| Step |
Hyp |
Ref |
Expression |
| 1 |
|
iprodgam.1 |
⊢ ( 𝜑 → 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) ) |
| 2 |
|
eflgam |
⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( exp ‘ ( log Γ ‘ 𝐴 ) ) = ( Γ ‘ 𝐴 ) ) |
| 3 |
1 2
|
syl |
⊢ ( 𝜑 → ( exp ‘ ( log Γ ‘ 𝐴 ) ) = ( Γ ‘ 𝐴 ) ) |
| 4 |
|
oveq1 |
⊢ ( 𝑧 = 𝐴 → ( 𝑧 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) = ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) ) |
| 5 |
|
oveq1 |
⊢ ( 𝑧 = 𝐴 → ( 𝑧 / 𝑘 ) = ( 𝐴 / 𝑘 ) ) |
| 6 |
5
|
fvoveq1d |
⊢ ( 𝑧 = 𝐴 → ( log ‘ ( ( 𝑧 / 𝑘 ) + 1 ) ) = ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) |
| 7 |
4 6
|
oveq12d |
⊢ ( 𝑧 = 𝐴 → ( ( 𝑧 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝑧 / 𝑘 ) + 1 ) ) ) = ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) |
| 8 |
7
|
sumeq2sdv |
⊢ ( 𝑧 = 𝐴 → Σ 𝑘 ∈ ℕ ( ( 𝑧 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝑧 / 𝑘 ) + 1 ) ) ) = Σ 𝑘 ∈ ℕ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) |
| 9 |
|
fveq2 |
⊢ ( 𝑧 = 𝐴 → ( log ‘ 𝑧 ) = ( log ‘ 𝐴 ) ) |
| 10 |
8 9
|
oveq12d |
⊢ ( 𝑧 = 𝐴 → ( Σ 𝑘 ∈ ℕ ( ( 𝑧 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝑧 / 𝑘 ) + 1 ) ) ) − ( log ‘ 𝑧 ) ) = ( Σ 𝑘 ∈ ℕ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) − ( log ‘ 𝐴 ) ) ) |
| 11 |
|
df-lgam |
⊢ log Γ = ( 𝑧 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) ↦ ( Σ 𝑘 ∈ ℕ ( ( 𝑧 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝑧 / 𝑘 ) + 1 ) ) ) − ( log ‘ 𝑧 ) ) ) |
| 12 |
|
ovex |
⊢ ( Σ 𝑘 ∈ ℕ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) − ( log ‘ 𝐴 ) ) ∈ V |
| 13 |
10 11 12
|
fvmpt |
⊢ ( 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) → ( log Γ ‘ 𝐴 ) = ( Σ 𝑘 ∈ ℕ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) − ( log ‘ 𝐴 ) ) ) |
| 14 |
1 13
|
syl |
⊢ ( 𝜑 → ( log Γ ‘ 𝐴 ) = ( Σ 𝑘 ∈ ℕ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) − ( log ‘ 𝐴 ) ) ) |
| 15 |
14
|
fveq2d |
⊢ ( 𝜑 → ( exp ‘ ( log Γ ‘ 𝐴 ) ) = ( exp ‘ ( Σ 𝑘 ∈ ℕ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) − ( log ‘ 𝐴 ) ) ) ) |
| 16 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
| 17 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
| 18 |
|
oveq1 |
⊢ ( 𝑗 = 𝑘 → ( 𝑗 + 1 ) = ( 𝑘 + 1 ) ) |
| 19 |
|
id |
⊢ ( 𝑗 = 𝑘 → 𝑗 = 𝑘 ) |
| 20 |
18 19
|
oveq12d |
⊢ ( 𝑗 = 𝑘 → ( ( 𝑗 + 1 ) / 𝑗 ) = ( ( 𝑘 + 1 ) / 𝑘 ) ) |
| 21 |
20
|
fveq2d |
⊢ ( 𝑗 = 𝑘 → ( log ‘ ( ( 𝑗 + 1 ) / 𝑗 ) ) = ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) |
| 22 |
21
|
oveq2d |
⊢ ( 𝑗 = 𝑘 → ( 𝐴 · ( log ‘ ( ( 𝑗 + 1 ) / 𝑗 ) ) ) = ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) ) |
| 23 |
|
oveq2 |
⊢ ( 𝑗 = 𝑘 → ( 𝐴 / 𝑗 ) = ( 𝐴 / 𝑘 ) ) |
| 24 |
23
|
fvoveq1d |
⊢ ( 𝑗 = 𝑘 → ( log ‘ ( ( 𝐴 / 𝑗 ) + 1 ) ) = ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) |
| 25 |
22 24
|
oveq12d |
⊢ ( 𝑗 = 𝑘 → ( ( 𝐴 · ( log ‘ ( ( 𝑗 + 1 ) / 𝑗 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑗 ) + 1 ) ) ) = ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) |
| 26 |
|
eqid |
⊢ ( 𝑗 ∈ ℕ ↦ ( ( 𝐴 · ( log ‘ ( ( 𝑗 + 1 ) / 𝑗 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑗 ) + 1 ) ) ) ) = ( 𝑗 ∈ ℕ ↦ ( ( 𝐴 · ( log ‘ ( ( 𝑗 + 1 ) / 𝑗 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑗 ) + 1 ) ) ) ) |
| 27 |
|
ovex |
⊢ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ∈ V |
| 28 |
25 26 27
|
fvmpt |
⊢ ( 𝑘 ∈ ℕ → ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐴 · ( log ‘ ( ( 𝑗 + 1 ) / 𝑗 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑗 ) + 1 ) ) ) ) ‘ 𝑘 ) = ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) |
| 29 |
28
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑗 ∈ ℕ ↦ ( ( 𝐴 · ( log ‘ ( ( 𝑗 + 1 ) / 𝑗 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑗 ) + 1 ) ) ) ) ‘ 𝑘 ) = ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) |
| 30 |
1
|
eldifad |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
| 31 |
30
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝐴 ∈ ℂ ) |
| 32 |
|
peano2nn |
⊢ ( 𝑘 ∈ ℕ → ( 𝑘 + 1 ) ∈ ℕ ) |
| 33 |
32
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑘 + 1 ) ∈ ℕ ) |
| 34 |
33
|
nncnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑘 + 1 ) ∈ ℂ ) |
| 35 |
|
nncn |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℂ ) |
| 36 |
35
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℂ ) |
| 37 |
|
nnne0 |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ≠ 0 ) |
| 38 |
37
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝑘 ≠ 0 ) |
| 39 |
34 36 38
|
divcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑘 + 1 ) / 𝑘 ) ∈ ℂ ) |
| 40 |
33
|
nnne0d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑘 + 1 ) ≠ 0 ) |
| 41 |
34 36 40 38
|
divne0d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑘 + 1 ) / 𝑘 ) ≠ 0 ) |
| 42 |
39 41
|
logcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ∈ ℂ ) |
| 43 |
31 42
|
mulcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) ∈ ℂ ) |
| 44 |
31 36 38
|
divcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐴 / 𝑘 ) ∈ ℂ ) |
| 45 |
|
1cnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 1 ∈ ℂ ) |
| 46 |
44 45
|
addcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝐴 / 𝑘 ) + 1 ) ∈ ℂ ) |
| 47 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝐴 ∈ ( ℂ ∖ ( ℤ ∖ ℕ ) ) ) |
| 48 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℕ ) |
| 49 |
47 48
|
dmgmdivn0 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝐴 / 𝑘 ) + 1 ) ≠ 0 ) |
| 50 |
46 49
|
logcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ∈ ℂ ) |
| 51 |
43 50
|
subcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ∈ ℂ ) |
| 52 |
26 1
|
lgamcvg |
⊢ ( 𝜑 → seq 1 ( + , ( 𝑗 ∈ ℕ ↦ ( ( 𝐴 · ( log ‘ ( ( 𝑗 + 1 ) / 𝑗 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑗 ) + 1 ) ) ) ) ) ⇝ ( ( log Γ ‘ 𝐴 ) + ( log ‘ 𝐴 ) ) ) |
| 53 |
|
seqex |
⊢ seq 1 ( + , ( 𝑗 ∈ ℕ ↦ ( ( 𝐴 · ( log ‘ ( ( 𝑗 + 1 ) / 𝑗 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑗 ) + 1 ) ) ) ) ) ∈ V |
| 54 |
|
ovex |
⊢ ( ( log Γ ‘ 𝐴 ) + ( log ‘ 𝐴 ) ) ∈ V |
| 55 |
53 54
|
breldm |
⊢ ( seq 1 ( + , ( 𝑗 ∈ ℕ ↦ ( ( 𝐴 · ( log ‘ ( ( 𝑗 + 1 ) / 𝑗 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑗 ) + 1 ) ) ) ) ) ⇝ ( ( log Γ ‘ 𝐴 ) + ( log ‘ 𝐴 ) ) → seq 1 ( + , ( 𝑗 ∈ ℕ ↦ ( ( 𝐴 · ( log ‘ ( ( 𝑗 + 1 ) / 𝑗 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑗 ) + 1 ) ) ) ) ) ∈ dom ⇝ ) |
| 56 |
52 55
|
syl |
⊢ ( 𝜑 → seq 1 ( + , ( 𝑗 ∈ ℕ ↦ ( ( 𝐴 · ( log ‘ ( ( 𝑗 + 1 ) / 𝑗 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑗 ) + 1 ) ) ) ) ) ∈ dom ⇝ ) |
| 57 |
16 17 29 51 56
|
isumcl |
⊢ ( 𝜑 → Σ 𝑘 ∈ ℕ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ∈ ℂ ) |
| 58 |
1
|
dmgmn0 |
⊢ ( 𝜑 → 𝐴 ≠ 0 ) |
| 59 |
30 58
|
logcld |
⊢ ( 𝜑 → ( log ‘ 𝐴 ) ∈ ℂ ) |
| 60 |
|
efsub |
⊢ ( ( Σ 𝑘 ∈ ℕ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ∈ ℂ ∧ ( log ‘ 𝐴 ) ∈ ℂ ) → ( exp ‘ ( Σ 𝑘 ∈ ℕ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) − ( log ‘ 𝐴 ) ) ) = ( ( exp ‘ Σ 𝑘 ∈ ℕ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) / ( exp ‘ ( log ‘ 𝐴 ) ) ) ) |
| 61 |
57 59 60
|
syl2anc |
⊢ ( 𝜑 → ( exp ‘ ( Σ 𝑘 ∈ ℕ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) − ( log ‘ 𝐴 ) ) ) = ( ( exp ‘ Σ 𝑘 ∈ ℕ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) / ( exp ‘ ( log ‘ 𝐴 ) ) ) ) |
| 62 |
16 17 29 51 56
|
iprodefisum |
⊢ ( 𝜑 → ∏ 𝑘 ∈ ℕ ( exp ‘ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) = ( exp ‘ Σ 𝑘 ∈ ℕ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) ) |
| 63 |
|
efsub |
⊢ ( ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) ∈ ℂ ∧ ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ∈ ℂ ) → ( exp ‘ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) = ( ( exp ‘ ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) ) / ( exp ‘ ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) ) |
| 64 |
43 50 63
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( exp ‘ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) = ( ( exp ‘ ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) ) / ( exp ‘ ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) ) |
| 65 |
36 45 36 38
|
divdird |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑘 + 1 ) / 𝑘 ) = ( ( 𝑘 / 𝑘 ) + ( 1 / 𝑘 ) ) ) |
| 66 |
36 38
|
dividd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝑘 / 𝑘 ) = 1 ) |
| 67 |
66
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑘 / 𝑘 ) + ( 1 / 𝑘 ) ) = ( 1 + ( 1 / 𝑘 ) ) ) |
| 68 |
65 67
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝑘 + 1 ) / 𝑘 ) = ( 1 + ( 1 / 𝑘 ) ) ) |
| 69 |
68
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) = ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ) |
| 70 |
69
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) = ( 𝐴 · ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ) ) |
| 71 |
70
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( exp ‘ ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) ) = ( exp ‘ ( 𝐴 · ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ) ) ) |
| 72 |
|
1rp |
⊢ 1 ∈ ℝ+ |
| 73 |
72
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 1 ∈ ℝ+ ) |
| 74 |
48
|
nnrpd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℝ+ ) |
| 75 |
74
|
rpreccld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 1 / 𝑘 ) ∈ ℝ+ ) |
| 76 |
73 75
|
rpaddcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 1 + ( 1 / 𝑘 ) ) ∈ ℝ+ ) |
| 77 |
76
|
rpcnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 1 + ( 1 / 𝑘 ) ) ∈ ℂ ) |
| 78 |
76
|
rpne0d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( 1 + ( 1 / 𝑘 ) ) ≠ 0 ) |
| 79 |
77 78 31
|
cxpefd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 1 + ( 1 / 𝑘 ) ) ↑𝑐 𝐴 ) = ( exp ‘ ( 𝐴 · ( log ‘ ( 1 + ( 1 / 𝑘 ) ) ) ) ) ) |
| 80 |
71 79
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( exp ‘ ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) ) = ( ( 1 + ( 1 / 𝑘 ) ) ↑𝑐 𝐴 ) ) |
| 81 |
|
eflog |
⊢ ( ( ( ( 𝐴 / 𝑘 ) + 1 ) ∈ ℂ ∧ ( ( 𝐴 / 𝑘 ) + 1 ) ≠ 0 ) → ( exp ‘ ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) = ( ( 𝐴 / 𝑘 ) + 1 ) ) |
| 82 |
46 49 81
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( exp ‘ ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) = ( ( 𝐴 / 𝑘 ) + 1 ) ) |
| 83 |
44 45
|
addcomd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( 𝐴 / 𝑘 ) + 1 ) = ( 1 + ( 𝐴 / 𝑘 ) ) ) |
| 84 |
82 83
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( exp ‘ ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) = ( 1 + ( 𝐴 / 𝑘 ) ) ) |
| 85 |
80 84
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( ( exp ‘ ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) ) / ( exp ‘ ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) = ( ( ( 1 + ( 1 / 𝑘 ) ) ↑𝑐 𝐴 ) / ( 1 + ( 𝐴 / 𝑘 ) ) ) ) |
| 86 |
64 85
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ ) → ( exp ‘ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) = ( ( ( 1 + ( 1 / 𝑘 ) ) ↑𝑐 𝐴 ) / ( 1 + ( 𝐴 / 𝑘 ) ) ) ) |
| 87 |
86
|
prodeq2dv |
⊢ ( 𝜑 → ∏ 𝑘 ∈ ℕ ( exp ‘ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) = ∏ 𝑘 ∈ ℕ ( ( ( 1 + ( 1 / 𝑘 ) ) ↑𝑐 𝐴 ) / ( 1 + ( 𝐴 / 𝑘 ) ) ) ) |
| 88 |
62 87
|
eqtr3d |
⊢ ( 𝜑 → ( exp ‘ Σ 𝑘 ∈ ℕ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) = ∏ 𝑘 ∈ ℕ ( ( ( 1 + ( 1 / 𝑘 ) ) ↑𝑐 𝐴 ) / ( 1 + ( 𝐴 / 𝑘 ) ) ) ) |
| 89 |
|
eflog |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( exp ‘ ( log ‘ 𝐴 ) ) = 𝐴 ) |
| 90 |
30 58 89
|
syl2anc |
⊢ ( 𝜑 → ( exp ‘ ( log ‘ 𝐴 ) ) = 𝐴 ) |
| 91 |
88 90
|
oveq12d |
⊢ ( 𝜑 → ( ( exp ‘ Σ 𝑘 ∈ ℕ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) ) / ( exp ‘ ( log ‘ 𝐴 ) ) ) = ( ∏ 𝑘 ∈ ℕ ( ( ( 1 + ( 1 / 𝑘 ) ) ↑𝑐 𝐴 ) / ( 1 + ( 𝐴 / 𝑘 ) ) ) / 𝐴 ) ) |
| 92 |
61 91
|
eqtrd |
⊢ ( 𝜑 → ( exp ‘ ( Σ 𝑘 ∈ ℕ ( ( 𝐴 · ( log ‘ ( ( 𝑘 + 1 ) / 𝑘 ) ) ) − ( log ‘ ( ( 𝐴 / 𝑘 ) + 1 ) ) ) − ( log ‘ 𝐴 ) ) ) = ( ∏ 𝑘 ∈ ℕ ( ( ( 1 + ( 1 / 𝑘 ) ) ↑𝑐 𝐴 ) / ( 1 + ( 𝐴 / 𝑘 ) ) ) / 𝐴 ) ) |
| 93 |
15 92
|
eqtrd |
⊢ ( 𝜑 → ( exp ‘ ( log Γ ‘ 𝐴 ) ) = ( ∏ 𝑘 ∈ ℕ ( ( ( 1 + ( 1 / 𝑘 ) ) ↑𝑐 𝐴 ) / ( 1 + ( 𝐴 / 𝑘 ) ) ) / 𝐴 ) ) |
| 94 |
3 93
|
eqtr3d |
⊢ ( 𝜑 → ( Γ ‘ 𝐴 ) = ( ∏ 𝑘 ∈ ℕ ( ( ( 1 + ( 1 / 𝑘 ) ) ↑𝑐 𝐴 ) / ( 1 + ( 𝐴 / 𝑘 ) ) ) / 𝐴 ) ) |