Step |
Hyp |
Ref |
Expression |
1 |
|
rpvmasum.z |
⊢ 𝑍 = ( ℤ/nℤ ‘ 𝑁 ) |
2 |
|
rpvmasum.l |
⊢ 𝐿 = ( ℤRHom ‘ 𝑍 ) |
3 |
|
rpvmasum.a |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
4 |
|
rpvmasum.g |
⊢ 𝐺 = ( DChr ‘ 𝑁 ) |
5 |
|
rpvmasum.d |
⊢ 𝐷 = ( Base ‘ 𝐺 ) |
6 |
|
rpvmasum.1 |
⊢ 1 = ( 0g ‘ 𝐺 ) |
7 |
|
dchrisum.b |
⊢ ( 𝜑 → 𝑋 ∈ 𝐷 ) |
8 |
|
dchrisum.n1 |
⊢ ( 𝜑 → 𝑋 ≠ 1 ) |
9 |
|
dchrvmasum.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ+ ) |
10 |
|
dchrvmasum2.2 |
⊢ ( 𝜑 → 1 ≤ 𝐴 ) |
11 |
|
2fveq3 |
⊢ ( 𝑛 = ( 𝑑 · 𝑚 ) → ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) = ( 𝑋 ‘ ( 𝐿 ‘ ( 𝑑 · 𝑚 ) ) ) ) |
12 |
|
id |
⊢ ( 𝑛 = ( 𝑑 · 𝑚 ) → 𝑛 = ( 𝑑 · 𝑚 ) ) |
13 |
11 12
|
oveq12d |
⊢ ( 𝑛 = ( 𝑑 · 𝑚 ) → ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) / 𝑛 ) = ( ( 𝑋 ‘ ( 𝐿 ‘ ( 𝑑 · 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) ) |
14 |
|
oveq2 |
⊢ ( 𝑛 = ( 𝑑 · 𝑚 ) → ( 𝐴 / 𝑛 ) = ( 𝐴 / ( 𝑑 · 𝑚 ) ) ) |
15 |
14
|
fveq2d |
⊢ ( 𝑛 = ( 𝑑 · 𝑚 ) → ( log ‘ ( 𝐴 / 𝑛 ) ) = ( log ‘ ( 𝐴 / ( 𝑑 · 𝑚 ) ) ) ) |
16 |
13 15
|
oveq12d |
⊢ ( 𝑛 = ( 𝑑 · 𝑚 ) → ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) / 𝑛 ) · ( log ‘ ( 𝐴 / 𝑛 ) ) ) = ( ( ( 𝑋 ‘ ( 𝐿 ‘ ( 𝑑 · 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) · ( log ‘ ( 𝐴 / ( 𝑑 · 𝑚 ) ) ) ) ) |
17 |
16
|
oveq2d |
⊢ ( 𝑛 = ( 𝑑 · 𝑚 ) → ( ( μ ‘ 𝑑 ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) / 𝑛 ) · ( log ‘ ( 𝐴 / 𝑛 ) ) ) ) = ( ( μ ‘ 𝑑 ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ ( 𝑑 · 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) · ( log ‘ ( 𝐴 / ( 𝑑 · 𝑚 ) ) ) ) ) ) |
18 |
9
|
rpred |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
19 |
|
elrabi |
⊢ ( 𝑑 ∈ { 𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛 } → 𝑑 ∈ ℕ ) |
20 |
19
|
ad2antll |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ 𝑑 ∈ { 𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛 } ) ) → 𝑑 ∈ ℕ ) |
21 |
|
mucl |
⊢ ( 𝑑 ∈ ℕ → ( μ ‘ 𝑑 ) ∈ ℤ ) |
22 |
20 21
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ 𝑑 ∈ { 𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛 } ) ) → ( μ ‘ 𝑑 ) ∈ ℤ ) |
23 |
22
|
zcnd |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ 𝑑 ∈ { 𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛 } ) ) → ( μ ‘ 𝑑 ) ∈ ℂ ) |
24 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝑋 ∈ 𝐷 ) |
25 |
|
elfzelz |
⊢ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) → 𝑛 ∈ ℤ ) |
26 |
25
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝑛 ∈ ℤ ) |
27 |
4 1 5 2 24 26
|
dchrzrhcl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) ∈ ℂ ) |
28 |
|
elfznn |
⊢ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) → 𝑛 ∈ ℕ ) |
29 |
28
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝑛 ∈ ℕ ) |
30 |
29
|
nncnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝑛 ∈ ℂ ) |
31 |
29
|
nnne0d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝑛 ≠ 0 ) |
32 |
27 30 31
|
divcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) / 𝑛 ) ∈ ℂ ) |
33 |
28
|
nnrpd |
⊢ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) → 𝑛 ∈ ℝ+ ) |
34 |
|
rpdivcl |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ) → ( 𝐴 / 𝑛 ) ∈ ℝ+ ) |
35 |
9 33 34
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( 𝐴 / 𝑛 ) ∈ ℝ+ ) |
36 |
35
|
relogcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( log ‘ ( 𝐴 / 𝑛 ) ) ∈ ℝ ) |
37 |
36
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( log ‘ ( 𝐴 / 𝑛 ) ) ∈ ℂ ) |
38 |
32 37
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) / 𝑛 ) · ( log ‘ ( 𝐴 / 𝑛 ) ) ) ∈ ℂ ) |
39 |
38
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ 𝑑 ∈ { 𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛 } ) ) → ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) / 𝑛 ) · ( log ‘ ( 𝐴 / 𝑛 ) ) ) ∈ ℂ ) |
40 |
23 39
|
mulcld |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ 𝑑 ∈ { 𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛 } ) ) → ( ( μ ‘ 𝑑 ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) / 𝑛 ) · ( log ‘ ( 𝐴 / 𝑛 ) ) ) ) ∈ ℂ ) |
41 |
17 18 40
|
dvdsflsumcom |
⊢ ( 𝜑 → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) Σ 𝑑 ∈ { 𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛 } ( ( μ ‘ 𝑑 ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) / 𝑛 ) · ( log ‘ ( 𝐴 / 𝑛 ) ) ) ) = Σ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ( ( μ ‘ 𝑑 ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ ( 𝑑 · 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) · ( log ‘ ( 𝐴 / ( 𝑑 · 𝑚 ) ) ) ) ) ) |
42 |
|
2fveq3 |
⊢ ( 𝑛 = 1 → ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) = ( 𝑋 ‘ ( 𝐿 ‘ 1 ) ) ) |
43 |
|
id |
⊢ ( 𝑛 = 1 → 𝑛 = 1 ) |
44 |
42 43
|
oveq12d |
⊢ ( 𝑛 = 1 → ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) / 𝑛 ) = ( ( 𝑋 ‘ ( 𝐿 ‘ 1 ) ) / 1 ) ) |
45 |
|
oveq2 |
⊢ ( 𝑛 = 1 → ( 𝐴 / 𝑛 ) = ( 𝐴 / 1 ) ) |
46 |
45
|
fveq2d |
⊢ ( 𝑛 = 1 → ( log ‘ ( 𝐴 / 𝑛 ) ) = ( log ‘ ( 𝐴 / 1 ) ) ) |
47 |
44 46
|
oveq12d |
⊢ ( 𝑛 = 1 → ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) / 𝑛 ) · ( log ‘ ( 𝐴 / 𝑛 ) ) ) = ( ( ( 𝑋 ‘ ( 𝐿 ‘ 1 ) ) / 1 ) · ( log ‘ ( 𝐴 / 1 ) ) ) ) |
48 |
|
fzfid |
⊢ ( 𝜑 → ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∈ Fin ) |
49 |
|
fz1ssnn |
⊢ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ⊆ ℕ |
50 |
49
|
a1i |
⊢ ( 𝜑 → ( 1 ... ( ⌊ ‘ 𝐴 ) ) ⊆ ℕ ) |
51 |
|
flge1nn |
⊢ ( ( 𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ) → ( ⌊ ‘ 𝐴 ) ∈ ℕ ) |
52 |
18 10 51
|
syl2anc |
⊢ ( 𝜑 → ( ⌊ ‘ 𝐴 ) ∈ ℕ ) |
53 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
54 |
52 53
|
eleqtrdi |
⊢ ( 𝜑 → ( ⌊ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 1 ) ) |
55 |
|
eluzfz1 |
⊢ ( ( ⌊ ‘ 𝐴 ) ∈ ( ℤ≥ ‘ 1 ) → 1 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) |
56 |
54 55
|
syl |
⊢ ( 𝜑 → 1 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) |
57 |
47 48 50 56 38
|
musumsum |
⊢ ( 𝜑 → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) Σ 𝑑 ∈ { 𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛 } ( ( μ ‘ 𝑑 ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) / 𝑛 ) · ( log ‘ ( 𝐴 / 𝑛 ) ) ) ) = ( ( ( 𝑋 ‘ ( 𝐿 ‘ 1 ) ) / 1 ) · ( log ‘ ( 𝐴 / 1 ) ) ) ) |
58 |
4 1 5 2 7
|
dchrzrh1 |
⊢ ( 𝜑 → ( 𝑋 ‘ ( 𝐿 ‘ 1 ) ) = 1 ) |
59 |
58
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝑋 ‘ ( 𝐿 ‘ 1 ) ) / 1 ) = ( 1 / 1 ) ) |
60 |
|
1div1e1 |
⊢ ( 1 / 1 ) = 1 |
61 |
59 60
|
eqtrdi |
⊢ ( 𝜑 → ( ( 𝑋 ‘ ( 𝐿 ‘ 1 ) ) / 1 ) = 1 ) |
62 |
9
|
rpcnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
63 |
62
|
div1d |
⊢ ( 𝜑 → ( 𝐴 / 1 ) = 𝐴 ) |
64 |
63
|
fveq2d |
⊢ ( 𝜑 → ( log ‘ ( 𝐴 / 1 ) ) = ( log ‘ 𝐴 ) ) |
65 |
61 64
|
oveq12d |
⊢ ( 𝜑 → ( ( ( 𝑋 ‘ ( 𝐿 ‘ 1 ) ) / 1 ) · ( log ‘ ( 𝐴 / 1 ) ) ) = ( 1 · ( log ‘ 𝐴 ) ) ) |
66 |
9
|
relogcld |
⊢ ( 𝜑 → ( log ‘ 𝐴 ) ∈ ℝ ) |
67 |
66
|
recnd |
⊢ ( 𝜑 → ( log ‘ 𝐴 ) ∈ ℂ ) |
68 |
67
|
mulid2d |
⊢ ( 𝜑 → ( 1 · ( log ‘ 𝐴 ) ) = ( log ‘ 𝐴 ) ) |
69 |
57 65 68
|
3eqtrrd |
⊢ ( 𝜑 → ( log ‘ 𝐴 ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) Σ 𝑑 ∈ { 𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛 } ( ( μ ‘ 𝑑 ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑛 ) ) / 𝑛 ) · ( log ‘ ( 𝐴 / 𝑛 ) ) ) ) ) |
70 |
|
fzfid |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ∈ Fin ) |
71 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝑋 ∈ 𝐷 ) |
72 |
|
elfzelz |
⊢ ( 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) → 𝑑 ∈ ℤ ) |
73 |
72
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝑑 ∈ ℤ ) |
74 |
4 1 5 2 71 73
|
dchrzrhcl |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) ∈ ℂ ) |
75 |
|
fznnfl |
⊢ ( 𝐴 ∈ ℝ → ( 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ↔ ( 𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴 ) ) ) |
76 |
18 75
|
syl |
⊢ ( 𝜑 → ( 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ↔ ( 𝑑 ∈ ℕ ∧ 𝑑 ≤ 𝐴 ) ) ) |
77 |
76
|
simprbda |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝑑 ∈ ℕ ) |
78 |
77 21
|
syl |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( μ ‘ 𝑑 ) ∈ ℤ ) |
79 |
78
|
zred |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( μ ‘ 𝑑 ) ∈ ℝ ) |
80 |
79 77
|
nndivred |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ( μ ‘ 𝑑 ) / 𝑑 ) ∈ ℝ ) |
81 |
80
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ( μ ‘ 𝑑 ) / 𝑑 ) ∈ ℂ ) |
82 |
74 81
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) · ( ( μ ‘ 𝑑 ) / 𝑑 ) ) ∈ ℂ ) |
83 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → 𝑋 ∈ 𝐷 ) |
84 |
|
elfzelz |
⊢ ( 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) → 𝑚 ∈ ℤ ) |
85 |
84
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → 𝑚 ∈ ℤ ) |
86 |
4 1 5 2 83 85
|
dchrzrhcl |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) ∈ ℂ ) |
87 |
|
elfznn |
⊢ ( 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) → 𝑑 ∈ ℕ ) |
88 |
87
|
nnrpd |
⊢ ( 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) → 𝑑 ∈ ℝ+ ) |
89 |
|
rpdivcl |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝑑 ∈ ℝ+ ) → ( 𝐴 / 𝑑 ) ∈ ℝ+ ) |
90 |
9 88 89
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( 𝐴 / 𝑑 ) ∈ ℝ+ ) |
91 |
|
elfznn |
⊢ ( 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) → 𝑚 ∈ ℕ ) |
92 |
91
|
nnrpd |
⊢ ( 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) → 𝑚 ∈ ℝ+ ) |
93 |
|
rpdivcl |
⊢ ( ( ( 𝐴 / 𝑑 ) ∈ ℝ+ ∧ 𝑚 ∈ ℝ+ ) → ( ( 𝐴 / 𝑑 ) / 𝑚 ) ∈ ℝ+ ) |
94 |
90 92 93
|
syl2an |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( 𝐴 / 𝑑 ) / 𝑚 ) ∈ ℝ+ ) |
95 |
94
|
relogcld |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) ∈ ℝ ) |
96 |
91
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → 𝑚 ∈ ℕ ) |
97 |
95 96
|
nndivred |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) / 𝑚 ) ∈ ℝ ) |
98 |
97
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) / 𝑚 ) ∈ ℂ ) |
99 |
86 98
|
mulcld |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) · ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) / 𝑚 ) ) ∈ ℂ ) |
100 |
70 82 99
|
fsummulc2 |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) · ( ( μ ‘ 𝑑 ) / 𝑑 ) ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) · ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) / 𝑚 ) ) ) = Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) · ( ( μ ‘ 𝑑 ) / 𝑑 ) ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) · ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) / 𝑚 ) ) ) ) |
101 |
74
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) ∈ ℂ ) |
102 |
79
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( μ ‘ 𝑑 ) ∈ ℝ ) |
103 |
102
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( μ ‘ 𝑑 ) ∈ ℂ ) |
104 |
77
|
nnrpd |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝑑 ∈ ℝ+ ) |
105 |
104
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → 𝑑 ∈ ℝ+ ) |
106 |
105
|
rpcnne0d |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( 𝑑 ∈ ℂ ∧ 𝑑 ≠ 0 ) ) |
107 |
|
div12 |
⊢ ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) ∈ ℂ ∧ ( μ ‘ 𝑑 ) ∈ ℂ ∧ ( 𝑑 ∈ ℂ ∧ 𝑑 ≠ 0 ) ) → ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) · ( ( μ ‘ 𝑑 ) / 𝑑 ) ) = ( ( μ ‘ 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) ) ) |
108 |
101 103 106 107
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) · ( ( μ ‘ 𝑑 ) / 𝑑 ) ) = ( ( μ ‘ 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) ) ) |
109 |
95
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) ∈ ℂ ) |
110 |
96
|
nnrpd |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → 𝑚 ∈ ℝ+ ) |
111 |
110
|
rpcnne0d |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( 𝑚 ∈ ℂ ∧ 𝑚 ≠ 0 ) ) |
112 |
|
div12 |
⊢ ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) ∈ ℂ ∧ ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) ∈ ℂ ∧ ( 𝑚 ∈ ℂ ∧ 𝑚 ≠ 0 ) ) → ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) · ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) / 𝑚 ) ) = ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) ) |
113 |
86 109 111 112
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) · ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) / 𝑚 ) ) = ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) ) |
114 |
108 113
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) · ( ( μ ‘ 𝑑 ) / 𝑑 ) ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) · ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) / 𝑚 ) ) ) = ( ( ( μ ‘ 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) ) · ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) ) ) |
115 |
105
|
rpcnd |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → 𝑑 ∈ ℂ ) |
116 |
105
|
rpne0d |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → 𝑑 ≠ 0 ) |
117 |
101 115 116
|
divcld |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) ∈ ℂ ) |
118 |
96
|
nncnd |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → 𝑚 ∈ ℂ ) |
119 |
96
|
nnne0d |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → 𝑚 ≠ 0 ) |
120 |
86 118 119
|
divcld |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ∈ ℂ ) |
121 |
117 120
|
mulcld |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) ∈ ℂ ) |
122 |
103 109 121
|
mulassd |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( ( μ ‘ 𝑑 ) · ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) ) = ( ( μ ‘ 𝑑 ) · ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) ) ) ) |
123 |
103 117 109 120
|
mul4d |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( ( μ ‘ 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) ) · ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) ) = ( ( ( μ ‘ 𝑑 ) · ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) ) ) |
124 |
72
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → 𝑑 ∈ ℤ ) |
125 |
4 1 5 2 83 124 85
|
dchrzrhmul |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( 𝑋 ‘ ( 𝐿 ‘ ( 𝑑 · 𝑚 ) ) ) = ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) · ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) ) ) |
126 |
125
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( 𝑋 ‘ ( 𝐿 ‘ ( 𝑑 · 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) = ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) · ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) ) |
127 |
|
divmuldiv |
⊢ ( ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) ∈ ℂ ∧ ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) ∈ ℂ ) ∧ ( ( 𝑑 ∈ ℂ ∧ 𝑑 ≠ 0 ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑚 ≠ 0 ) ) ) → ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) = ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) · ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) ) |
128 |
101 86 106 111 127
|
syl22anc |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) = ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) · ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) ) |
129 |
126 128
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( 𝑋 ‘ ( 𝐿 ‘ ( 𝑑 · 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) = ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) ) |
130 |
62
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → 𝐴 ∈ ℂ ) |
131 |
|
divdiv1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝑑 ∈ ℂ ∧ 𝑑 ≠ 0 ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑚 ≠ 0 ) ) → ( ( 𝐴 / 𝑑 ) / 𝑚 ) = ( 𝐴 / ( 𝑑 · 𝑚 ) ) ) |
132 |
130 106 111 131
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( 𝐴 / 𝑑 ) / 𝑚 ) = ( 𝐴 / ( 𝑑 · 𝑚 ) ) ) |
133 |
132
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( 𝐴 / ( 𝑑 · 𝑚 ) ) = ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) |
134 |
133
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( log ‘ ( 𝐴 / ( 𝑑 · 𝑚 ) ) ) = ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) ) |
135 |
129 134
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( ( 𝑋 ‘ ( 𝐿 ‘ ( 𝑑 · 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) · ( log ‘ ( 𝐴 / ( 𝑑 · 𝑚 ) ) ) ) = ( ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) · ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) ) ) |
136 |
121 109
|
mulcomd |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) · ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) ) = ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) ) ) |
137 |
135 136
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( ( 𝑋 ‘ ( 𝐿 ‘ ( 𝑑 · 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) · ( log ‘ ( 𝐴 / ( 𝑑 · 𝑚 ) ) ) ) = ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) ) ) |
138 |
137
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( μ ‘ 𝑑 ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ ( 𝑑 · 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) · ( log ‘ ( 𝐴 / ( 𝑑 · 𝑚 ) ) ) ) ) = ( ( μ ‘ 𝑑 ) · ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) ) ) ) |
139 |
122 123 138
|
3eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( ( μ ‘ 𝑑 ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) / 𝑑 ) ) · ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) / 𝑚 ) ) ) = ( ( μ ‘ 𝑑 ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ ( 𝑑 · 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) · ( log ‘ ( 𝐴 / ( 𝑑 · 𝑚 ) ) ) ) ) ) |
140 |
114 139
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ) → ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) · ( ( μ ‘ 𝑑 ) / 𝑑 ) ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) · ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) / 𝑚 ) ) ) = ( ( μ ‘ 𝑑 ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ ( 𝑑 · 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) · ( log ‘ ( 𝐴 / ( 𝑑 · 𝑚 ) ) ) ) ) ) |
141 |
140
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) · ( ( μ ‘ 𝑑 ) / 𝑑 ) ) · ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) · ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) / 𝑚 ) ) ) = Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ( ( μ ‘ 𝑑 ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ ( 𝑑 · 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) · ( log ‘ ( 𝐴 / ( 𝑑 · 𝑚 ) ) ) ) ) ) |
142 |
100 141
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) · ( ( μ ‘ 𝑑 ) / 𝑑 ) ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) · ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) / 𝑚 ) ) ) = Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ( ( μ ‘ 𝑑 ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ ( 𝑑 · 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) · ( log ‘ ( 𝐴 / ( 𝑑 · 𝑚 ) ) ) ) ) ) |
143 |
142
|
sumeq2dv |
⊢ ( 𝜑 → Σ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) · ( ( μ ‘ 𝑑 ) / 𝑑 ) ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) · ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) / 𝑚 ) ) ) = Σ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ( ( μ ‘ 𝑑 ) · ( ( ( 𝑋 ‘ ( 𝐿 ‘ ( 𝑑 · 𝑚 ) ) ) / ( 𝑑 · 𝑚 ) ) · ( log ‘ ( 𝐴 / ( 𝑑 · 𝑚 ) ) ) ) ) ) |
144 |
41 69 143
|
3eqtr4d |
⊢ ( 𝜑 → ( log ‘ 𝐴 ) = Σ 𝑑 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑑 ) ) · ( ( μ ‘ 𝑑 ) / 𝑑 ) ) · Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑑 ) ) ) ( ( 𝑋 ‘ ( 𝐿 ‘ 𝑚 ) ) · ( ( log ‘ ( ( 𝐴 / 𝑑 ) / 𝑚 ) ) / 𝑚 ) ) ) ) |