Step |
Hyp |
Ref |
Expression |
1 |
|
pntsval.1 |
⊢ 𝑆 = ( 𝑎 ∈ ℝ ↦ Σ 𝑖 ∈ ( 1 ... ( ⌊ ‘ 𝑎 ) ) ( ( Λ ‘ 𝑖 ) · ( ( log ‘ 𝑖 ) + ( ψ ‘ ( 𝑎 / 𝑖 ) ) ) ) ) |
2 |
1
|
pntsval |
⊢ ( 𝐴 ∈ ℝ → ( 𝑆 ‘ 𝐴 ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( Λ ‘ 𝑛 ) · ( ( log ‘ 𝑛 ) + ( ψ ‘ ( 𝐴 / 𝑛 ) ) ) ) ) |
3 |
|
elfznn |
⊢ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) → 𝑛 ∈ ℕ ) |
4 |
3
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝑛 ∈ ℕ ) |
5 |
|
vmacl |
⊢ ( 𝑛 ∈ ℕ → ( Λ ‘ 𝑛 ) ∈ ℝ ) |
6 |
4 5
|
syl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( Λ ‘ 𝑛 ) ∈ ℝ ) |
7 |
6
|
recnd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( Λ ‘ 𝑛 ) ∈ ℂ ) |
8 |
4
|
nnrpd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝑛 ∈ ℝ+ ) |
9 |
8
|
relogcld |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( log ‘ 𝑛 ) ∈ ℝ ) |
10 |
9
|
recnd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( log ‘ 𝑛 ) ∈ ℂ ) |
11 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝐴 ∈ ℝ ) |
12 |
11 4
|
nndivred |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( 𝐴 / 𝑛 ) ∈ ℝ ) |
13 |
|
chpcl |
⊢ ( ( 𝐴 / 𝑛 ) ∈ ℝ → ( ψ ‘ ( 𝐴 / 𝑛 ) ) ∈ ℝ ) |
14 |
12 13
|
syl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ψ ‘ ( 𝐴 / 𝑛 ) ) ∈ ℝ ) |
15 |
14
|
recnd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ψ ‘ ( 𝐴 / 𝑛 ) ) ∈ ℂ ) |
16 |
7 10 15
|
adddid |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ( Λ ‘ 𝑛 ) · ( ( log ‘ 𝑛 ) + ( ψ ‘ ( 𝐴 / 𝑛 ) ) ) ) = ( ( ( Λ ‘ 𝑛 ) · ( log ‘ 𝑛 ) ) + ( ( Λ ‘ 𝑛 ) · ( ψ ‘ ( 𝐴 / 𝑛 ) ) ) ) ) |
17 |
16
|
sumeq2dv |
⊢ ( 𝐴 ∈ ℝ → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( Λ ‘ 𝑛 ) · ( ( log ‘ 𝑛 ) + ( ψ ‘ ( 𝐴 / 𝑛 ) ) ) ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( ( Λ ‘ 𝑛 ) · ( log ‘ 𝑛 ) ) + ( ( Λ ‘ 𝑛 ) · ( ψ ‘ ( 𝐴 / 𝑛 ) ) ) ) ) |
18 |
|
fveq2 |
⊢ ( 𝑛 = 𝑚 → ( Λ ‘ 𝑛 ) = ( Λ ‘ 𝑚 ) ) |
19 |
|
oveq2 |
⊢ ( 𝑛 = 𝑚 → ( 𝐴 / 𝑛 ) = ( 𝐴 / 𝑚 ) ) |
20 |
19
|
fveq2d |
⊢ ( 𝑛 = 𝑚 → ( ψ ‘ ( 𝐴 / 𝑛 ) ) = ( ψ ‘ ( 𝐴 / 𝑚 ) ) ) |
21 |
18 20
|
oveq12d |
⊢ ( 𝑛 = 𝑚 → ( ( Λ ‘ 𝑛 ) · ( ψ ‘ ( 𝐴 / 𝑛 ) ) ) = ( ( Λ ‘ 𝑚 ) · ( ψ ‘ ( 𝐴 / 𝑚 ) ) ) ) |
22 |
21
|
cbvsumv |
⊢ Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( Λ ‘ 𝑛 ) · ( ψ ‘ ( 𝐴 / 𝑛 ) ) ) = Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( Λ ‘ 𝑚 ) · ( ψ ‘ ( 𝐴 / 𝑚 ) ) ) |
23 |
|
fzfid |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ∈ Fin ) |
24 |
|
elfznn |
⊢ ( 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) → 𝑚 ∈ ℕ ) |
25 |
24
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝑚 ∈ ℕ ) |
26 |
|
vmacl |
⊢ ( 𝑚 ∈ ℕ → ( Λ ‘ 𝑚 ) ∈ ℝ ) |
27 |
25 26
|
syl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( Λ ‘ 𝑚 ) ∈ ℝ ) |
28 |
27
|
recnd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( Λ ‘ 𝑚 ) ∈ ℂ ) |
29 |
|
elfznn |
⊢ ( 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) → 𝑘 ∈ ℕ ) |
30 |
29
|
adantl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ) → 𝑘 ∈ ℕ ) |
31 |
|
vmacl |
⊢ ( 𝑘 ∈ ℕ → ( Λ ‘ 𝑘 ) ∈ ℝ ) |
32 |
30 31
|
syl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ) → ( Λ ‘ 𝑘 ) ∈ ℝ ) |
33 |
32
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ) → ( Λ ‘ 𝑘 ) ∈ ℂ ) |
34 |
23 28 33
|
fsummulc2 |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ( Λ ‘ 𝑚 ) · Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ( Λ ‘ 𝑘 ) ) = Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ( ( Λ ‘ 𝑚 ) · ( Λ ‘ 𝑘 ) ) ) |
35 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → 𝐴 ∈ ℝ ) |
36 |
35 25
|
nndivred |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( 𝐴 / 𝑚 ) ∈ ℝ ) |
37 |
|
chpval |
⊢ ( ( 𝐴 / 𝑚 ) ∈ ℝ → ( ψ ‘ ( 𝐴 / 𝑚 ) ) = Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ( Λ ‘ 𝑘 ) ) |
38 |
36 37
|
syl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ψ ‘ ( 𝐴 / 𝑚 ) ) = Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ( Λ ‘ 𝑘 ) ) |
39 |
38
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ( Λ ‘ 𝑚 ) · ( ψ ‘ ( 𝐴 / 𝑚 ) ) ) = ( ( Λ ‘ 𝑚 ) · Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ( Λ ‘ 𝑘 ) ) ) |
40 |
30
|
nncnd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ) → 𝑘 ∈ ℂ ) |
41 |
24
|
ad2antlr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ) → 𝑚 ∈ ℕ ) |
42 |
41
|
nncnd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ) → 𝑚 ∈ ℂ ) |
43 |
41
|
nnne0d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ) → 𝑚 ≠ 0 ) |
44 |
40 42 43
|
divcan3d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ) → ( ( 𝑚 · 𝑘 ) / 𝑚 ) = 𝑘 ) |
45 |
44
|
fveq2d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ) → ( Λ ‘ ( ( 𝑚 · 𝑘 ) / 𝑚 ) ) = ( Λ ‘ 𝑘 ) ) |
46 |
45
|
oveq2d |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ) → ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( ( 𝑚 · 𝑘 ) / 𝑚 ) ) ) = ( ( Λ ‘ 𝑚 ) · ( Λ ‘ 𝑘 ) ) ) |
47 |
46
|
sumeq2dv |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( ( 𝑚 · 𝑘 ) / 𝑚 ) ) ) = Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ( ( Λ ‘ 𝑚 ) · ( Λ ‘ 𝑘 ) ) ) |
48 |
34 39 47
|
3eqtr4d |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ( Λ ‘ 𝑚 ) · ( ψ ‘ ( 𝐴 / 𝑚 ) ) ) = Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( ( 𝑚 · 𝑘 ) / 𝑚 ) ) ) ) |
49 |
48
|
sumeq2dv |
⊢ ( 𝐴 ∈ ℝ → Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( Λ ‘ 𝑚 ) · ( ψ ‘ ( 𝐴 / 𝑚 ) ) ) = Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( ( 𝑚 · 𝑘 ) / 𝑚 ) ) ) ) |
50 |
|
fvoveq1 |
⊢ ( 𝑛 = ( 𝑚 · 𝑘 ) → ( Λ ‘ ( 𝑛 / 𝑚 ) ) = ( Λ ‘ ( ( 𝑚 · 𝑘 ) / 𝑚 ) ) ) |
51 |
50
|
oveq2d |
⊢ ( 𝑛 = ( 𝑚 · 𝑘 ) → ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( 𝑛 / 𝑚 ) ) ) = ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( ( 𝑚 · 𝑘 ) / 𝑚 ) ) ) ) |
52 |
|
id |
⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ ) |
53 |
|
ssrab2 |
⊢ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ⊆ ℕ |
54 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ) → 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ) |
55 |
53 54
|
sseldi |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ) → 𝑚 ∈ ℕ ) |
56 |
55 26
|
syl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ) → ( Λ ‘ 𝑚 ) ∈ ℝ ) |
57 |
|
dvdsdivcl |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ) → ( 𝑛 / 𝑚 ) ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ) |
58 |
4 57
|
sylan |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ) → ( 𝑛 / 𝑚 ) ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ) |
59 |
53 58
|
sseldi |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ) → ( 𝑛 / 𝑚 ) ∈ ℕ ) |
60 |
|
vmacl |
⊢ ( ( 𝑛 / 𝑚 ) ∈ ℕ → ( Λ ‘ ( 𝑛 / 𝑚 ) ) ∈ ℝ ) |
61 |
59 60
|
syl |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ) → ( Λ ‘ ( 𝑛 / 𝑚 ) ) ∈ ℝ ) |
62 |
56 61
|
remulcld |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ) → ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( 𝑛 / 𝑚 ) ) ) ∈ ℝ ) |
63 |
62
|
recnd |
⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) ∧ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ) → ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( 𝑛 / 𝑚 ) ) ) ∈ ℂ ) |
64 |
63
|
anasss |
⊢ ( ( 𝐴 ∈ ℝ ∧ ( 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∧ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ) ) → ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( 𝑛 / 𝑚 ) ) ) ∈ ℂ ) |
65 |
51 52 64
|
dvdsflsumcom |
⊢ ( 𝐴 ∈ ℝ → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) Σ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( 𝑛 / 𝑚 ) ) ) = Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) Σ 𝑘 ∈ ( 1 ... ( ⌊ ‘ ( 𝐴 / 𝑚 ) ) ) ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( ( 𝑚 · 𝑘 ) / 𝑚 ) ) ) ) |
66 |
49 65
|
eqtr4d |
⊢ ( 𝐴 ∈ ℝ → Σ 𝑚 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( Λ ‘ 𝑚 ) · ( ψ ‘ ( 𝐴 / 𝑚 ) ) ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) Σ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( 𝑛 / 𝑚 ) ) ) ) |
67 |
22 66
|
syl5eq |
⊢ ( 𝐴 ∈ ℝ → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( Λ ‘ 𝑛 ) · ( ψ ‘ ( 𝐴 / 𝑛 ) ) ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) Σ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( 𝑛 / 𝑚 ) ) ) ) |
68 |
67
|
oveq2d |
⊢ ( 𝐴 ∈ ℝ → ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( Λ ‘ 𝑛 ) · ( log ‘ 𝑛 ) ) + Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( Λ ‘ 𝑛 ) · ( ψ ‘ ( 𝐴 / 𝑛 ) ) ) ) = ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( Λ ‘ 𝑛 ) · ( log ‘ 𝑛 ) ) + Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) Σ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( 𝑛 / 𝑚 ) ) ) ) ) |
69 |
|
fzfid |
⊢ ( 𝐴 ∈ ℝ → ( 1 ... ( ⌊ ‘ 𝐴 ) ) ∈ Fin ) |
70 |
7 10
|
mulcld |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ( Λ ‘ 𝑛 ) · ( log ‘ 𝑛 ) ) ∈ ℂ ) |
71 |
7 15
|
mulcld |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( ( Λ ‘ 𝑛 ) · ( ψ ‘ ( 𝐴 / 𝑛 ) ) ) ∈ ℂ ) |
72 |
69 70 71
|
fsumadd |
⊢ ( 𝐴 ∈ ℝ → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( ( Λ ‘ 𝑛 ) · ( log ‘ 𝑛 ) ) + ( ( Λ ‘ 𝑛 ) · ( ψ ‘ ( 𝐴 / 𝑛 ) ) ) ) = ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( Λ ‘ 𝑛 ) · ( log ‘ 𝑛 ) ) + Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( Λ ‘ 𝑛 ) · ( ψ ‘ ( 𝐴 / 𝑛 ) ) ) ) ) |
73 |
|
fzfid |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → ( 1 ... 𝑛 ) ∈ Fin ) |
74 |
|
dvdsssfz1 |
⊢ ( 𝑛 ∈ ℕ → { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ⊆ ( 1 ... 𝑛 ) ) |
75 |
4 74
|
syl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ⊆ ( 1 ... 𝑛 ) ) |
76 |
73 75
|
ssfid |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ∈ Fin ) |
77 |
76 62
|
fsumrecl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → Σ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( 𝑛 / 𝑚 ) ) ) ∈ ℝ ) |
78 |
77
|
recnd |
⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ) → Σ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( 𝑛 / 𝑚 ) ) ) ∈ ℂ ) |
79 |
69 70 78
|
fsumadd |
⊢ ( 𝐴 ∈ ℝ → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( ( Λ ‘ 𝑛 ) · ( log ‘ 𝑛 ) ) + Σ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( 𝑛 / 𝑚 ) ) ) ) = ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( Λ ‘ 𝑛 ) · ( log ‘ 𝑛 ) ) + Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) Σ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( 𝑛 / 𝑚 ) ) ) ) ) |
80 |
68 72 79
|
3eqtr4d |
⊢ ( 𝐴 ∈ ℝ → Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( ( Λ ‘ 𝑛 ) · ( log ‘ 𝑛 ) ) + ( ( Λ ‘ 𝑛 ) · ( ψ ‘ ( 𝐴 / 𝑛 ) ) ) ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( ( Λ ‘ 𝑛 ) · ( log ‘ 𝑛 ) ) + Σ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( 𝑛 / 𝑚 ) ) ) ) ) |
81 |
2 17 80
|
3eqtrd |
⊢ ( 𝐴 ∈ ℝ → ( 𝑆 ‘ 𝐴 ) = Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝐴 ) ) ( ( ( Λ ‘ 𝑛 ) · ( log ‘ 𝑛 ) ) + Σ 𝑚 ∈ { 𝑦 ∈ ℕ ∣ 𝑦 ∥ 𝑛 } ( ( Λ ‘ 𝑚 ) · ( Λ ‘ ( 𝑛 / 𝑚 ) ) ) ) ) |