Step | Hyp | Ref
| Expression |
1 | | ostth2.3 |
. . . . 5
⊢ (𝜑 → 1 < (𝐹‘𝑁)) |
2 | | 1re 10976 |
. . . . . 6
⊢ 1 ∈
ℝ |
3 | | ostth.1 |
. . . . . . 7
⊢ (𝜑 → 𝐹 ∈ 𝐴) |
4 | | ostth2.2 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈
(ℤ≥‘2)) |
5 | | eluz2b2 12660 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘2) ↔ (𝑁 ∈ ℕ ∧ 1 < 𝑁)) |
6 | 4, 5 | sylib 217 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 1 < 𝑁)) |
7 | 6 | simpld 495 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℕ) |
8 | | nnq 12701 |
. . . . . . . 8
⊢ (𝑁 ∈ ℕ → 𝑁 ∈
ℚ) |
9 | 7, 8 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℚ) |
10 | | qabsabv.a |
. . . . . . . 8
⊢ 𝐴 = (AbsVal‘𝑄) |
11 | | qrng.q |
. . . . . . . . 9
⊢ 𝑄 = (ℂfld
↾s ℚ) |
12 | 11 | qrngbas 26765 |
. . . . . . . 8
⊢ ℚ =
(Base‘𝑄) |
13 | 10, 12 | abvcl 20082 |
. . . . . . 7
⊢ ((𝐹 ∈ 𝐴 ∧ 𝑁 ∈ ℚ) → (𝐹‘𝑁) ∈ ℝ) |
14 | 3, 9, 13 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (𝐹‘𝑁) ∈ ℝ) |
15 | | ltnle 11055 |
. . . . . 6
⊢ ((1
∈ ℝ ∧ (𝐹‘𝑁) ∈ ℝ) → (1 < (𝐹‘𝑁) ↔ ¬ (𝐹‘𝑁) ≤ 1)) |
16 | 2, 14, 15 | sylancr 587 |
. . . . 5
⊢ (𝜑 → (1 < (𝐹‘𝑁) ↔ ¬ (𝐹‘𝑁) ≤ 1)) |
17 | 1, 16 | mpbid 231 |
. . . 4
⊢ (𝜑 → ¬ (𝐹‘𝑁) ≤ 1) |
18 | | ostth2.7 |
. . . . . . . . . . . . 13
⊢ 𝑇 = if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)) |
19 | | ostth2.5 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑀 ∈
(ℤ≥‘2)) |
20 | | eluz2b2 12660 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑀 ∈
(ℤ≥‘2) ↔ (𝑀 ∈ ℕ ∧ 1 < 𝑀)) |
21 | 19, 20 | sylib 217 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 1 < 𝑀)) |
22 | 21 | simpld 495 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 ∈ ℕ) |
23 | | nnq 12701 |
. . . . . . . . . . . . . . . 16
⊢ (𝑀 ∈ ℕ → 𝑀 ∈
ℚ) |
24 | 22, 23 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 ∈ ℚ) |
25 | 10, 12 | abvcl 20082 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹 ∈ 𝐴 ∧ 𝑀 ∈ ℚ) → (𝐹‘𝑀) ∈ ℝ) |
26 | 3, 24, 25 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐹‘𝑀) ∈ ℝ) |
27 | | ifcl 4510 |
. . . . . . . . . . . . . 14
⊢ ((1
∈ ℝ ∧ (𝐹‘𝑀) ∈ ℝ) → if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)) ∈ ℝ) |
28 | 2, 26, 27 | sylancr 587 |
. . . . . . . . . . . . 13
⊢ (𝜑 → if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)) ∈ ℝ) |
29 | 18, 28 | eqeltrid 2845 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑇 ∈ ℝ) |
30 | | 0red 10979 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 ∈
ℝ) |
31 | 2 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 1 ∈
ℝ) |
32 | | 0lt1 11497 |
. . . . . . . . . . . . . 14
⊢ 0 <
1 |
33 | 32 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 < 1) |
34 | | max2 12920 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹‘𝑀) ∈ ℝ ∧ 1 ∈ ℝ)
→ 1 ≤ if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀))) |
35 | 26, 2, 34 | sylancl 586 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 1 ≤ if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀))) |
36 | 35, 18 | breqtrrdi 5121 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 1 ≤ 𝑇) |
37 | 30, 31, 29, 33, 36 | ltletrd 11135 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 < 𝑇) |
38 | 29, 37 | elrpd 12768 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑇 ∈
ℝ+) |
39 | | ostth2.8 |
. . . . . . . . . . . 12
⊢ 𝑈 = ((log‘𝑁) / (log‘𝑀)) |
40 | 7 | nnrpd 12769 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈
ℝ+) |
41 | 40 | relogcld 25776 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (log‘𝑁) ∈
ℝ) |
42 | 22 | nnred 11988 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑀 ∈ ℝ) |
43 | 21 | simprd 496 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 1 < 𝑀) |
44 | 42, 43 | rplogcld 25782 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (log‘𝑀) ∈
ℝ+) |
45 | 41, 44 | rerpdivcld 12802 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((log‘𝑁) / (log‘𝑀)) ∈ ℝ) |
46 | 39, 45 | eqeltrid 2845 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑈 ∈ ℝ) |
47 | 38, 46 | rpcxpcld 25885 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑇↑𝑐𝑈) ∈
ℝ+) |
48 | 14, 47 | rerpdivcld 12802 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐹‘𝑁) / (𝑇↑𝑐𝑈)) ∈ ℝ) |
49 | 42, 29 | remulcld 11006 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑀 · 𝑇) ∈ ℝ) |
50 | | peano2re 11148 |
. . . . . . . . . . 11
⊢ (𝑈 ∈ ℝ → (𝑈 + 1) ∈
ℝ) |
51 | 46, 50 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑈 + 1) ∈ ℝ) |
52 | 49, 51 | remulcld 11006 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑀 · 𝑇) · (𝑈 + 1)) ∈ ℝ) |
53 | | padic.j |
. . . . . . . . . 10
⊢ 𝐽 = (𝑞 ∈ ℙ ↦ (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, (𝑞↑-(𝑞 pCnt 𝑥))))) |
54 | | ostth.k |
. . . . . . . . . 10
⊢ 𝐾 = (𝑥 ∈ ℚ ↦ if(𝑥 = 0, 0, 1)) |
55 | | ostth2.4 |
. . . . . . . . . 10
⊢ 𝑅 = ((log‘(𝐹‘𝑁)) / (log‘𝑁)) |
56 | | ostth2.6 |
. . . . . . . . . 10
⊢ 𝑆 = ((log‘(𝐹‘𝑀)) / (log‘𝑀)) |
57 | 11, 10, 53, 54, 3, 4, 1, 55, 19, 56, 18, 39 | ostth2lem3 26781 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((𝐹‘𝑁) / (𝑇↑𝑐𝑈))↑𝑛) ≤ (𝑛 · ((𝑀 · 𝑇) · (𝑈 + 1)))) |
58 | 48, 52, 57 | ostth2lem1 26764 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘𝑁) / (𝑇↑𝑐𝑈)) ≤ 1) |
59 | 14, 31, 47 | ledivmuld 12824 |
. . . . . . . 8
⊢ (𝜑 → (((𝐹‘𝑁) / (𝑇↑𝑐𝑈)) ≤ 1 ↔ (𝐹‘𝑁) ≤ ((𝑇↑𝑐𝑈) · 1))) |
60 | 58, 59 | mpbid 231 |
. . . . . . 7
⊢ (𝜑 → (𝐹‘𝑁) ≤ ((𝑇↑𝑐𝑈) · 1)) |
61 | 47 | rpcnd 12773 |
. . . . . . . 8
⊢ (𝜑 → (𝑇↑𝑐𝑈) ∈ ℂ) |
62 | 61 | mulid1d 10993 |
. . . . . . 7
⊢ (𝜑 → ((𝑇↑𝑐𝑈) · 1) = (𝑇↑𝑐𝑈)) |
63 | 60, 62 | breqtrd 5105 |
. . . . . 6
⊢ (𝜑 → (𝐹‘𝑁) ≤ (𝑇↑𝑐𝑈)) |
64 | 63 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ (𝐹‘𝑀) ≤ 1) → (𝐹‘𝑁) ≤ (𝑇↑𝑐𝑈)) |
65 | | iftrue 4471 |
. . . . . . . 8
⊢ ((𝐹‘𝑀) ≤ 1 → if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)) = 1) |
66 | 18, 65 | eqtrid 2792 |
. . . . . . 7
⊢ ((𝐹‘𝑀) ≤ 1 → 𝑇 = 1) |
67 | 66 | oveq1d 7286 |
. . . . . 6
⊢ ((𝐹‘𝑀) ≤ 1 → (𝑇↑𝑐𝑈) = (1↑𝑐𝑈)) |
68 | 46 | recnd 11004 |
. . . . . . 7
⊢ (𝜑 → 𝑈 ∈ ℂ) |
69 | 68 | 1cxpd 25860 |
. . . . . 6
⊢ (𝜑 →
(1↑𝑐𝑈) = 1) |
70 | 67, 69 | sylan9eqr 2802 |
. . . . 5
⊢ ((𝜑 ∧ (𝐹‘𝑀) ≤ 1) → (𝑇↑𝑐𝑈) = 1) |
71 | 64, 70 | breqtrd 5105 |
. . . 4
⊢ ((𝜑 ∧ (𝐹‘𝑀) ≤ 1) → (𝐹‘𝑁) ≤ 1) |
72 | 17, 71 | mtand 813 |
. . 3
⊢ (𝜑 → ¬ (𝐹‘𝑀) ≤ 1) |
73 | | ltnle 11055 |
. . . 4
⊢ ((1
∈ ℝ ∧ (𝐹‘𝑀) ∈ ℝ) → (1 < (𝐹‘𝑀) ↔ ¬ (𝐹‘𝑀) ≤ 1)) |
74 | 2, 26, 73 | sylancr 587 |
. . 3
⊢ (𝜑 → (1 < (𝐹‘𝑀) ↔ ¬ (𝐹‘𝑀) ≤ 1)) |
75 | 72, 74 | mpbird 256 |
. 2
⊢ (𝜑 → 1 < (𝐹‘𝑀)) |
76 | 30, 31, 14, 33, 1 | lttrd 11136 |
. . . . . . . . 9
⊢ (𝜑 → 0 < (𝐹‘𝑁)) |
77 | 14, 76 | elrpd 12768 |
. . . . . . . 8
⊢ (𝜑 → (𝐹‘𝑁) ∈
ℝ+) |
78 | 77 | reeflogd 25777 |
. . . . . . 7
⊢ (𝜑 →
(exp‘(log‘(𝐹‘𝑁))) = (𝐹‘𝑁)) |
79 | | iffalse 4474 |
. . . . . . . . . . 11
⊢ (¬
(𝐹‘𝑀) ≤ 1 → if((𝐹‘𝑀) ≤ 1, 1, (𝐹‘𝑀)) = (𝐹‘𝑀)) |
80 | 18, 79 | eqtrid 2792 |
. . . . . . . . . 10
⊢ (¬
(𝐹‘𝑀) ≤ 1 → 𝑇 = (𝐹‘𝑀)) |
81 | 72, 80 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑇 = (𝐹‘𝑀)) |
82 | 81 | oveq1d 7286 |
. . . . . . . 8
⊢ (𝜑 → (𝑇↑𝑐𝑈) = ((𝐹‘𝑀)↑𝑐𝑈)) |
83 | 26 | recnd 11004 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ) |
84 | 30, 31, 26, 33, 75 | lttrd 11136 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 < (𝐹‘𝑀)) |
85 | 26, 84 | elrpd 12768 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹‘𝑀) ∈
ℝ+) |
86 | 85 | rpne0d 12776 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘𝑀) ≠ 0) |
87 | 83, 86, 68 | cxpefd 25865 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹‘𝑀)↑𝑐𝑈) = (exp‘(𝑈 · (log‘(𝐹‘𝑀))))) |
88 | 82, 87 | eqtr2d 2781 |
. . . . . . 7
⊢ (𝜑 → (exp‘(𝑈 · (log‘(𝐹‘𝑀)))) = (𝑇↑𝑐𝑈)) |
89 | 63, 78, 88 | 3brtr4d 5111 |
. . . . . 6
⊢ (𝜑 →
(exp‘(log‘(𝐹‘𝑁))) ≤ (exp‘(𝑈 · (log‘(𝐹‘𝑀))))) |
90 | 77 | relogcld 25776 |
. . . . . . 7
⊢ (𝜑 → (log‘(𝐹‘𝑁)) ∈ ℝ) |
91 | 85 | relogcld 25776 |
. . . . . . . 8
⊢ (𝜑 → (log‘(𝐹‘𝑀)) ∈ ℝ) |
92 | 46, 91 | remulcld 11006 |
. . . . . . 7
⊢ (𝜑 → (𝑈 · (log‘(𝐹‘𝑀))) ∈ ℝ) |
93 | | efle 15825 |
. . . . . . 7
⊢
(((log‘(𝐹‘𝑁)) ∈ ℝ ∧ (𝑈 · (log‘(𝐹‘𝑀))) ∈ ℝ) →
((log‘(𝐹‘𝑁)) ≤ (𝑈 · (log‘(𝐹‘𝑀))) ↔ (exp‘(log‘(𝐹‘𝑁))) ≤ (exp‘(𝑈 · (log‘(𝐹‘𝑀)))))) |
94 | 90, 92, 93 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → ((log‘(𝐹‘𝑁)) ≤ (𝑈 · (log‘(𝐹‘𝑀))) ↔ (exp‘(log‘(𝐹‘𝑁))) ≤ (exp‘(𝑈 · (log‘(𝐹‘𝑀)))))) |
95 | 89, 94 | mpbird 256 |
. . . . 5
⊢ (𝜑 → (log‘(𝐹‘𝑁)) ≤ (𝑈 · (log‘(𝐹‘𝑀)))) |
96 | 41 | recnd 11004 |
. . . . . . . 8
⊢ (𝜑 → (log‘𝑁) ∈
ℂ) |
97 | 91 | recnd 11004 |
. . . . . . . 8
⊢ (𝜑 → (log‘(𝐹‘𝑀)) ∈ ℂ) |
98 | 44 | rpcnd 12773 |
. . . . . . . 8
⊢ (𝜑 → (log‘𝑀) ∈
ℂ) |
99 | 44 | rpne0d 12776 |
. . . . . . . 8
⊢ (𝜑 → (log‘𝑀) ≠ 0) |
100 | 96, 97, 98, 99 | div12d 11787 |
. . . . . . 7
⊢ (𝜑 → ((log‘𝑁) · ((log‘(𝐹‘𝑀)) / (log‘𝑀))) = ((log‘(𝐹‘𝑀)) · ((log‘𝑁) / (log‘𝑀)))) |
101 | 39 | oveq2i 7282 |
. . . . . . 7
⊢
((log‘(𝐹‘𝑀)) · 𝑈) = ((log‘(𝐹‘𝑀)) · ((log‘𝑁) / (log‘𝑀))) |
102 | 100, 101 | eqtr4di 2798 |
. . . . . 6
⊢ (𝜑 → ((log‘𝑁) · ((log‘(𝐹‘𝑀)) / (log‘𝑀))) = ((log‘(𝐹‘𝑀)) · 𝑈)) |
103 | 97, 68 | mulcomd 10997 |
. . . . . 6
⊢ (𝜑 → ((log‘(𝐹‘𝑀)) · 𝑈) = (𝑈 · (log‘(𝐹‘𝑀)))) |
104 | 102, 103 | eqtrd 2780 |
. . . . 5
⊢ (𝜑 → ((log‘𝑁) · ((log‘(𝐹‘𝑀)) / (log‘𝑀))) = (𝑈 · (log‘(𝐹‘𝑀)))) |
105 | 95, 104 | breqtrrd 5107 |
. . . 4
⊢ (𝜑 → (log‘(𝐹‘𝑁)) ≤ ((log‘𝑁) · ((log‘(𝐹‘𝑀)) / (log‘𝑀)))) |
106 | 91, 44 | rerpdivcld 12802 |
. . . . 5
⊢ (𝜑 → ((log‘(𝐹‘𝑀)) / (log‘𝑀)) ∈ ℝ) |
107 | 7 | nnred 11988 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℝ) |
108 | 6 | simprd 496 |
. . . . . 6
⊢ (𝜑 → 1 < 𝑁) |
109 | 107, 108 | rplogcld 25782 |
. . . . 5
⊢ (𝜑 → (log‘𝑁) ∈
ℝ+) |
110 | 90, 106, 109 | ledivmuld 12824 |
. . . 4
⊢ (𝜑 → (((log‘(𝐹‘𝑁)) / (log‘𝑁)) ≤ ((log‘(𝐹‘𝑀)) / (log‘𝑀)) ↔ (log‘(𝐹‘𝑁)) ≤ ((log‘𝑁) · ((log‘(𝐹‘𝑀)) / (log‘𝑀))))) |
111 | 105, 110 | mpbird 256 |
. . 3
⊢ (𝜑 → ((log‘(𝐹‘𝑁)) / (log‘𝑁)) ≤ ((log‘(𝐹‘𝑀)) / (log‘𝑀))) |
112 | 111, 55, 56 | 3brtr4g 5113 |
. 2
⊢ (𝜑 → 𝑅 ≤ 𝑆) |
113 | 75, 112 | jca 512 |
1
⊢ (𝜑 → (1 < (𝐹‘𝑀) ∧ 𝑅 ≤ 𝑆)) |