Step |
Hyp |
Ref |
Expression |
1 |
|
binomcxp.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ+ ) |
2 |
|
binomcxp.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
binomcxp.lt |
⊢ ( 𝜑 → ( abs ‘ 𝐵 ) < ( abs ‘ 𝐴 ) ) |
4 |
|
binomcxp.c |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
5 |
1
|
rpcnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
6 |
2
|
recnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
7 |
|
binom |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℕ0 ) → ( ( 𝐴 + 𝐵 ) ↑ 𝐶 ) = Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C 𝑘 ) · ( ( 𝐴 ↑ ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) |
8 |
7
|
3expia |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐶 ∈ ℕ0 → ( ( 𝐴 + 𝐵 ) ↑ 𝐶 ) = Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C 𝑘 ) · ( ( 𝐴 ↑ ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) ) |
9 |
5 6 8
|
syl2anc |
⊢ ( 𝜑 → ( 𝐶 ∈ ℕ0 → ( ( 𝐴 + 𝐵 ) ↑ 𝐶 ) = Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C 𝑘 ) · ( ( 𝐴 ↑ ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) ) |
10 |
9
|
imp |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → ( ( 𝐴 + 𝐵 ) ↑ 𝐶 ) = Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C 𝑘 ) · ( ( 𝐴 ↑ ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) |
11 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → 𝐴 ∈ ℂ ) |
12 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → 𝐵 ∈ ℂ ) |
13 |
11 12
|
addcld |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → ( 𝐴 + 𝐵 ) ∈ ℂ ) |
14 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → 𝐶 ∈ ℕ0 ) |
15 |
|
cxpexp |
⊢ ( ( ( 𝐴 + 𝐵 ) ∈ ℂ ∧ 𝐶 ∈ ℕ0 ) → ( ( 𝐴 + 𝐵 ) ↑𝑐 𝐶 ) = ( ( 𝐴 + 𝐵 ) ↑ 𝐶 ) ) |
16 |
13 14 15
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → ( ( 𝐴 + 𝐵 ) ↑𝑐 𝐶 ) = ( ( 𝐴 + 𝐵 ) ↑ 𝐶 ) ) |
17 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... 𝐶 ) → 𝑘 ∈ ℕ0 ) |
18 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝐶 ∈ ℕ0 ) |
19 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℕ0 ) |
20 |
18 19
|
bccbc |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐶 C𝑐 𝑘 ) = ( 𝐶 C 𝑘 ) ) |
21 |
17 20
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝐶 ) ) → ( 𝐶 C𝑐 𝑘 ) = ( 𝐶 C 𝑘 ) ) |
22 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝐶 ) ) → 𝐴 ∈ ℂ ) |
23 |
|
elfzle2 |
⊢ ( 𝑘 ∈ ( 0 ... 𝐶 ) → 𝑘 ≤ 𝐶 ) |
24 |
23
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝐶 ) ) → 𝑘 ≤ 𝐶 ) |
25 |
|
nn0sub |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( 𝑘 ≤ 𝐶 ↔ ( 𝐶 − 𝑘 ) ∈ ℕ0 ) ) |
26 |
25
|
ancoms |
⊢ ( ( 𝐶 ∈ ℕ0 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 ≤ 𝐶 ↔ ( 𝐶 − 𝑘 ) ∈ ℕ0 ) ) |
27 |
26
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 ≤ 𝐶 ↔ ( 𝐶 − 𝑘 ) ∈ ℕ0 ) ) |
28 |
17 27
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝐶 ) ) → ( 𝑘 ≤ 𝐶 ↔ ( 𝐶 − 𝑘 ) ∈ ℕ0 ) ) |
29 |
24 28
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝐶 ) ) → ( 𝐶 − 𝑘 ) ∈ ℕ0 ) |
30 |
|
cxpexp |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐶 − 𝑘 ) ∈ ℕ0 ) → ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) = ( 𝐴 ↑ ( 𝐶 − 𝑘 ) ) ) |
31 |
22 29 30
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝐶 ) ) → ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) = ( 𝐴 ↑ ( 𝐶 − 𝑘 ) ) ) |
32 |
31
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝐶 ) ) → ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) = ( ( 𝐴 ↑ ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) |
33 |
21 32
|
oveq12d |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... 𝐶 ) ) → ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) = ( ( 𝐶 C 𝑘 ) · ( ( 𝐴 ↑ ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) |
34 |
33
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) = Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C 𝑘 ) · ( ( 𝐴 ↑ ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) |
35 |
10 16 34
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → ( ( 𝐴 + 𝐵 ) ↑𝑐 𝐶 ) = Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) |
36 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → 𝐶 ∈ ℂ ) |
37 |
13 36
|
cxpcld |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → ( ( 𝐴 + 𝐵 ) ↑𝑐 𝐶 ) ∈ ℂ ) |
38 |
35 37
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ∈ ℂ ) |
39 |
38
|
addid1d |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → ( Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) + 0 ) = Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) |
40 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
41 |
|
eqid |
⊢ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) = ( ℤ≥ ‘ ( 𝐶 + 1 ) ) |
42 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
43 |
42
|
a1i |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → 1 ∈ ℕ0 ) |
44 |
14 43
|
nn0addcld |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → ( 𝐶 + 1 ) ∈ ℕ0 ) |
45 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) · ( 𝐵 ↑ 𝑗 ) ) ) ) = ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) · ( 𝐵 ↑ 𝑗 ) ) ) ) ) |
46 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = 𝑘 ) → 𝑗 = 𝑘 ) |
47 |
46
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = 𝑘 ) → ( 𝐶 C𝑐 𝑗 ) = ( 𝐶 C𝑐 𝑘 ) ) |
48 |
46
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = 𝑘 ) → ( 𝐶 − 𝑗 ) = ( 𝐶 − 𝑘 ) ) |
49 |
48
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = 𝑘 ) → ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) = ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) ) |
50 |
46
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = 𝑘 ) → ( 𝐵 ↑ 𝑗 ) = ( 𝐵 ↑ 𝑘 ) ) |
51 |
49 50
|
oveq12d |
⊢ ( ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = 𝑘 ) → ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) · ( 𝐵 ↑ 𝑗 ) ) = ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) |
52 |
47 51
|
oveq12d |
⊢ ( ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = 𝑘 ) → ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) · ( 𝐵 ↑ 𝑗 ) ) ) = ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) |
53 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝐶 ∈ ℂ ) |
54 |
53 19
|
bcccl |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐶 C𝑐 𝑘 ) ∈ ℂ ) |
55 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝐴 ∈ ℂ ) |
56 |
19
|
nn0cnd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℂ ) |
57 |
53 56
|
subcld |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐶 − 𝑘 ) ∈ ℂ ) |
58 |
55 57
|
cxpcld |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) ∈ ℂ ) |
59 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝐵 ∈ ℂ ) |
60 |
59 19
|
expcld |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐵 ↑ 𝑘 ) ∈ ℂ ) |
61 |
58 60
|
mulcld |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ∈ ℂ ) |
62 |
54 61
|
mulcld |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ∈ ℂ ) |
63 |
45 52 19 62
|
fvmptd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) · ( 𝐵 ↑ 𝑗 ) ) ) ) ‘ 𝑘 ) = ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) |
64 |
|
peano2nn0 |
⊢ ( 𝐶 ∈ ℕ0 → ( 𝐶 + 1 ) ∈ ℕ0 ) |
65 |
64
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → ( 𝐶 + 1 ) ∈ ℕ0 ) |
66 |
|
c0ex |
⊢ 0 ∈ V |
67 |
66
|
fconst |
⊢ ( ℕ0 × { 0 } ) : ℕ0 ⟶ { 0 } |
68 |
67
|
a1i |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → ( ℕ0 × { 0 } ) : ℕ0 ⟶ { 0 } ) |
69 |
|
0red |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → 0 ∈ ℝ ) |
70 |
69
|
snssd |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → { 0 } ⊆ ℝ ) |
71 |
68 70
|
fssd |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → ( ℕ0 × { 0 } ) : ℕ0 ⟶ ℝ ) |
72 |
71
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ℕ0 × { 0 } ) ‘ 𝑘 ) ∈ ℝ ) |
73 |
63 62
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) · ( 𝐵 ↑ 𝑗 ) ) ) ) ‘ 𝑘 ) ∈ ℂ ) |
74 |
|
climrel |
⊢ Rel ⇝ |
75 |
40
|
xpeq1i |
⊢ ( ℕ0 × { 0 } ) = ( ( ℤ≥ ‘ 0 ) × { 0 } ) |
76 |
|
seqeq3 |
⊢ ( ( ℕ0 × { 0 } ) = ( ( ℤ≥ ‘ 0 ) × { 0 } ) → seq 0 ( + , ( ℕ0 × { 0 } ) ) = seq 0 ( + , ( ( ℤ≥ ‘ 0 ) × { 0 } ) ) ) |
77 |
75 76
|
ax-mp |
⊢ seq 0 ( + , ( ℕ0 × { 0 } ) ) = seq 0 ( + , ( ( ℤ≥ ‘ 0 ) × { 0 } ) ) |
78 |
|
0z |
⊢ 0 ∈ ℤ |
79 |
|
serclim0 |
⊢ ( 0 ∈ ℤ → seq 0 ( + , ( ( ℤ≥ ‘ 0 ) × { 0 } ) ) ⇝ 0 ) |
80 |
78 79
|
ax-mp |
⊢ seq 0 ( + , ( ( ℤ≥ ‘ 0 ) × { 0 } ) ) ⇝ 0 |
81 |
77 80
|
eqbrtri |
⊢ seq 0 ( + , ( ℕ0 × { 0 } ) ) ⇝ 0 |
82 |
|
releldm |
⊢ ( ( Rel ⇝ ∧ seq 0 ( + , ( ℕ0 × { 0 } ) ) ⇝ 0 ) → seq 0 ( + , ( ℕ0 × { 0 } ) ) ∈ dom ⇝ ) |
83 |
74 81 82
|
mp2an |
⊢ seq 0 ( + , ( ℕ0 × { 0 } ) ) ∈ dom ⇝ |
84 |
83
|
a1i |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → seq 0 ( + , ( ℕ0 × { 0 } ) ) ∈ dom ⇝ ) |
85 |
|
eluznn0 |
⊢ ( ( ( 𝐶 + 1 ) ∈ ℕ0 ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → 𝑘 ∈ ℕ0 ) |
86 |
65 85
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → 𝑘 ∈ ℕ0 ) |
87 |
86 63
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) · ( 𝐵 ↑ 𝑗 ) ) ) ) ‘ 𝑘 ) = ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) |
88 |
|
0zd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → 0 ∈ ℤ ) |
89 |
86
|
nn0zd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → 𝑘 ∈ ℤ ) |
90 |
|
1zzd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → 1 ∈ ℤ ) |
91 |
89 90
|
zsubcld |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( 𝑘 − 1 ) ∈ ℤ ) |
92 |
14
|
nn0zd |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → 𝐶 ∈ ℤ ) |
93 |
92
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → 𝐶 ∈ ℤ ) |
94 |
14
|
nn0ge0d |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → 0 ≤ 𝐶 ) |
95 |
94
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → 0 ≤ 𝐶 ) |
96 |
|
eluzle |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) → ( 𝐶 + 1 ) ≤ 𝑘 ) |
97 |
96
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( 𝐶 + 1 ) ≤ 𝑘 ) |
98 |
93
|
zred |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → 𝐶 ∈ ℝ ) |
99 |
|
1red |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → 1 ∈ ℝ ) |
100 |
86
|
nn0red |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → 𝑘 ∈ ℝ ) |
101 |
|
leaddsub |
⊢ ( ( 𝐶 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( ( 𝐶 + 1 ) ≤ 𝑘 ↔ 𝐶 ≤ ( 𝑘 − 1 ) ) ) |
102 |
98 99 100 101
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( ( 𝐶 + 1 ) ≤ 𝑘 ↔ 𝐶 ≤ ( 𝑘 − 1 ) ) ) |
103 |
97 102
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → 𝐶 ≤ ( 𝑘 − 1 ) ) |
104 |
88 91 93 95 103
|
elfzd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → 𝐶 ∈ ( 0 ... ( 𝑘 − 1 ) ) ) |
105 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → 𝐶 ∈ ℂ ) |
106 |
105 86
|
bcc0 |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( ( 𝐶 C𝑐 𝑘 ) = 0 ↔ 𝐶 ∈ ( 0 ... ( 𝑘 − 1 ) ) ) ) |
107 |
104 106
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( 𝐶 C𝑐 𝑘 ) = 0 ) |
108 |
107
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) = ( 0 · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) |
109 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → 𝐴 ∈ ℂ ) |
110 |
|
eluzelcn |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) → 𝑘 ∈ ℂ ) |
111 |
110
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → 𝑘 ∈ ℂ ) |
112 |
105 111
|
subcld |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( 𝐶 − 𝑘 ) ∈ ℂ ) |
113 |
109 112
|
cxpcld |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) ∈ ℂ ) |
114 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → 𝐵 ∈ ℂ ) |
115 |
114 86
|
expcld |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( 𝐵 ↑ 𝑘 ) ∈ ℂ ) |
116 |
113 115
|
mulcld |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ∈ ℂ ) |
117 |
116
|
mul02d |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( 0 · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) = 0 ) |
118 |
108 117
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) = 0 ) |
119 |
87 118
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) · ( 𝐵 ↑ 𝑗 ) ) ) ) ‘ 𝑘 ) = 0 ) |
120 |
119
|
abs00bd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( abs ‘ ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) · ( 𝐵 ↑ 𝑗 ) ) ) ) ‘ 𝑘 ) ) = 0 ) |
121 |
|
0re |
⊢ 0 ∈ ℝ |
122 |
120 121
|
eqeltrdi |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( abs ‘ ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) · ( 𝐵 ↑ 𝑗 ) ) ) ) ‘ 𝑘 ) ) ∈ ℝ ) |
123 |
|
eqle |
⊢ ( ( ( abs ‘ ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) · ( 𝐵 ↑ 𝑗 ) ) ) ) ‘ 𝑘 ) ) ∈ ℝ ∧ ( abs ‘ ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) · ( 𝐵 ↑ 𝑗 ) ) ) ) ‘ 𝑘 ) ) = 0 ) → ( abs ‘ ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) · ( 𝐵 ↑ 𝑗 ) ) ) ) ‘ 𝑘 ) ) ≤ 0 ) |
124 |
122 120 123
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( abs ‘ ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) · ( 𝐵 ↑ 𝑗 ) ) ) ) ‘ 𝑘 ) ) ≤ 0 ) |
125 |
72
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ℕ0 × { 0 } ) ‘ 𝑘 ) ∈ ℂ ) |
126 |
86 125
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( ( ℕ0 × { 0 } ) ‘ 𝑘 ) ∈ ℂ ) |
127 |
126
|
mul02d |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( 0 · ( ( ℕ0 × { 0 } ) ‘ 𝑘 ) ) = 0 ) |
128 |
124 127
|
breqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ) → ( abs ‘ ( ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) · ( 𝐵 ↑ 𝑗 ) ) ) ) ‘ 𝑘 ) ) ≤ ( 0 · ( ( ℕ0 × { 0 } ) ‘ 𝑘 ) ) ) |
129 |
40 65 72 73 84 69 128
|
cvgcmpce |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → seq 0 ( + , ( 𝑗 ∈ ℕ0 ↦ ( ( 𝐶 C𝑐 𝑗 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑗 ) ) · ( 𝐵 ↑ 𝑗 ) ) ) ) ) ∈ dom ⇝ ) |
130 |
40 41 44 63 62 129
|
isumsplit |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → Σ 𝑘 ∈ ℕ0 ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) = ( Σ 𝑘 ∈ ( 0 ... ( ( 𝐶 + 1 ) − 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) + Σ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) ) |
131 |
|
1cnd |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → 1 ∈ ℂ ) |
132 |
36 131
|
pncand |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → ( ( 𝐶 + 1 ) − 1 ) = 𝐶 ) |
133 |
132
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → ( 0 ... ( ( 𝐶 + 1 ) − 1 ) ) = ( 0 ... 𝐶 ) ) |
134 |
133
|
sumeq1d |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → Σ 𝑘 ∈ ( 0 ... ( ( 𝐶 + 1 ) − 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) = Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) |
135 |
134
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → ( Σ 𝑘 ∈ ( 0 ... ( ( 𝐶 + 1 ) − 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) + Σ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) = ( Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) + Σ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) ) |
136 |
118
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → Σ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) = Σ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) 0 ) |
137 |
|
ssid |
⊢ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ⊆ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) |
138 |
137
|
orci |
⊢ ( ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ⊆ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ∨ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ∈ Fin ) |
139 |
|
sumz |
⊢ ( ( ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ⊆ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ∨ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ∈ Fin ) → Σ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) 0 = 0 ) |
140 |
138 139
|
ax-mp |
⊢ Σ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) 0 = 0 |
141 |
136 140
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → Σ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) = 0 ) |
142 |
141
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → ( Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) + Σ 𝑘 ∈ ( ℤ≥ ‘ ( 𝐶 + 1 ) ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) = ( Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) + 0 ) ) |
143 |
130 135 142
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → Σ 𝑘 ∈ ℕ0 ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) = ( Σ 𝑘 ∈ ( 0 ... 𝐶 ) ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) + 0 ) ) |
144 |
39 143 35
|
3eqtr4rd |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ℕ0 ) → ( ( 𝐴 + 𝐵 ) ↑𝑐 𝐶 ) = Σ 𝑘 ∈ ℕ0 ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐴 ↑𝑐 ( 𝐶 − 𝑘 ) ) · ( 𝐵 ↑ 𝑘 ) ) ) ) |