Step |
Hyp |
Ref |
Expression |
1 |
|
derang.d |
⊢ 𝐷 = ( 𝑥 ∈ Fin ↦ ( ♯ ‘ { 𝑓 ∣ ( 𝑓 : 𝑥 –1-1-onto→ 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) ≠ 𝑦 ) } ) ) |
2 |
|
subfac.n |
⊢ 𝑆 = ( 𝑛 ∈ ℕ0 ↦ ( 𝐷 ‘ ( 1 ... 𝑛 ) ) ) |
3 |
|
fveq2 |
⊢ ( 𝑥 = 0 → ( 𝑆 ‘ 𝑥 ) = ( 𝑆 ‘ 0 ) ) |
4 |
1 2
|
subfac0 |
⊢ ( 𝑆 ‘ 0 ) = 1 |
5 |
3 4
|
eqtrdi |
⊢ ( 𝑥 = 0 → ( 𝑆 ‘ 𝑥 ) = 1 ) |
6 |
|
fveq2 |
⊢ ( 𝑥 = 0 → ( ! ‘ 𝑥 ) = ( ! ‘ 0 ) ) |
7 |
|
fac0 |
⊢ ( ! ‘ 0 ) = 1 |
8 |
6 7
|
eqtrdi |
⊢ ( 𝑥 = 0 → ( ! ‘ 𝑥 ) = 1 ) |
9 |
|
oveq2 |
⊢ ( 𝑥 = 0 → ( 0 ... 𝑥 ) = ( 0 ... 0 ) ) |
10 |
9
|
sumeq1d |
⊢ ( 𝑥 = 0 → Σ 𝑘 ∈ ( 0 ... 𝑥 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... 0 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) |
11 |
8 10
|
oveq12d |
⊢ ( 𝑥 = 0 → ( ( ! ‘ 𝑥 ) · Σ 𝑘 ∈ ( 0 ... 𝑥 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( 1 · Σ 𝑘 ∈ ( 0 ... 0 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) |
12 |
5 11
|
eqeq12d |
⊢ ( 𝑥 = 0 → ( ( 𝑆 ‘ 𝑥 ) = ( ( ! ‘ 𝑥 ) · Σ 𝑘 ∈ ( 0 ... 𝑥 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ↔ 1 = ( 1 · Σ 𝑘 ∈ ( 0 ... 0 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
13 |
|
fv0p1e1 |
⊢ ( 𝑥 = 0 → ( 𝑆 ‘ ( 𝑥 + 1 ) ) = ( 𝑆 ‘ 1 ) ) |
14 |
1 2
|
subfac1 |
⊢ ( 𝑆 ‘ 1 ) = 0 |
15 |
13 14
|
eqtrdi |
⊢ ( 𝑥 = 0 → ( 𝑆 ‘ ( 𝑥 + 1 ) ) = 0 ) |
16 |
|
fv0p1e1 |
⊢ ( 𝑥 = 0 → ( ! ‘ ( 𝑥 + 1 ) ) = ( ! ‘ 1 ) ) |
17 |
|
fac1 |
⊢ ( ! ‘ 1 ) = 1 |
18 |
16 17
|
eqtrdi |
⊢ ( 𝑥 = 0 → ( ! ‘ ( 𝑥 + 1 ) ) = 1 ) |
19 |
|
oveq1 |
⊢ ( 𝑥 = 0 → ( 𝑥 + 1 ) = ( 0 + 1 ) ) |
20 |
|
0p1e1 |
⊢ ( 0 + 1 ) = 1 |
21 |
19 20
|
eqtrdi |
⊢ ( 𝑥 = 0 → ( 𝑥 + 1 ) = 1 ) |
22 |
21
|
oveq2d |
⊢ ( 𝑥 = 0 → ( 0 ... ( 𝑥 + 1 ) ) = ( 0 ... 1 ) ) |
23 |
22
|
sumeq1d |
⊢ ( 𝑥 = 0 → Σ 𝑘 ∈ ( 0 ... ( 𝑥 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... 1 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) |
24 |
18 23
|
oveq12d |
⊢ ( 𝑥 = 0 → ( ( ! ‘ ( 𝑥 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑥 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( 1 · Σ 𝑘 ∈ ( 0 ... 1 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) |
25 |
15 24
|
eqeq12d |
⊢ ( 𝑥 = 0 → ( ( 𝑆 ‘ ( 𝑥 + 1 ) ) = ( ( ! ‘ ( 𝑥 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑥 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ↔ 0 = ( 1 · Σ 𝑘 ∈ ( 0 ... 1 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
26 |
12 25
|
anbi12d |
⊢ ( 𝑥 = 0 → ( ( ( 𝑆 ‘ 𝑥 ) = ( ( ! ‘ 𝑥 ) · Σ 𝑘 ∈ ( 0 ... 𝑥 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ ( 𝑆 ‘ ( 𝑥 + 1 ) ) = ( ( ! ‘ ( 𝑥 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑥 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ↔ ( 1 = ( 1 · Σ 𝑘 ∈ ( 0 ... 0 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ 0 = ( 1 · Σ 𝑘 ∈ ( 0 ... 1 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) ) |
27 |
|
fveq2 |
⊢ ( 𝑥 = 𝑚 → ( 𝑆 ‘ 𝑥 ) = ( 𝑆 ‘ 𝑚 ) ) |
28 |
|
fveq2 |
⊢ ( 𝑥 = 𝑚 → ( ! ‘ 𝑥 ) = ( ! ‘ 𝑚 ) ) |
29 |
|
oveq2 |
⊢ ( 𝑥 = 𝑚 → ( 0 ... 𝑥 ) = ( 0 ... 𝑚 ) ) |
30 |
29
|
sumeq1d |
⊢ ( 𝑥 = 𝑚 → Σ 𝑘 ∈ ( 0 ... 𝑥 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) |
31 |
28 30
|
oveq12d |
⊢ ( 𝑥 = 𝑚 → ( ( ! ‘ 𝑥 ) · Σ 𝑘 ∈ ( 0 ... 𝑥 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) |
32 |
27 31
|
eqeq12d |
⊢ ( 𝑥 = 𝑚 → ( ( 𝑆 ‘ 𝑥 ) = ( ( ! ‘ 𝑥 ) · Σ 𝑘 ∈ ( 0 ... 𝑥 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ↔ ( 𝑆 ‘ 𝑚 ) = ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
33 |
|
fvoveq1 |
⊢ ( 𝑥 = 𝑚 → ( 𝑆 ‘ ( 𝑥 + 1 ) ) = ( 𝑆 ‘ ( 𝑚 + 1 ) ) ) |
34 |
|
fvoveq1 |
⊢ ( 𝑥 = 𝑚 → ( ! ‘ ( 𝑥 + 1 ) ) = ( ! ‘ ( 𝑚 + 1 ) ) ) |
35 |
|
oveq1 |
⊢ ( 𝑥 = 𝑚 → ( 𝑥 + 1 ) = ( 𝑚 + 1 ) ) |
36 |
35
|
oveq2d |
⊢ ( 𝑥 = 𝑚 → ( 0 ... ( 𝑥 + 1 ) ) = ( 0 ... ( 𝑚 + 1 ) ) ) |
37 |
36
|
sumeq1d |
⊢ ( 𝑥 = 𝑚 → Σ 𝑘 ∈ ( 0 ... ( 𝑥 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) |
38 |
34 37
|
oveq12d |
⊢ ( 𝑥 = 𝑚 → ( ( ! ‘ ( 𝑥 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑥 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) |
39 |
33 38
|
eqeq12d |
⊢ ( 𝑥 = 𝑚 → ( ( 𝑆 ‘ ( 𝑥 + 1 ) ) = ( ( ! ‘ ( 𝑥 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑥 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ↔ ( 𝑆 ‘ ( 𝑚 + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
40 |
32 39
|
anbi12d |
⊢ ( 𝑥 = 𝑚 → ( ( ( 𝑆 ‘ 𝑥 ) = ( ( ! ‘ 𝑥 ) · Σ 𝑘 ∈ ( 0 ... 𝑥 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ ( 𝑆 ‘ ( 𝑥 + 1 ) ) = ( ( ! ‘ ( 𝑥 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑥 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ↔ ( ( 𝑆 ‘ 𝑚 ) = ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ ( 𝑆 ‘ ( 𝑚 + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) ) |
41 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝑚 + 1 ) → ( 𝑆 ‘ 𝑥 ) = ( 𝑆 ‘ ( 𝑚 + 1 ) ) ) |
42 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝑚 + 1 ) → ( ! ‘ 𝑥 ) = ( ! ‘ ( 𝑚 + 1 ) ) ) |
43 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑚 + 1 ) → ( 0 ... 𝑥 ) = ( 0 ... ( 𝑚 + 1 ) ) ) |
44 |
43
|
sumeq1d |
⊢ ( 𝑥 = ( 𝑚 + 1 ) → Σ 𝑘 ∈ ( 0 ... 𝑥 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) |
45 |
42 44
|
oveq12d |
⊢ ( 𝑥 = ( 𝑚 + 1 ) → ( ( ! ‘ 𝑥 ) · Σ 𝑘 ∈ ( 0 ... 𝑥 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) |
46 |
41 45
|
eqeq12d |
⊢ ( 𝑥 = ( 𝑚 + 1 ) → ( ( 𝑆 ‘ 𝑥 ) = ( ( ! ‘ 𝑥 ) · Σ 𝑘 ∈ ( 0 ... 𝑥 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ↔ ( 𝑆 ‘ ( 𝑚 + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
47 |
|
fvoveq1 |
⊢ ( 𝑥 = ( 𝑚 + 1 ) → ( 𝑆 ‘ ( 𝑥 + 1 ) ) = ( 𝑆 ‘ ( ( 𝑚 + 1 ) + 1 ) ) ) |
48 |
|
fvoveq1 |
⊢ ( 𝑥 = ( 𝑚 + 1 ) → ( ! ‘ ( 𝑥 + 1 ) ) = ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) ) |
49 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝑚 + 1 ) → ( 𝑥 + 1 ) = ( ( 𝑚 + 1 ) + 1 ) ) |
50 |
49
|
oveq2d |
⊢ ( 𝑥 = ( 𝑚 + 1 ) → ( 0 ... ( 𝑥 + 1 ) ) = ( 0 ... ( ( 𝑚 + 1 ) + 1 ) ) ) |
51 |
50
|
sumeq1d |
⊢ ( 𝑥 = ( 𝑚 + 1 ) → Σ 𝑘 ∈ ( 0 ... ( 𝑥 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... ( ( 𝑚 + 1 ) + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) |
52 |
48 51
|
oveq12d |
⊢ ( 𝑥 = ( 𝑚 + 1 ) → ( ( ! ‘ ( 𝑥 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑥 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( ( 𝑚 + 1 ) + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) |
53 |
47 52
|
eqeq12d |
⊢ ( 𝑥 = ( 𝑚 + 1 ) → ( ( 𝑆 ‘ ( 𝑥 + 1 ) ) = ( ( ! ‘ ( 𝑥 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑥 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ↔ ( 𝑆 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( ( 𝑚 + 1 ) + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
54 |
46 53
|
anbi12d |
⊢ ( 𝑥 = ( 𝑚 + 1 ) → ( ( ( 𝑆 ‘ 𝑥 ) = ( ( ! ‘ 𝑥 ) · Σ 𝑘 ∈ ( 0 ... 𝑥 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ ( 𝑆 ‘ ( 𝑥 + 1 ) ) = ( ( ! ‘ ( 𝑥 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑥 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ↔ ( ( 𝑆 ‘ ( 𝑚 + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ ( 𝑆 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( ( 𝑚 + 1 ) + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) ) |
55 |
|
fveq2 |
⊢ ( 𝑥 = 𝑁 → ( 𝑆 ‘ 𝑥 ) = ( 𝑆 ‘ 𝑁 ) ) |
56 |
|
fveq2 |
⊢ ( 𝑥 = 𝑁 → ( ! ‘ 𝑥 ) = ( ! ‘ 𝑁 ) ) |
57 |
|
oveq2 |
⊢ ( 𝑥 = 𝑁 → ( 0 ... 𝑥 ) = ( 0 ... 𝑁 ) ) |
58 |
57
|
sumeq1d |
⊢ ( 𝑥 = 𝑁 → Σ 𝑘 ∈ ( 0 ... 𝑥 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) |
59 |
56 58
|
oveq12d |
⊢ ( 𝑥 = 𝑁 → ( ( ! ‘ 𝑥 ) · Σ 𝑘 ∈ ( 0 ... 𝑥 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( ! ‘ 𝑁 ) · Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) |
60 |
55 59
|
eqeq12d |
⊢ ( 𝑥 = 𝑁 → ( ( 𝑆 ‘ 𝑥 ) = ( ( ! ‘ 𝑥 ) · Σ 𝑘 ∈ ( 0 ... 𝑥 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ↔ ( 𝑆 ‘ 𝑁 ) = ( ( ! ‘ 𝑁 ) · Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
61 |
|
fvoveq1 |
⊢ ( 𝑥 = 𝑁 → ( 𝑆 ‘ ( 𝑥 + 1 ) ) = ( 𝑆 ‘ ( 𝑁 + 1 ) ) ) |
62 |
|
fvoveq1 |
⊢ ( 𝑥 = 𝑁 → ( ! ‘ ( 𝑥 + 1 ) ) = ( ! ‘ ( 𝑁 + 1 ) ) ) |
63 |
|
oveq1 |
⊢ ( 𝑥 = 𝑁 → ( 𝑥 + 1 ) = ( 𝑁 + 1 ) ) |
64 |
63
|
oveq2d |
⊢ ( 𝑥 = 𝑁 → ( 0 ... ( 𝑥 + 1 ) ) = ( 0 ... ( 𝑁 + 1 ) ) ) |
65 |
64
|
sumeq1d |
⊢ ( 𝑥 = 𝑁 → Σ 𝑘 ∈ ( 0 ... ( 𝑥 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... ( 𝑁 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) |
66 |
62 65
|
oveq12d |
⊢ ( 𝑥 = 𝑁 → ( ( ! ‘ ( 𝑥 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑥 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( ! ‘ ( 𝑁 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑁 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) |
67 |
61 66
|
eqeq12d |
⊢ ( 𝑥 = 𝑁 → ( ( 𝑆 ‘ ( 𝑥 + 1 ) ) = ( ( ! ‘ ( 𝑥 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑥 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ↔ ( 𝑆 ‘ ( 𝑁 + 1 ) ) = ( ( ! ‘ ( 𝑁 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑁 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
68 |
60 67
|
anbi12d |
⊢ ( 𝑥 = 𝑁 → ( ( ( 𝑆 ‘ 𝑥 ) = ( ( ! ‘ 𝑥 ) · Σ 𝑘 ∈ ( 0 ... 𝑥 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ ( 𝑆 ‘ ( 𝑥 + 1 ) ) = ( ( ! ‘ ( 𝑥 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑥 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ↔ ( ( 𝑆 ‘ 𝑁 ) = ( ( ! ‘ 𝑁 ) · Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ ( 𝑆 ‘ ( 𝑁 + 1 ) ) = ( ( ! ‘ ( 𝑁 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑁 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) ) |
69 |
|
0z |
⊢ 0 ∈ ℤ |
70 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
71 |
|
oveq2 |
⊢ ( 𝑘 = 0 → ( - 1 ↑ 𝑘 ) = ( - 1 ↑ 0 ) ) |
72 |
|
neg1cn |
⊢ - 1 ∈ ℂ |
73 |
|
exp0 |
⊢ ( - 1 ∈ ℂ → ( - 1 ↑ 0 ) = 1 ) |
74 |
72 73
|
ax-mp |
⊢ ( - 1 ↑ 0 ) = 1 |
75 |
71 74
|
eqtrdi |
⊢ ( 𝑘 = 0 → ( - 1 ↑ 𝑘 ) = 1 ) |
76 |
|
fveq2 |
⊢ ( 𝑘 = 0 → ( ! ‘ 𝑘 ) = ( ! ‘ 0 ) ) |
77 |
76 7
|
eqtrdi |
⊢ ( 𝑘 = 0 → ( ! ‘ 𝑘 ) = 1 ) |
78 |
75 77
|
oveq12d |
⊢ ( 𝑘 = 0 → ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = ( 1 / 1 ) ) |
79 |
70
|
div1i |
⊢ ( 1 / 1 ) = 1 |
80 |
78 79
|
eqtrdi |
⊢ ( 𝑘 = 0 → ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = 1 ) |
81 |
80
|
fsum1 |
⊢ ( ( 0 ∈ ℤ ∧ 1 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... 0 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = 1 ) |
82 |
69 70 81
|
mp2an |
⊢ Σ 𝑘 ∈ ( 0 ... 0 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = 1 |
83 |
82
|
oveq2i |
⊢ ( 1 · Σ 𝑘 ∈ ( 0 ... 0 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( 1 · 1 ) |
84 |
|
1t1e1 |
⊢ ( 1 · 1 ) = 1 |
85 |
83 84
|
eqtr2i |
⊢ 1 = ( 1 · Σ 𝑘 ∈ ( 0 ... 0 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) |
86 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
87 |
|
1e0p1 |
⊢ 1 = ( 0 + 1 ) |
88 |
|
oveq2 |
⊢ ( 𝑘 = 1 → ( - 1 ↑ 𝑘 ) = ( - 1 ↑ 1 ) ) |
89 |
|
exp1 |
⊢ ( - 1 ∈ ℂ → ( - 1 ↑ 1 ) = - 1 ) |
90 |
72 89
|
ax-mp |
⊢ ( - 1 ↑ 1 ) = - 1 |
91 |
88 90
|
eqtrdi |
⊢ ( 𝑘 = 1 → ( - 1 ↑ 𝑘 ) = - 1 ) |
92 |
|
fveq2 |
⊢ ( 𝑘 = 1 → ( ! ‘ 𝑘 ) = ( ! ‘ 1 ) ) |
93 |
92 17
|
eqtrdi |
⊢ ( 𝑘 = 1 → ( ! ‘ 𝑘 ) = 1 ) |
94 |
91 93
|
oveq12d |
⊢ ( 𝑘 = 1 → ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = ( - 1 / 1 ) ) |
95 |
72
|
div1i |
⊢ ( - 1 / 1 ) = - 1 |
96 |
94 95
|
eqtrdi |
⊢ ( 𝑘 = 1 → ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = - 1 ) |
97 |
|
neg1rr |
⊢ - 1 ∈ ℝ |
98 |
|
reexpcl |
⊢ ( ( - 1 ∈ ℝ ∧ 𝑘 ∈ ℕ0 ) → ( - 1 ↑ 𝑘 ) ∈ ℝ ) |
99 |
97 98
|
mpan |
⊢ ( 𝑘 ∈ ℕ0 → ( - 1 ↑ 𝑘 ) ∈ ℝ ) |
100 |
|
faccl |
⊢ ( 𝑘 ∈ ℕ0 → ( ! ‘ 𝑘 ) ∈ ℕ ) |
101 |
99 100
|
nndivred |
⊢ ( 𝑘 ∈ ℕ0 → ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ∈ ℝ ) |
102 |
101
|
recnd |
⊢ ( 𝑘 ∈ ℕ0 → ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ∈ ℂ ) |
103 |
102
|
adantl |
⊢ ( ( ⊤ ∧ 𝑘 ∈ ℕ0 ) → ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ∈ ℂ ) |
104 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
105 |
104 82
|
pm3.2i |
⊢ ( 0 ∈ ℕ0 ∧ Σ 𝑘 ∈ ( 0 ... 0 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = 1 ) |
106 |
105
|
a1i |
⊢ ( ⊤ → ( 0 ∈ ℕ0 ∧ Σ 𝑘 ∈ ( 0 ... 0 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = 1 ) ) |
107 |
|
1pneg1e0 |
⊢ ( 1 + - 1 ) = 0 |
108 |
107
|
a1i |
⊢ ( ⊤ → ( 1 + - 1 ) = 0 ) |
109 |
86 87 96 103 106 108
|
fsump1i |
⊢ ( ⊤ → ( 1 ∈ ℕ0 ∧ Σ 𝑘 ∈ ( 0 ... 1 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = 0 ) ) |
110 |
109
|
mptru |
⊢ ( 1 ∈ ℕ0 ∧ Σ 𝑘 ∈ ( 0 ... 1 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = 0 ) |
111 |
110
|
simpri |
⊢ Σ 𝑘 ∈ ( 0 ... 1 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = 0 |
112 |
111
|
oveq2i |
⊢ ( 1 · Σ 𝑘 ∈ ( 0 ... 1 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( 1 · 0 ) |
113 |
70
|
mul01i |
⊢ ( 1 · 0 ) = 0 |
114 |
112 113
|
eqtr2i |
⊢ 0 = ( 1 · Σ 𝑘 ∈ ( 0 ... 1 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) |
115 |
85 114
|
pm3.2i |
⊢ ( 1 = ( 1 · Σ 𝑘 ∈ ( 0 ... 0 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ 0 = ( 1 · Σ 𝑘 ∈ ( 0 ... 1 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) |
116 |
|
simpr |
⊢ ( ( ( 𝑆 ‘ 𝑚 ) = ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ ( 𝑆 ‘ ( 𝑚 + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) → ( 𝑆 ‘ ( 𝑚 + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) |
117 |
116
|
a1i |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( 𝑆 ‘ 𝑚 ) = ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ ( 𝑆 ‘ ( 𝑚 + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) → ( 𝑆 ‘ ( 𝑚 + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
118 |
|
oveq12 |
⊢ ( ( ( 𝑆 ‘ ( 𝑚 + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ ( 𝑆 ‘ 𝑚 ) = ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) → ( ( 𝑆 ‘ ( 𝑚 + 1 ) ) + ( 𝑆 ‘ 𝑚 ) ) = ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
119 |
118
|
ancoms |
⊢ ( ( ( 𝑆 ‘ 𝑚 ) = ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ ( 𝑆 ‘ ( 𝑚 + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) → ( ( 𝑆 ‘ ( 𝑚 + 1 ) ) + ( 𝑆 ‘ 𝑚 ) ) = ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
120 |
119
|
oveq2d |
⊢ ( ( ( 𝑆 ‘ 𝑚 ) = ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ ( 𝑆 ‘ ( 𝑚 + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) → ( ( 𝑚 + 1 ) · ( ( 𝑆 ‘ ( 𝑚 + 1 ) ) + ( 𝑆 ‘ 𝑚 ) ) ) = ( ( 𝑚 + 1 ) · ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) ) |
121 |
|
nn0p1nn |
⊢ ( 𝑚 ∈ ℕ0 → ( 𝑚 + 1 ) ∈ ℕ ) |
122 |
1 2
|
subfacp1 |
⊢ ( ( 𝑚 + 1 ) ∈ ℕ → ( 𝑆 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( 𝑚 + 1 ) · ( ( 𝑆 ‘ ( 𝑚 + 1 ) ) + ( 𝑆 ‘ ( ( 𝑚 + 1 ) − 1 ) ) ) ) ) |
123 |
121 122
|
syl |
⊢ ( 𝑚 ∈ ℕ0 → ( 𝑆 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( 𝑚 + 1 ) · ( ( 𝑆 ‘ ( 𝑚 + 1 ) ) + ( 𝑆 ‘ ( ( 𝑚 + 1 ) − 1 ) ) ) ) ) |
124 |
|
nn0cn |
⊢ ( 𝑚 ∈ ℕ0 → 𝑚 ∈ ℂ ) |
125 |
|
pncan |
⊢ ( ( 𝑚 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑚 + 1 ) − 1 ) = 𝑚 ) |
126 |
124 70 125
|
sylancl |
⊢ ( 𝑚 ∈ ℕ0 → ( ( 𝑚 + 1 ) − 1 ) = 𝑚 ) |
127 |
126
|
fveq2d |
⊢ ( 𝑚 ∈ ℕ0 → ( 𝑆 ‘ ( ( 𝑚 + 1 ) − 1 ) ) = ( 𝑆 ‘ 𝑚 ) ) |
128 |
127
|
oveq2d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( 𝑆 ‘ ( 𝑚 + 1 ) ) + ( 𝑆 ‘ ( ( 𝑚 + 1 ) − 1 ) ) ) = ( ( 𝑆 ‘ ( 𝑚 + 1 ) ) + ( 𝑆 ‘ 𝑚 ) ) ) |
129 |
128
|
oveq2d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( 𝑚 + 1 ) · ( ( 𝑆 ‘ ( 𝑚 + 1 ) ) + ( 𝑆 ‘ ( ( 𝑚 + 1 ) − 1 ) ) ) ) = ( ( 𝑚 + 1 ) · ( ( 𝑆 ‘ ( 𝑚 + 1 ) ) + ( 𝑆 ‘ 𝑚 ) ) ) ) |
130 |
123 129
|
eqtrd |
⊢ ( 𝑚 ∈ ℕ0 → ( 𝑆 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( 𝑚 + 1 ) · ( ( 𝑆 ‘ ( 𝑚 + 1 ) ) + ( 𝑆 ‘ 𝑚 ) ) ) ) |
131 |
|
peano2nn0 |
⊢ ( 𝑚 ∈ ℕ0 → ( 𝑚 + 1 ) ∈ ℕ0 ) |
132 |
|
peano2nn0 |
⊢ ( ( 𝑚 + 1 ) ∈ ℕ0 → ( ( 𝑚 + 1 ) + 1 ) ∈ ℕ0 ) |
133 |
131 132
|
syl |
⊢ ( 𝑚 ∈ ℕ0 → ( ( 𝑚 + 1 ) + 1 ) ∈ ℕ0 ) |
134 |
|
faccl |
⊢ ( ( ( 𝑚 + 1 ) + 1 ) ∈ ℕ0 → ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) ∈ ℕ ) |
135 |
133 134
|
syl |
⊢ ( 𝑚 ∈ ℕ0 → ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) ∈ ℕ ) |
136 |
135
|
nncnd |
⊢ ( 𝑚 ∈ ℕ0 → ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) ∈ ℂ ) |
137 |
|
fzfid |
⊢ ( 𝑚 ∈ ℕ0 → ( 0 ... ( 𝑚 + 1 ) ) ∈ Fin ) |
138 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) → 𝑘 ∈ ℕ0 ) |
139 |
138
|
adantl |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ) → 𝑘 ∈ ℕ0 ) |
140 |
139 102
|
syl |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ) → ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ∈ ℂ ) |
141 |
137 140
|
fsumcl |
⊢ ( 𝑚 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ∈ ℂ ) |
142 |
|
expcl |
⊢ ( ( - 1 ∈ ℂ ∧ ( ( 𝑚 + 1 ) + 1 ) ∈ ℕ0 ) → ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) ∈ ℂ ) |
143 |
72 133 142
|
sylancr |
⊢ ( 𝑚 ∈ ℕ0 → ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) ∈ ℂ ) |
144 |
135
|
nnne0d |
⊢ ( 𝑚 ∈ ℕ0 → ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) ≠ 0 ) |
145 |
143 136 144
|
divcld |
⊢ ( 𝑚 ∈ ℕ0 → ( ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) / ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) ) ∈ ℂ ) |
146 |
136 141 145
|
adddid |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · ( Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) + ( ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) / ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) ) ) ) = ( ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · ( ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) / ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) ) ) ) ) |
147 |
|
id |
⊢ ( 𝑚 ∈ ℕ0 → 𝑚 ∈ ℕ0 ) |
148 |
147 86
|
eleqtrdi |
⊢ ( 𝑚 ∈ ℕ0 → 𝑚 ∈ ( ℤ≥ ‘ 0 ) ) |
149 |
|
oveq2 |
⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( - 1 ↑ 𝑘 ) = ( - 1 ↑ ( 𝑚 + 1 ) ) ) |
150 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ! ‘ 𝑘 ) = ( ! ‘ ( 𝑚 + 1 ) ) ) |
151 |
149 150
|
oveq12d |
⊢ ( 𝑘 = ( 𝑚 + 1 ) → ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = ( ( - 1 ↑ ( 𝑚 + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ) |
152 |
148 140 151
|
fsump1 |
⊢ ( 𝑚 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = ( Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) + ( ( - 1 ↑ ( 𝑚 + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ) ) |
153 |
152
|
oveq2d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · ( Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) + ( ( - 1 ↑ ( 𝑚 + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ) ) ) |
154 |
|
fzfid |
⊢ ( 𝑚 ∈ ℕ0 → ( 0 ... 𝑚 ) ∈ Fin ) |
155 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... 𝑚 ) → 𝑘 ∈ ℕ0 ) |
156 |
155
|
adantl |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑘 ∈ ( 0 ... 𝑚 ) ) → 𝑘 ∈ ℕ0 ) |
157 |
156 102
|
syl |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑘 ∈ ( 0 ... 𝑚 ) ) → ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ∈ ℂ ) |
158 |
154 157
|
fsumcl |
⊢ ( 𝑚 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ∈ ℂ ) |
159 |
|
expcl |
⊢ ( ( - 1 ∈ ℂ ∧ ( 𝑚 + 1 ) ∈ ℕ0 ) → ( - 1 ↑ ( 𝑚 + 1 ) ) ∈ ℂ ) |
160 |
72 131 159
|
sylancr |
⊢ ( 𝑚 ∈ ℕ0 → ( - 1 ↑ ( 𝑚 + 1 ) ) ∈ ℂ ) |
161 |
|
faccl |
⊢ ( ( 𝑚 + 1 ) ∈ ℕ0 → ( ! ‘ ( 𝑚 + 1 ) ) ∈ ℕ ) |
162 |
131 161
|
syl |
⊢ ( 𝑚 ∈ ℕ0 → ( ! ‘ ( 𝑚 + 1 ) ) ∈ ℕ ) |
163 |
162
|
nncnd |
⊢ ( 𝑚 ∈ ℕ0 → ( ! ‘ ( 𝑚 + 1 ) ) ∈ ℂ ) |
164 |
162
|
nnne0d |
⊢ ( 𝑚 ∈ ℕ0 → ( ! ‘ ( 𝑚 + 1 ) ) ≠ 0 ) |
165 |
160 163 164
|
divcld |
⊢ ( 𝑚 ∈ ℕ0 → ( ( - 1 ↑ ( 𝑚 + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ∈ ℂ ) |
166 |
136 158 165
|
adddid |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · ( Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) + ( ( - 1 ↑ ( 𝑚 + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ) ) = ( ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · ( ( - 1 ↑ ( 𝑚 + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ) ) ) |
167 |
|
facp1 |
⊢ ( ( 𝑚 + 1 ) ∈ ℕ0 → ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) ) |
168 |
131 167
|
syl |
⊢ ( 𝑚 ∈ ℕ0 → ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) ) |
169 |
|
facp1 |
⊢ ( 𝑚 ∈ ℕ0 → ( ! ‘ ( 𝑚 + 1 ) ) = ( ( ! ‘ 𝑚 ) · ( 𝑚 + 1 ) ) ) |
170 |
|
faccl |
⊢ ( 𝑚 ∈ ℕ0 → ( ! ‘ 𝑚 ) ∈ ℕ ) |
171 |
170
|
nncnd |
⊢ ( 𝑚 ∈ ℕ0 → ( ! ‘ 𝑚 ) ∈ ℂ ) |
172 |
121
|
nncnd |
⊢ ( 𝑚 ∈ ℕ0 → ( 𝑚 + 1 ) ∈ ℂ ) |
173 |
171 172
|
mulcomd |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ 𝑚 ) · ( 𝑚 + 1 ) ) = ( ( 𝑚 + 1 ) · ( ! ‘ 𝑚 ) ) ) |
174 |
169 173
|
eqtrd |
⊢ ( 𝑚 ∈ ℕ0 → ( ! ‘ ( 𝑚 + 1 ) ) = ( ( 𝑚 + 1 ) · ( ! ‘ 𝑚 ) ) ) |
175 |
174
|
oveq1d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) = ( ( ( 𝑚 + 1 ) · ( ! ‘ 𝑚 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) ) |
176 |
133
|
nn0cnd |
⊢ ( 𝑚 ∈ ℕ0 → ( ( 𝑚 + 1 ) + 1 ) ∈ ℂ ) |
177 |
172 171 176
|
mulassd |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( 𝑚 + 1 ) · ( ! ‘ 𝑚 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) = ( ( 𝑚 + 1 ) · ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) ) ) |
178 |
168 175 177
|
3eqtrd |
⊢ ( 𝑚 ∈ ℕ0 → ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( 𝑚 + 1 ) · ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) ) ) |
179 |
178
|
oveq1d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( ( 𝑚 + 1 ) · ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) |
180 |
136 160 163 164
|
div12d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · ( ( - 1 ↑ ( 𝑚 + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ) = ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ) ) |
181 |
168
|
oveq1d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) = ( ( ( ! ‘ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ) |
182 |
176 163 164
|
divcan3d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ! ‘ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) = ( ( 𝑚 + 1 ) + 1 ) ) |
183 |
181 182
|
eqtrd |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) = ( ( 𝑚 + 1 ) + 1 ) ) |
184 |
183
|
oveq2d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ) = ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) ) |
185 |
180 184
|
eqtrd |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · ( ( - 1 ↑ ( 𝑚 + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ) = ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) ) |
186 |
179 185
|
oveq12d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · ( ( - 1 ↑ ( 𝑚 + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ) ) = ( ( ( ( 𝑚 + 1 ) · ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) ) ) |
187 |
153 166 186
|
3eqtrd |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( ( ( 𝑚 + 1 ) · ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) ) ) |
188 |
143 136 144
|
divcan2d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · ( ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) / ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) ) ) = ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) ) |
189 |
187 188
|
oveq12d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · ( ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) / ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) ) ) ) = ( ( ( ( ( 𝑚 + 1 ) · ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) ) + ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) ) ) |
190 |
171 176
|
mulcld |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) ∈ ℂ ) |
191 |
172 190 158
|
mulassd |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( 𝑚 + 1 ) · ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( 𝑚 + 1 ) · ( ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
192 |
72
|
a1i |
⊢ ( 𝑚 ∈ ℕ0 → - 1 ∈ ℂ ) |
193 |
160 176 192
|
adddid |
⊢ ( 𝑚 ∈ ℕ0 → ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( ( 𝑚 + 1 ) + 1 ) + - 1 ) ) = ( ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) + ( ( - 1 ↑ ( 𝑚 + 1 ) ) · - 1 ) ) ) |
194 |
|
negsub |
⊢ ( ( ( ( 𝑚 + 1 ) + 1 ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( ( 𝑚 + 1 ) + 1 ) + - 1 ) = ( ( ( 𝑚 + 1 ) + 1 ) − 1 ) ) |
195 |
176 70 194
|
sylancl |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( 𝑚 + 1 ) + 1 ) + - 1 ) = ( ( ( 𝑚 + 1 ) + 1 ) − 1 ) ) |
196 |
|
pncan |
⊢ ( ( ( 𝑚 + 1 ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( ( 𝑚 + 1 ) + 1 ) − 1 ) = ( 𝑚 + 1 ) ) |
197 |
172 70 196
|
sylancl |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( 𝑚 + 1 ) + 1 ) − 1 ) = ( 𝑚 + 1 ) ) |
198 |
195 197
|
eqtrd |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( 𝑚 + 1 ) + 1 ) + - 1 ) = ( 𝑚 + 1 ) ) |
199 |
198
|
oveq2d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( ( 𝑚 + 1 ) + 1 ) + - 1 ) ) = ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( 𝑚 + 1 ) ) ) |
200 |
193 199
|
eqtr3d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) + ( ( - 1 ↑ ( 𝑚 + 1 ) ) · - 1 ) ) = ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( 𝑚 + 1 ) ) ) |
201 |
|
expp1 |
⊢ ( ( - 1 ∈ ℂ ∧ ( 𝑚 + 1 ) ∈ ℕ0 ) → ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( - 1 ↑ ( 𝑚 + 1 ) ) · - 1 ) ) |
202 |
72 131 201
|
sylancr |
⊢ ( 𝑚 ∈ ℕ0 → ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( - 1 ↑ ( 𝑚 + 1 ) ) · - 1 ) ) |
203 |
202
|
oveq2d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) + ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) ) = ( ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) + ( ( - 1 ↑ ( 𝑚 + 1 ) ) · - 1 ) ) ) |
204 |
172 160
|
mulcomd |
⊢ ( 𝑚 ∈ ℕ0 → ( ( 𝑚 + 1 ) · ( - 1 ↑ ( 𝑚 + 1 ) ) ) = ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( 𝑚 + 1 ) ) ) |
205 |
200 203 204
|
3eqtr4d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) + ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) ) = ( ( 𝑚 + 1 ) · ( - 1 ↑ ( 𝑚 + 1 ) ) ) ) |
206 |
191 205
|
oveq12d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ( 𝑚 + 1 ) · ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) + ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) ) ) = ( ( ( 𝑚 + 1 ) · ( ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) + ( ( 𝑚 + 1 ) · ( - 1 ↑ ( 𝑚 + 1 ) ) ) ) ) |
207 |
172 190
|
mulcld |
⊢ ( 𝑚 ∈ ℕ0 → ( ( 𝑚 + 1 ) · ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) ) ∈ ℂ ) |
208 |
207 158
|
mulcld |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( 𝑚 + 1 ) · ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∈ ℂ ) |
209 |
160 176
|
mulcld |
⊢ ( 𝑚 ∈ ℕ0 → ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) ∈ ℂ ) |
210 |
208 209 143
|
addassd |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ( ( 𝑚 + 1 ) · ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) ) + ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) ) = ( ( ( ( 𝑚 + 1 ) · ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) + ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) ) ) ) |
211 |
190 158
|
mulcld |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∈ ℂ ) |
212 |
172 211 160
|
adddid |
⊢ ( 𝑚 ∈ ℕ0 → ( ( 𝑚 + 1 ) · ( ( ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( - 1 ↑ ( 𝑚 + 1 ) ) ) ) = ( ( ( 𝑚 + 1 ) · ( ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) + ( ( 𝑚 + 1 ) · ( - 1 ↑ ( 𝑚 + 1 ) ) ) ) ) |
213 |
206 210 212
|
3eqtr4d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ( ( 𝑚 + 1 ) · ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( - 1 ↑ ( 𝑚 + 1 ) ) · ( ( 𝑚 + 1 ) + 1 ) ) ) + ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) ) = ( ( 𝑚 + 1 ) · ( ( ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( - 1 ↑ ( 𝑚 + 1 ) ) ) ) ) |
214 |
146 189 213
|
3eqtrd |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · ( Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) + ( ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) / ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) ) ) ) = ( ( 𝑚 + 1 ) · ( ( ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( - 1 ↑ ( 𝑚 + 1 ) ) ) ) ) |
215 |
131 86
|
eleqtrdi |
⊢ ( 𝑚 ∈ ℕ0 → ( 𝑚 + 1 ) ∈ ( ℤ≥ ‘ 0 ) ) |
216 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... ( ( 𝑚 + 1 ) + 1 ) ) → 𝑘 ∈ ℕ0 ) |
217 |
216
|
adantl |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑘 ∈ ( 0 ... ( ( 𝑚 + 1 ) + 1 ) ) ) → 𝑘 ∈ ℕ0 ) |
218 |
217 102
|
syl |
⊢ ( ( 𝑚 ∈ ℕ0 ∧ 𝑘 ∈ ( 0 ... ( ( 𝑚 + 1 ) + 1 ) ) ) → ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ∈ ℂ ) |
219 |
|
oveq2 |
⊢ ( 𝑘 = ( ( 𝑚 + 1 ) + 1 ) → ( - 1 ↑ 𝑘 ) = ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) ) |
220 |
|
fveq2 |
⊢ ( 𝑘 = ( ( 𝑚 + 1 ) + 1 ) → ( ! ‘ 𝑘 ) = ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) ) |
221 |
219 220
|
oveq12d |
⊢ ( 𝑘 = ( ( 𝑚 + 1 ) + 1 ) → ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = ( ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) / ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) ) ) |
222 |
215 218 221
|
fsump1 |
⊢ ( 𝑚 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... ( ( 𝑚 + 1 ) + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) = ( Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) + ( ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) / ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) ) ) ) |
223 |
222
|
oveq2d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( ( 𝑚 + 1 ) + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · ( Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) + ( ( - 1 ↑ ( ( 𝑚 + 1 ) + 1 ) ) / ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) ) ) ) ) |
224 |
163 158
|
mulcld |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∈ ℂ ) |
225 |
171 158
|
mulcld |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∈ ℂ ) |
226 |
224 160 225
|
add32d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( - 1 ↑ ( 𝑚 + 1 ) ) ) + ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) = ( ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) + ( - 1 ↑ ( 𝑚 + 1 ) ) ) ) |
227 |
152
|
oveq2d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · ( Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) + ( ( - 1 ↑ ( 𝑚 + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ) ) ) |
228 |
163 158 165
|
adddid |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( 𝑚 + 1 ) ) · ( Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) + ( ( - 1 ↑ ( 𝑚 + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ) ) = ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ ( 𝑚 + 1 ) ) · ( ( - 1 ↑ ( 𝑚 + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ) ) ) |
229 |
160 163 164
|
divcan2d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( 𝑚 + 1 ) ) · ( ( - 1 ↑ ( 𝑚 + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ) = ( - 1 ↑ ( 𝑚 + 1 ) ) ) |
230 |
229
|
oveq2d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ ( 𝑚 + 1 ) ) · ( ( - 1 ↑ ( 𝑚 + 1 ) ) / ( ! ‘ ( 𝑚 + 1 ) ) ) ) ) = ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( - 1 ↑ ( 𝑚 + 1 ) ) ) ) |
231 |
227 228 230
|
3eqtrd |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( - 1 ↑ ( 𝑚 + 1 ) ) ) ) |
232 |
231
|
oveq1d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) = ( ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( - 1 ↑ ( 𝑚 + 1 ) ) ) + ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
233 |
70
|
a1i |
⊢ ( 𝑚 ∈ ℕ0 → 1 ∈ ℂ ) |
234 |
171 172 233
|
adddid |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) = ( ( ( ! ‘ 𝑚 ) · ( 𝑚 + 1 ) ) + ( ( ! ‘ 𝑚 ) · 1 ) ) ) |
235 |
169
|
eqcomd |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ 𝑚 ) · ( 𝑚 + 1 ) ) = ( ! ‘ ( 𝑚 + 1 ) ) ) |
236 |
171
|
mulid1d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ 𝑚 ) · 1 ) = ( ! ‘ 𝑚 ) ) |
237 |
235 236
|
oveq12d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ! ‘ 𝑚 ) · ( 𝑚 + 1 ) ) + ( ( ! ‘ 𝑚 ) · 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) + ( ! ‘ 𝑚 ) ) ) |
238 |
234 237
|
eqtrd |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) + ( ! ‘ 𝑚 ) ) ) |
239 |
238
|
oveq1d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( ( ! ‘ ( 𝑚 + 1 ) ) + ( ! ‘ 𝑚 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) |
240 |
163 171 158
|
adddird |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ! ‘ ( 𝑚 + 1 ) ) + ( ! ‘ 𝑚 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
241 |
239 240
|
eqtrd |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
242 |
241
|
oveq1d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( - 1 ↑ ( 𝑚 + 1 ) ) ) = ( ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) + ( - 1 ↑ ( 𝑚 + 1 ) ) ) ) |
243 |
226 232 242
|
3eqtr4d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) = ( ( ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( - 1 ↑ ( 𝑚 + 1 ) ) ) ) |
244 |
243
|
oveq2d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( 𝑚 + 1 ) · ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) = ( ( 𝑚 + 1 ) · ( ( ( ( ! ‘ 𝑚 ) · ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( - 1 ↑ ( 𝑚 + 1 ) ) ) ) ) |
245 |
214 223 244
|
3eqtr4d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( ( 𝑚 + 1 ) + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) = ( ( 𝑚 + 1 ) · ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) ) |
246 |
130 245
|
eqeq12d |
⊢ ( 𝑚 ∈ ℕ0 → ( ( 𝑆 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( ( 𝑚 + 1 ) + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ↔ ( ( 𝑚 + 1 ) · ( ( 𝑆 ‘ ( 𝑚 + 1 ) ) + ( 𝑆 ‘ 𝑚 ) ) ) = ( ( 𝑚 + 1 ) · ( ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) + ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) ) ) |
247 |
120 246
|
syl5ibr |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( 𝑆 ‘ 𝑚 ) = ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ ( 𝑆 ‘ ( 𝑚 + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) → ( 𝑆 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( ( 𝑚 + 1 ) + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
248 |
117 247
|
jcad |
⊢ ( 𝑚 ∈ ℕ0 → ( ( ( 𝑆 ‘ 𝑚 ) = ( ( ! ‘ 𝑚 ) · Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ ( 𝑆 ‘ ( 𝑚 + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) → ( ( 𝑆 ‘ ( 𝑚 + 1 ) ) = ( ( ! ‘ ( 𝑚 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑚 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ ( 𝑆 ‘ ( ( 𝑚 + 1 ) + 1 ) ) = ( ( ! ‘ ( ( 𝑚 + 1 ) + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( ( 𝑚 + 1 ) + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) ) |
249 |
26 40 54 68 115 248
|
nn0ind |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝑆 ‘ 𝑁 ) = ( ( ! ‘ 𝑁 ) · Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ∧ ( 𝑆 ‘ ( 𝑁 + 1 ) ) = ( ( ! ‘ ( 𝑁 + 1 ) ) · Σ 𝑘 ∈ ( 0 ... ( 𝑁 + 1 ) ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) ) |
250 |
249
|
simpld |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑆 ‘ 𝑁 ) = ( ( ! ‘ 𝑁 ) · Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( - 1 ↑ 𝑘 ) / ( ! ‘ 𝑘 ) ) ) ) |