Step |
Hyp |
Ref |
Expression |
1 |
|
etransclem44.a |
⊢ ( 𝜑 → 𝐴 : ℕ0 ⟶ ℤ ) |
2 |
|
etransclem44.a0 |
⊢ ( 𝜑 → ( 𝐴 ‘ 0 ) ≠ 0 ) |
3 |
|
etransclem44.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ0 ) |
4 |
|
etransclem44.p |
⊢ ( 𝜑 → 𝑃 ∈ ℙ ) |
5 |
|
etransclem44.ap |
⊢ ( 𝜑 → ( abs ‘ ( 𝐴 ‘ 0 ) ) < 𝑃 ) |
6 |
|
etransclem44.mp |
⊢ ( 𝜑 → ( ! ‘ 𝑀 ) < 𝑃 ) |
7 |
|
etransclem44.f |
⊢ 𝐹 = ( 𝑥 ∈ ℝ ↦ ( ( 𝑥 ↑ ( 𝑃 − 1 ) ) · ∏ 𝑗 ∈ ( 1 ... 𝑀 ) ( ( 𝑥 − 𝑗 ) ↑ 𝑃 ) ) ) |
8 |
|
etransclem44.k |
⊢ 𝐾 = ( Σ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) |
9 |
8
|
a1i |
⊢ ( 𝜑 → 𝐾 = ( Σ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) |
10 |
|
nfv |
⊢ Ⅎ 𝑘 𝜑 |
11 |
|
nfcv |
⊢ Ⅎ 𝑘 ( ( 𝐴 ‘ 0 ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ) |
12 |
|
fzfi |
⊢ ( 0 ... 𝑀 ) ∈ Fin |
13 |
|
fzfi |
⊢ ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ∈ Fin |
14 |
|
xpfi |
⊢ ( ( ( 0 ... 𝑀 ) ∈ Fin ∧ ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ∈ Fin ) → ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∈ Fin ) |
15 |
12 13 14
|
mp2an |
⊢ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∈ Fin |
16 |
15
|
a1i |
⊢ ( 𝜑 → ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∈ Fin ) |
17 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) → 𝐴 : ℕ0 ⟶ ℤ ) |
18 |
|
fzssnn0 |
⊢ ( 0 ... 𝑀 ) ⊆ ℕ0 |
19 |
|
xp1st |
⊢ ( 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) → ( 1st ‘ 𝑘 ) ∈ ( 0 ... 𝑀 ) ) |
20 |
18 19
|
sseldi |
⊢ ( 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) → ( 1st ‘ 𝑘 ) ∈ ℕ0 ) |
21 |
20
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) → ( 1st ‘ 𝑘 ) ∈ ℕ0 ) |
22 |
17 21
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) → ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) ∈ ℤ ) |
23 |
|
reelprrecn |
⊢ ℝ ∈ { ℝ , ℂ } |
24 |
23
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) → ℝ ∈ { ℝ , ℂ } ) |
25 |
|
reopn |
⊢ ℝ ∈ ( topGen ‘ ran (,) ) |
26 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
27 |
26
|
tgioo2 |
⊢ ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) |
28 |
25 27
|
eleqtri |
⊢ ℝ ∈ ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) |
29 |
28
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) → ℝ ∈ ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) ) |
30 |
|
prmnn |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℕ ) |
31 |
4 30
|
syl |
⊢ ( 𝜑 → 𝑃 ∈ ℕ ) |
32 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) → 𝑃 ∈ ℕ ) |
33 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) → 𝑀 ∈ ℕ0 ) |
34 |
|
xp2nd |
⊢ ( 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) → ( 2nd ‘ 𝑘 ) ∈ ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) |
35 |
|
elfznn0 |
⊢ ( ( 2nd ‘ 𝑘 ) ∈ ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) → ( 2nd ‘ 𝑘 ) ∈ ℕ0 ) |
36 |
34 35
|
syl |
⊢ ( 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) → ( 2nd ‘ 𝑘 ) ∈ ℕ0 ) |
37 |
36
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) → ( 2nd ‘ 𝑘 ) ∈ ℕ0 ) |
38 |
21
|
nn0red |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) → ( 1st ‘ 𝑘 ) ∈ ℝ ) |
39 |
21
|
nn0zd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) → ( 1st ‘ 𝑘 ) ∈ ℤ ) |
40 |
24 29 32 33 7 37 38 39
|
etransclem42 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) → ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ∈ ℤ ) |
41 |
22 40
|
zmulcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) → ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) ∈ ℤ ) |
42 |
41
|
zcnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) → ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) ∈ ℂ ) |
43 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
44 |
3 43
|
eleqtrdi |
⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ 0 ) ) |
45 |
|
eluzfz1 |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 0 ) → 0 ∈ ( 0 ... 𝑀 ) ) |
46 |
44 45
|
syl |
⊢ ( 𝜑 → 0 ∈ ( 0 ... 𝑀 ) ) |
47 |
|
0zd |
⊢ ( 𝜑 → 0 ∈ ℤ ) |
48 |
3
|
nn0zd |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
49 |
31
|
nnzd |
⊢ ( 𝜑 → 𝑃 ∈ ℤ ) |
50 |
48 49
|
zmulcld |
⊢ ( 𝜑 → ( 𝑀 · 𝑃 ) ∈ ℤ ) |
51 |
|
nnm1nn0 |
⊢ ( 𝑃 ∈ ℕ → ( 𝑃 − 1 ) ∈ ℕ0 ) |
52 |
31 51
|
syl |
⊢ ( 𝜑 → ( 𝑃 − 1 ) ∈ ℕ0 ) |
53 |
52
|
nn0zd |
⊢ ( 𝜑 → ( 𝑃 − 1 ) ∈ ℤ ) |
54 |
50 53
|
zaddcld |
⊢ ( 𝜑 → ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ∈ ℤ ) |
55 |
52
|
nn0ge0d |
⊢ ( 𝜑 → 0 ≤ ( 𝑃 − 1 ) ) |
56 |
31
|
nnnn0d |
⊢ ( 𝜑 → 𝑃 ∈ ℕ0 ) |
57 |
3 56
|
nn0mulcld |
⊢ ( 𝜑 → ( 𝑀 · 𝑃 ) ∈ ℕ0 ) |
58 |
57
|
nn0ge0d |
⊢ ( 𝜑 → 0 ≤ ( 𝑀 · 𝑃 ) ) |
59 |
52
|
nn0red |
⊢ ( 𝜑 → ( 𝑃 − 1 ) ∈ ℝ ) |
60 |
50
|
zred |
⊢ ( 𝜑 → ( 𝑀 · 𝑃 ) ∈ ℝ ) |
61 |
59 60
|
addge02d |
⊢ ( 𝜑 → ( 0 ≤ ( 𝑀 · 𝑃 ) ↔ ( 𝑃 − 1 ) ≤ ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) |
62 |
58 61
|
mpbid |
⊢ ( 𝜑 → ( 𝑃 − 1 ) ≤ ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) |
63 |
47 54 53 55 62
|
elfzd |
⊢ ( 𝜑 → ( 𝑃 − 1 ) ∈ ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) |
64 |
|
opelxp |
⊢ ( 〈 0 , ( 𝑃 − 1 ) 〉 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ↔ ( 0 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑃 − 1 ) ∈ ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) |
65 |
46 63 64
|
sylanbrc |
⊢ ( 𝜑 → 〈 0 , ( 𝑃 − 1 ) 〉 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) |
66 |
|
fveq2 |
⊢ ( 𝑘 = 〈 0 , ( 𝑃 − 1 ) 〉 → ( 1st ‘ 𝑘 ) = ( 1st ‘ 〈 0 , ( 𝑃 − 1 ) 〉 ) ) |
67 |
|
0re |
⊢ 0 ∈ ℝ |
68 |
|
ovex |
⊢ ( 𝑃 − 1 ) ∈ V |
69 |
|
op1stg |
⊢ ( ( 0 ∈ ℝ ∧ ( 𝑃 − 1 ) ∈ V ) → ( 1st ‘ 〈 0 , ( 𝑃 − 1 ) 〉 ) = 0 ) |
70 |
67 68 69
|
mp2an |
⊢ ( 1st ‘ 〈 0 , ( 𝑃 − 1 ) 〉 ) = 0 |
71 |
66 70
|
eqtrdi |
⊢ ( 𝑘 = 〈 0 , ( 𝑃 − 1 ) 〉 → ( 1st ‘ 𝑘 ) = 0 ) |
72 |
71
|
fveq2d |
⊢ ( 𝑘 = 〈 0 , ( 𝑃 − 1 ) 〉 → ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) = ( 𝐴 ‘ 0 ) ) |
73 |
|
fveq2 |
⊢ ( 𝑘 = 〈 0 , ( 𝑃 − 1 ) 〉 → ( 2nd ‘ 𝑘 ) = ( 2nd ‘ 〈 0 , ( 𝑃 − 1 ) 〉 ) ) |
74 |
|
op2ndg |
⊢ ( ( 0 ∈ ℝ ∧ ( 𝑃 − 1 ) ∈ V ) → ( 2nd ‘ 〈 0 , ( 𝑃 − 1 ) 〉 ) = ( 𝑃 − 1 ) ) |
75 |
67 68 74
|
mp2an |
⊢ ( 2nd ‘ 〈 0 , ( 𝑃 − 1 ) 〉 ) = ( 𝑃 − 1 ) |
76 |
73 75
|
eqtrdi |
⊢ ( 𝑘 = 〈 0 , ( 𝑃 − 1 ) 〉 → ( 2nd ‘ 𝑘 ) = ( 𝑃 − 1 ) ) |
77 |
76
|
fveq2d |
⊢ ( 𝑘 = 〈 0 , ( 𝑃 − 1 ) 〉 → ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) = ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ) |
78 |
77 71
|
fveq12d |
⊢ ( 𝑘 = 〈 0 , ( 𝑃 − 1 ) 〉 → ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) = ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ) |
79 |
72 78
|
oveq12d |
⊢ ( 𝑘 = 〈 0 , ( 𝑃 − 1 ) 〉 → ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) = ( ( 𝐴 ‘ 0 ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ) ) |
80 |
10 11 16 42 65 79
|
fsumsplit1 |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) = ( ( ( 𝐴 ‘ 0 ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ) + Σ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) ) ) |
81 |
80
|
oveq1d |
⊢ ( 𝜑 → ( Σ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) = ( ( ( ( 𝐴 ‘ 0 ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ) + Σ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) |
82 |
18 46
|
sseldi |
⊢ ( 𝜑 → 0 ∈ ℕ0 ) |
83 |
1 82
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐴 ‘ 0 ) ∈ ℤ ) |
84 |
23
|
a1i |
⊢ ( 𝜑 → ℝ ∈ { ℝ , ℂ } ) |
85 |
28
|
a1i |
⊢ ( 𝜑 → ℝ ∈ ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) ) |
86 |
67
|
a1i |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
87 |
84 85 31 3 7 52 86 47
|
etransclem42 |
⊢ ( 𝜑 → ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ∈ ℤ ) |
88 |
83 87
|
zmulcld |
⊢ ( 𝜑 → ( ( 𝐴 ‘ 0 ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ) ∈ ℤ ) |
89 |
88
|
zcnd |
⊢ ( 𝜑 → ( ( 𝐴 ‘ 0 ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ) ∈ ℂ ) |
90 |
|
difss |
⊢ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ⊆ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) |
91 |
|
ssfi |
⊢ ( ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∈ Fin ∧ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ⊆ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) → ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ∈ Fin ) |
92 |
15 90 91
|
mp2an |
⊢ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ∈ Fin |
93 |
92
|
a1i |
⊢ ( 𝜑 → ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ∈ Fin ) |
94 |
|
eldifi |
⊢ ( 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) → 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) |
95 |
94 41
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) ∈ ℤ ) |
96 |
93 95
|
fsumzcl |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) ∈ ℤ ) |
97 |
96
|
zcnd |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) ∈ ℂ ) |
98 |
52
|
faccld |
⊢ ( 𝜑 → ( ! ‘ ( 𝑃 − 1 ) ) ∈ ℕ ) |
99 |
98
|
nncnd |
⊢ ( 𝜑 → ( ! ‘ ( 𝑃 − 1 ) ) ∈ ℂ ) |
100 |
98
|
nnne0d |
⊢ ( 𝜑 → ( ! ‘ ( 𝑃 − 1 ) ) ≠ 0 ) |
101 |
89 97 99 100
|
divdird |
⊢ ( 𝜑 → ( ( ( ( 𝐴 ‘ 0 ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ) + Σ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) = ( ( ( ( 𝐴 ‘ 0 ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) + ( Σ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ) |
102 |
9 81 101
|
3eqtrd |
⊢ ( 𝜑 → 𝐾 = ( ( ( ( 𝐴 ‘ 0 ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) + ( Σ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ) |
103 |
31
|
nnne0d |
⊢ ( 𝜑 → 𝑃 ≠ 0 ) |
104 |
83
|
zcnd |
⊢ ( 𝜑 → ( 𝐴 ‘ 0 ) ∈ ℂ ) |
105 |
87
|
zcnd |
⊢ ( 𝜑 → ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ∈ ℂ ) |
106 |
104 105 99 100
|
divassd |
⊢ ( 𝜑 → ( ( ( 𝐴 ‘ 0 ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) = ( ( 𝐴 ‘ 0 ) · ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ) |
107 |
|
etransclem5 |
⊢ ( 𝑘 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑦 ∈ ℝ ↦ ( ( 𝑦 − 𝑘 ) ↑ if ( 𝑘 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) = ( 𝑗 ∈ ( 0 ... 𝑀 ) ↦ ( 𝑥 ∈ ℝ ↦ ( ( 𝑥 − 𝑗 ) ↑ if ( 𝑗 = 0 , ( 𝑃 − 1 ) , 𝑃 ) ) ) ) |
108 |
|
etransclem11 |
⊢ ( 𝑚 ∈ ℕ0 ↦ { 𝑑 ∈ ( ( 0 ... 𝑚 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( 𝑑 ‘ 𝑘 ) = 𝑚 } ) = ( 𝑛 ∈ ℕ0 ↦ { 𝑐 ∈ ( ( 0 ... 𝑛 ) ↑m ( 0 ... 𝑀 ) ) ∣ Σ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑐 ‘ 𝑗 ) = 𝑛 } ) |
109 |
84 85 31 3 7 52 107 108 46 86
|
etransclem37 |
⊢ ( 𝜑 → ( ! ‘ ( 𝑃 − 1 ) ) ∥ ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ) |
110 |
98
|
nnzd |
⊢ ( 𝜑 → ( ! ‘ ( 𝑃 − 1 ) ) ∈ ℤ ) |
111 |
|
dvdsval2 |
⊢ ( ( ( ! ‘ ( 𝑃 − 1 ) ) ∈ ℤ ∧ ( ! ‘ ( 𝑃 − 1 ) ) ≠ 0 ∧ ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ∈ ℤ ) → ( ( ! ‘ ( 𝑃 − 1 ) ) ∥ ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ↔ ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ∈ ℤ ) ) |
112 |
110 100 87 111
|
syl3anc |
⊢ ( 𝜑 → ( ( ! ‘ ( 𝑃 − 1 ) ) ∥ ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ↔ ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ∈ ℤ ) ) |
113 |
109 112
|
mpbid |
⊢ ( 𝜑 → ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ∈ ℤ ) |
114 |
83 113
|
zmulcld |
⊢ ( 𝜑 → ( ( 𝐴 ‘ 0 ) · ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ∈ ℤ ) |
115 |
106 114
|
eqeltrd |
⊢ ( 𝜑 → ( ( ( 𝐴 ‘ 0 ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ∈ ℤ ) |
116 |
94 42
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) ∈ ℂ ) |
117 |
93 99 116 100
|
fsumdivc |
⊢ ( 𝜑 → ( Σ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) = Σ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ( ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) |
118 |
22
|
zcnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) → ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) ∈ ℂ ) |
119 |
94 118
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) ∈ ℂ ) |
120 |
94 40
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ∈ ℤ ) |
121 |
120
|
zcnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ∈ ℂ ) |
122 |
99
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( ! ‘ ( 𝑃 − 1 ) ) ∈ ℂ ) |
123 |
100
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( ! ‘ ( 𝑃 − 1 ) ) ≠ 0 ) |
124 |
119 121 122 123
|
divassd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) = ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ) |
125 |
94 22
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) ∈ ℤ ) |
126 |
23
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ℝ ∈ { ℝ , ℂ } ) |
127 |
28
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ℝ ∈ ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) ) |
128 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → 𝑃 ∈ ℕ ) |
129 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → 𝑀 ∈ ℕ0 ) |
130 |
94
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) |
131 |
130 36
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( 2nd ‘ 𝑘 ) ∈ ℕ0 ) |
132 |
130 19
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( 1st ‘ 𝑘 ) ∈ ( 0 ... 𝑀 ) ) |
133 |
94 38
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( 1st ‘ 𝑘 ) ∈ ℝ ) |
134 |
126 127 128 129 7 131 107 108 132 133
|
etransclem37 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( ! ‘ ( 𝑃 − 1 ) ) ∥ ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) |
135 |
110
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( ! ‘ ( 𝑃 − 1 ) ) ∈ ℤ ) |
136 |
|
dvdsval2 |
⊢ ( ( ( ! ‘ ( 𝑃 − 1 ) ) ∈ ℤ ∧ ( ! ‘ ( 𝑃 − 1 ) ) ≠ 0 ∧ ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ∈ ℤ ) → ( ( ! ‘ ( 𝑃 − 1 ) ) ∥ ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ↔ ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ∈ ℤ ) ) |
137 |
135 123 120 136
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( ( ! ‘ ( 𝑃 − 1 ) ) ∥ ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ↔ ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ∈ ℤ ) ) |
138 |
134 137
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ∈ ℤ ) |
139 |
125 138
|
zmulcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ∈ ℤ ) |
140 |
124 139
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ∈ ℤ ) |
141 |
93 140
|
fsumzcl |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ( ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ∈ ℤ ) |
142 |
117 141
|
eqeltrd |
⊢ ( 𝜑 → ( Σ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ∈ ℤ ) |
143 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
144 |
|
zabscl |
⊢ ( ( 𝐴 ‘ 0 ) ∈ ℤ → ( abs ‘ ( 𝐴 ‘ 0 ) ) ∈ ℤ ) |
145 |
83 144
|
syl |
⊢ ( 𝜑 → ( abs ‘ ( 𝐴 ‘ 0 ) ) ∈ ℤ ) |
146 |
|
nn0abscl |
⊢ ( ( 𝐴 ‘ 0 ) ∈ ℤ → ( abs ‘ ( 𝐴 ‘ 0 ) ) ∈ ℕ0 ) |
147 |
83 146
|
syl |
⊢ ( 𝜑 → ( abs ‘ ( 𝐴 ‘ 0 ) ) ∈ ℕ0 ) |
148 |
104 2
|
absne0d |
⊢ ( 𝜑 → ( abs ‘ ( 𝐴 ‘ 0 ) ) ≠ 0 ) |
149 |
|
elnnne0 |
⊢ ( ( abs ‘ ( 𝐴 ‘ 0 ) ) ∈ ℕ ↔ ( ( abs ‘ ( 𝐴 ‘ 0 ) ) ∈ ℕ0 ∧ ( abs ‘ ( 𝐴 ‘ 0 ) ) ≠ 0 ) ) |
150 |
147 148 149
|
sylanbrc |
⊢ ( 𝜑 → ( abs ‘ ( 𝐴 ‘ 0 ) ) ∈ ℕ ) |
151 |
150
|
nnge1d |
⊢ ( 𝜑 → 1 ≤ ( abs ‘ ( 𝐴 ‘ 0 ) ) ) |
152 |
|
zltlem1 |
⊢ ( ( ( abs ‘ ( 𝐴 ‘ 0 ) ) ∈ ℤ ∧ 𝑃 ∈ ℤ ) → ( ( abs ‘ ( 𝐴 ‘ 0 ) ) < 𝑃 ↔ ( abs ‘ ( 𝐴 ‘ 0 ) ) ≤ ( 𝑃 − 1 ) ) ) |
153 |
145 49 152
|
syl2anc |
⊢ ( 𝜑 → ( ( abs ‘ ( 𝐴 ‘ 0 ) ) < 𝑃 ↔ ( abs ‘ ( 𝐴 ‘ 0 ) ) ≤ ( 𝑃 − 1 ) ) ) |
154 |
5 153
|
mpbid |
⊢ ( 𝜑 → ( abs ‘ ( 𝐴 ‘ 0 ) ) ≤ ( 𝑃 − 1 ) ) |
155 |
143 53 145 151 154
|
elfzd |
⊢ ( 𝜑 → ( abs ‘ ( 𝐴 ‘ 0 ) ) ∈ ( 1 ... ( 𝑃 − 1 ) ) ) |
156 |
|
fzm1ndvds |
⊢ ( ( 𝑃 ∈ ℕ ∧ ( abs ‘ ( 𝐴 ‘ 0 ) ) ∈ ( 1 ... ( 𝑃 − 1 ) ) ) → ¬ 𝑃 ∥ ( abs ‘ ( 𝐴 ‘ 0 ) ) ) |
157 |
31 155 156
|
syl2anc |
⊢ ( 𝜑 → ¬ 𝑃 ∥ ( abs ‘ ( 𝐴 ‘ 0 ) ) ) |
158 |
|
dvdsabsb |
⊢ ( ( 𝑃 ∈ ℤ ∧ ( 𝐴 ‘ 0 ) ∈ ℤ ) → ( 𝑃 ∥ ( 𝐴 ‘ 0 ) ↔ 𝑃 ∥ ( abs ‘ ( 𝐴 ‘ 0 ) ) ) ) |
159 |
49 83 158
|
syl2anc |
⊢ ( 𝜑 → ( 𝑃 ∥ ( 𝐴 ‘ 0 ) ↔ 𝑃 ∥ ( abs ‘ ( 𝐴 ‘ 0 ) ) ) ) |
160 |
157 159
|
mtbird |
⊢ ( 𝜑 → ¬ 𝑃 ∥ ( 𝐴 ‘ 0 ) ) |
161 |
3 4 6 7
|
etransclem41 |
⊢ ( 𝜑 → ¬ 𝑃 ∥ ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) |
162 |
160 161
|
jca |
⊢ ( 𝜑 → ( ¬ 𝑃 ∥ ( 𝐴 ‘ 0 ) ∧ ¬ 𝑃 ∥ ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ) |
163 |
|
pm4.56 |
⊢ ( ( ¬ 𝑃 ∥ ( 𝐴 ‘ 0 ) ∧ ¬ 𝑃 ∥ ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ↔ ¬ ( 𝑃 ∥ ( 𝐴 ‘ 0 ) ∨ 𝑃 ∥ ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ) |
164 |
162 163
|
sylib |
⊢ ( 𝜑 → ¬ ( 𝑃 ∥ ( 𝐴 ‘ 0 ) ∨ 𝑃 ∥ ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ) |
165 |
|
euclemma |
⊢ ( ( 𝑃 ∈ ℙ ∧ ( 𝐴 ‘ 0 ) ∈ ℤ ∧ ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ∈ ℤ ) → ( 𝑃 ∥ ( ( 𝐴 ‘ 0 ) · ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ↔ ( 𝑃 ∥ ( 𝐴 ‘ 0 ) ∨ 𝑃 ∥ ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ) ) |
166 |
4 83 113 165
|
syl3anc |
⊢ ( 𝜑 → ( 𝑃 ∥ ( ( 𝐴 ‘ 0 ) · ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ↔ ( 𝑃 ∥ ( 𝐴 ‘ 0 ) ∨ 𝑃 ∥ ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ) ) |
167 |
164 166
|
mtbird |
⊢ ( 𝜑 → ¬ 𝑃 ∥ ( ( 𝐴 ‘ 0 ) · ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ) |
168 |
106
|
breq2d |
⊢ ( 𝜑 → ( 𝑃 ∥ ( ( ( 𝐴 ‘ 0 ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ↔ 𝑃 ∥ ( ( 𝐴 ‘ 0 ) · ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ) ) |
169 |
167 168
|
mtbird |
⊢ ( 𝜑 → ¬ 𝑃 ∥ ( ( ( 𝐴 ‘ 0 ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) |
170 |
49
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → 𝑃 ∈ ℤ ) |
171 |
170 125 138
|
3jca |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ( 𝑃 ∈ ℤ ∧ ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) ∈ ℤ ∧ ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ∈ ℤ ) ) |
172 |
|
eldifn |
⊢ ( 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) → ¬ 𝑘 ∈ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) |
173 |
94
|
adantr |
⊢ ( ( 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ∧ ( ( 2nd ‘ 𝑘 ) = ( 𝑃 − 1 ) ∧ ( 1st ‘ 𝑘 ) = 0 ) ) → 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ) |
174 |
|
1st2nd2 |
⊢ ( 𝑘 ∈ ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) → 𝑘 = 〈 ( 1st ‘ 𝑘 ) , ( 2nd ‘ 𝑘 ) 〉 ) |
175 |
173 174
|
syl |
⊢ ( ( 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ∧ ( ( 2nd ‘ 𝑘 ) = ( 𝑃 − 1 ) ∧ ( 1st ‘ 𝑘 ) = 0 ) ) → 𝑘 = 〈 ( 1st ‘ 𝑘 ) , ( 2nd ‘ 𝑘 ) 〉 ) |
176 |
|
simpr |
⊢ ( ( ( 2nd ‘ 𝑘 ) = ( 𝑃 − 1 ) ∧ ( 1st ‘ 𝑘 ) = 0 ) → ( 1st ‘ 𝑘 ) = 0 ) |
177 |
|
simpl |
⊢ ( ( ( 2nd ‘ 𝑘 ) = ( 𝑃 − 1 ) ∧ ( 1st ‘ 𝑘 ) = 0 ) → ( 2nd ‘ 𝑘 ) = ( 𝑃 − 1 ) ) |
178 |
176 177
|
opeq12d |
⊢ ( ( ( 2nd ‘ 𝑘 ) = ( 𝑃 − 1 ) ∧ ( 1st ‘ 𝑘 ) = 0 ) → 〈 ( 1st ‘ 𝑘 ) , ( 2nd ‘ 𝑘 ) 〉 = 〈 0 , ( 𝑃 − 1 ) 〉 ) |
179 |
178
|
adantl |
⊢ ( ( 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ∧ ( ( 2nd ‘ 𝑘 ) = ( 𝑃 − 1 ) ∧ ( 1st ‘ 𝑘 ) = 0 ) ) → 〈 ( 1st ‘ 𝑘 ) , ( 2nd ‘ 𝑘 ) 〉 = 〈 0 , ( 𝑃 − 1 ) 〉 ) |
180 |
175 179
|
eqtrd |
⊢ ( ( 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ∧ ( ( 2nd ‘ 𝑘 ) = ( 𝑃 − 1 ) ∧ ( 1st ‘ 𝑘 ) = 0 ) ) → 𝑘 = 〈 0 , ( 𝑃 − 1 ) 〉 ) |
181 |
|
velsn |
⊢ ( 𝑘 ∈ { 〈 0 , ( 𝑃 − 1 ) 〉 } ↔ 𝑘 = 〈 0 , ( 𝑃 − 1 ) 〉 ) |
182 |
180 181
|
sylibr |
⊢ ( ( 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ∧ ( ( 2nd ‘ 𝑘 ) = ( 𝑃 − 1 ) ∧ ( 1st ‘ 𝑘 ) = 0 ) ) → 𝑘 ∈ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) |
183 |
172 182
|
mtand |
⊢ ( 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) → ¬ ( ( 2nd ‘ 𝑘 ) = ( 𝑃 − 1 ) ∧ ( 1st ‘ 𝑘 ) = 0 ) ) |
184 |
183
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → ¬ ( ( 2nd ‘ 𝑘 ) = ( 𝑃 − 1 ) ∧ ( 1st ‘ 𝑘 ) = 0 ) ) |
185 |
128 129 7 131 132 184 108
|
etransclem38 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → 𝑃 ∥ ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) |
186 |
|
dvdsmultr2 |
⊢ ( ( 𝑃 ∈ ℤ ∧ ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) ∈ ℤ ∧ ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ∈ ℤ ) → ( 𝑃 ∥ ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) → 𝑃 ∥ ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ) ) |
187 |
171 185 186
|
sylc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → 𝑃 ∥ ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ) |
188 |
187 124
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ) → 𝑃 ∥ ( ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) |
189 |
93 49 140 188
|
fsumdvds |
⊢ ( 𝜑 → 𝑃 ∥ Σ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ( ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) |
190 |
189 117
|
breqtrrd |
⊢ ( 𝜑 → 𝑃 ∥ ( Σ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) |
191 |
49 103 115 142 169 190
|
etransclem9 |
⊢ ( 𝜑 → ( ( ( ( 𝐴 ‘ 0 ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 𝑃 − 1 ) ) ‘ 0 ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) + ( Σ 𝑘 ∈ ( ( ( 0 ... 𝑀 ) × ( 0 ... ( ( 𝑀 · 𝑃 ) + ( 𝑃 − 1 ) ) ) ) ∖ { 〈 0 , ( 𝑃 − 1 ) 〉 } ) ( ( 𝐴 ‘ ( 1st ‘ 𝑘 ) ) · ( ( ( ℝ D𝑛 𝐹 ) ‘ ( 2nd ‘ 𝑘 ) ) ‘ ( 1st ‘ 𝑘 ) ) ) / ( ! ‘ ( 𝑃 − 1 ) ) ) ) ≠ 0 ) |
192 |
102 191
|
eqnetrd |
⊢ ( 𝜑 → 𝐾 ≠ 0 ) |