤ 1, 1, (𝐹‘𝑁)) = if((𝐹‘𝑁) ≤ 1, 1, (𝐹‘𝑁)) |
94 | | eqid 2738 |
. . . . . . . . . . 11
⊢
((log‘𝑧) /
(log‘𝑁)) =
((log‘𝑧) /
(log‘𝑁)) |
95 | 10, 9, 82, 83, 84, 68, 92, 87, 85, 1, 93, 94 | ostth2lem4 26364 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (1 < (𝐹‘𝑁) ∧ ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ≤ 𝑅)) |
96 | 95 | simprd 499 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ≤ 𝑅) |
97 | 21 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 𝑅 ∈
ℝ) |
98 | 59 | adantl 485 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 𝑧 ∈
ℤ) |
99 | 98, 60 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 𝑧 ∈
ℚ) |
100 | 9, 11 | abvcl 19707 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ 𝐴 ∧ 𝑧 ∈ ℚ) → (𝐹‘𝑧) ∈ ℝ) |
101 | 84, 99, 100 | syl2anc 587 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (𝐹‘𝑧) ∈
ℝ) |
102 | 9, 11, 27 | abvgt0 19711 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ 𝐴 ∧ 𝑧 ∈ ℚ ∧ 𝑧 ≠ 0) → 0 < (𝐹‘𝑧)) |
103 | 84, 99, 78, 102 | syl3anc 1372 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 0 < (𝐹‘𝑧)) |
104 | 101, 103 | elrpd 12504 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (𝐹‘𝑧) ∈
ℝ+) |
105 | 104 | relogcld 25358 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (log‘(𝐹‘𝑧)) ∈ ℝ) |
106 | 71 | nnrpd 12505 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 𝑧 ∈
ℝ+) |
107 | 106 | relogcld 25358 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (log‘𝑧) ∈
ℝ) |
108 | | ef0 15529 |
. . . . . . . . . . . . . 14
⊢
(exp‘0) = 1 |
109 | 70 | simprd 499 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 1 < 𝑧) |
110 | 106 | reeflogd 25359 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (exp‘(log‘𝑧)) = 𝑧) |
111 | 109, 110 | breqtrrd 5055 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 1 < (exp‘(log‘𝑧))) |
112 | 108, 111 | eqbrtrid 5062 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (exp‘0) < (exp‘(log‘𝑧))) |
113 | | 0re 10714 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
ℝ |
114 | | eflt 15555 |
. . . . . . . . . . . . . 14
⊢ ((0
∈ ℝ ∧ (log‘𝑧) ∈ ℝ) → (0 <
(log‘𝑧) ↔
(exp‘0) < (exp‘(log‘𝑧)))) |
115 | 113, 107,
114 | sylancr 590 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (0 < (log‘𝑧) ↔ (exp‘0) <
(exp‘(log‘𝑧)))) |
116 | 112, 115 | mpbird 260 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 0 < (log‘𝑧)) |
117 | 116 | gt0ne0d 11275 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (log‘𝑧) ≠
0) |
118 | 105, 107,
117 | redivcld 11539 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ∈ ℝ) |
119 | 97, 118 | letri3d 10853 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (𝑅 =
((log‘(𝐹‘𝑧)) / (log‘𝑧)) ↔ (𝑅 ≤ ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ∧ ((log‘(𝐹‘𝑧)) / (log‘𝑧)) ≤ 𝑅))) |
120 | 91, 96, 119 | mpbir2and 713 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ 𝑅 =
((log‘(𝐹‘𝑧)) / (log‘𝑧))) |
121 | 120 | oveq1d 7179 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (𝑅 ·
(log‘𝑧)) =
(((log‘(𝐹‘𝑧)) / (log‘𝑧)) · (log‘𝑧))) |
122 | 105 | recnd 10740 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (log‘(𝐹‘𝑧)) ∈ ℂ) |
123 | 107 | recnd 10740 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (log‘𝑧) ∈
ℂ) |
124 | 122, 123,
117 | divcan1d 11488 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (((log‘(𝐹‘𝑧)) / (log‘𝑧)) · (log‘𝑧)) = (log‘(𝐹‘𝑧))) |
125 | 121, 124 | eqtrd 2773 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (𝑅 ·
(log‘𝑧)) =
(log‘(𝐹‘𝑧))) |
126 | 125 | fveq2d 6672 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (exp‘(𝑅
· (log‘𝑧))) =
(exp‘(log‘(𝐹‘𝑧)))) |
127 | 104 | reeflogd 25359 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (exp‘(log‘(𝐹‘𝑧))) = (𝐹‘𝑧)) |
128 | 81, 126, 127 | 3eqtrd 2777 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (𝑧↑𝑐𝑅) = (𝐹‘𝑧)) |
129 | 67, 76, 128 | 3eqtrrd 2778 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ (ℤ≥‘2))
→ (𝐹‘𝑧) = ((𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑𝑐𝑅))‘𝑧)) |
130 | 10, 9, 2, 58, 129 | ostthlem1 26355 |
. 2
⊢ (𝜑 → 𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑𝑐𝑅))) |
131 | | oveq2 7172 |
. . . 4
⊢ (𝑎 = 𝑅 → ((abs‘𝑦)↑𝑐𝑎) = ((abs‘𝑦)↑𝑐𝑅)) |
132 | 131 | mpteq2dv 5123 |
. . 3
⊢ (𝑎 = 𝑅 → (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑𝑐𝑎)) = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑𝑐𝑅))) |
133 | 132 | rspceeqv 3539 |
. 2
⊢ ((𝑅 ∈ (0(,]1) ∧ 𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑𝑐𝑅))) → ∃𝑎 ∈ (0(,]1)𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑𝑐𝑎))) |
134 | 51, 130, 133 | syl2anc 587 |
1
⊢ (𝜑 → ∃𝑎 ∈ (0(,]1)𝐹 = (𝑦 ∈ ℚ ↦ ((abs‘𝑦)↑𝑐𝑎))) |