Step |
Hyp |
Ref |
Expression |
1 |
|
plyaddlem.1 |
⊢ ( 𝜑 → 𝐹 ∈ ( Poly ‘ 𝑆 ) ) |
2 |
|
plyaddlem.2 |
⊢ ( 𝜑 → 𝐺 ∈ ( Poly ‘ 𝑆 ) ) |
3 |
|
plyaddlem.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ0 ) |
4 |
|
plyaddlem.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
5 |
|
plyaddlem.a |
⊢ ( 𝜑 → 𝐴 : ℕ0 ⟶ ℂ ) |
6 |
|
plyaddlem.b |
⊢ ( 𝜑 → 𝐵 : ℕ0 ⟶ ℂ ) |
7 |
|
plyaddlem.a2 |
⊢ ( 𝜑 → ( 𝐴 “ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = { 0 } ) |
8 |
|
plyaddlem.b2 |
⊢ ( 𝜑 → ( 𝐵 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ) |
9 |
|
plyaddlem.f |
⊢ ( 𝜑 → 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
10 |
|
plyaddlem.g |
⊢ ( 𝜑 → 𝐺 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
11 |
|
cnex |
⊢ ℂ ∈ V |
12 |
11
|
a1i |
⊢ ( 𝜑 → ℂ ∈ V ) |
13 |
|
sumex |
⊢ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ V |
14 |
13
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ V ) |
15 |
|
sumex |
⊢ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ V |
16 |
15
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ V ) |
17 |
12 14 16 9 10
|
offval2 |
⊢ ( 𝜑 → ( 𝐹 ∘f · 𝐺 ) = ( 𝑧 ∈ ℂ ↦ ( Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) |
18 |
|
fveq2 |
⊢ ( 𝑚 = 𝑛 → ( 𝐵 ‘ 𝑚 ) = ( 𝐵 ‘ 𝑛 ) ) |
19 |
|
oveq2 |
⊢ ( 𝑚 = 𝑛 → ( 𝑧 ↑ 𝑚 ) = ( 𝑧 ↑ 𝑛 ) ) |
20 |
18 19
|
oveq12d |
⊢ ( 𝑚 = 𝑛 → ( ( 𝐵 ‘ 𝑚 ) · ( 𝑧 ↑ 𝑚 ) ) = ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) |
21 |
20
|
oveq2d |
⊢ ( 𝑚 = 𝑛 → ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑚 ) · ( 𝑧 ↑ 𝑚 ) ) ) = ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) ) |
22 |
|
fveq2 |
⊢ ( 𝑚 = ( 𝑛 − 𝑘 ) → ( 𝐵 ‘ 𝑚 ) = ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) |
23 |
|
oveq2 |
⊢ ( 𝑚 = ( 𝑛 − 𝑘 ) → ( 𝑧 ↑ 𝑚 ) = ( 𝑧 ↑ ( 𝑛 − 𝑘 ) ) ) |
24 |
22 23
|
oveq12d |
⊢ ( 𝑚 = ( 𝑛 − 𝑘 ) → ( ( 𝐵 ‘ 𝑚 ) · ( 𝑧 ↑ 𝑚 ) ) = ( ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) · ( 𝑧 ↑ ( 𝑛 − 𝑘 ) ) ) ) |
25 |
24
|
oveq2d |
⊢ ( 𝑚 = ( 𝑛 − 𝑘 ) → ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑚 ) · ( 𝑧 ↑ 𝑚 ) ) ) = ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) · ( 𝑧 ↑ ( 𝑛 − 𝑘 ) ) ) ) ) |
26 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) → 𝑘 ∈ ℕ0 ) |
27 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → 𝐴 : ℕ0 ⟶ ℂ ) |
28 |
27
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ‘ 𝑘 ) ∈ ℂ ) |
29 |
|
expcl |
⊢ ( ( 𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝑧 ↑ 𝑘 ) ∈ ℂ ) |
30 |
29
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑧 ↑ 𝑘 ) ∈ ℂ ) |
31 |
28 30
|
mulcld |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ ℂ ) |
32 |
26 31
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ ℂ ) |
33 |
|
elfznn0 |
⊢ ( 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) → 𝑛 ∈ ℕ0 ) |
34 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → 𝐵 : ℕ0 ⟶ ℂ ) |
35 |
34
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝐵 ‘ 𝑛 ) ∈ ℂ ) |
36 |
|
expcl |
⊢ ( ( 𝑧 ∈ ℂ ∧ 𝑛 ∈ ℕ0 ) → ( 𝑧 ↑ 𝑛 ) ∈ ℂ ) |
37 |
36
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑧 ↑ 𝑛 ) ∈ ℂ ) |
38 |
35 37
|
mulcld |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ∈ ℂ ) |
39 |
33 38
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ) → ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ∈ ℂ ) |
40 |
32 39
|
anim12dan |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ ( 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ∧ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ) ) → ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ ℂ ∧ ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ∈ ℂ ) ) |
41 |
|
mulcl |
⊢ ( ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ ℂ ∧ ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ∈ ℂ ) → ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) ∈ ℂ ) |
42 |
40 41
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ ( 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ∧ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ) ) → ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) ∈ ℂ ) |
43 |
21 25 42
|
fsum0diag2 |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) Σ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) = Σ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) · ( 𝑧 ↑ ( 𝑛 − 𝑘 ) ) ) ) ) |
44 |
3
|
nn0cnd |
⊢ ( 𝜑 → 𝑀 ∈ ℂ ) |
45 |
44
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → 𝑀 ∈ ℂ ) |
46 |
4
|
nn0cnd |
⊢ ( 𝜑 → 𝑁 ∈ ℂ ) |
47 |
46
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → 𝑁 ∈ ℂ ) |
48 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... 𝑀 ) → 𝑘 ∈ ℕ0 ) |
49 |
48
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → 𝑘 ∈ ℕ0 ) |
50 |
49
|
nn0cnd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → 𝑘 ∈ ℂ ) |
51 |
45 47 50
|
addsubd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( ( 𝑀 + 𝑁 ) − 𝑘 ) = ( ( 𝑀 − 𝑘 ) + 𝑁 ) ) |
52 |
|
fznn0sub |
⊢ ( 𝑘 ∈ ( 0 ... 𝑀 ) → ( 𝑀 − 𝑘 ) ∈ ℕ0 ) |
53 |
52
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( 𝑀 − 𝑘 ) ∈ ℕ0 ) |
54 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
55 |
53 54
|
eleqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( 𝑀 − 𝑘 ) ∈ ( ℤ≥ ‘ 0 ) ) |
56 |
4
|
nn0zd |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
57 |
56
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → 𝑁 ∈ ℤ ) |
58 |
|
eluzadd |
⊢ ( ( ( 𝑀 − 𝑘 ) ∈ ( ℤ≥ ‘ 0 ) ∧ 𝑁 ∈ ℤ ) → ( ( 𝑀 − 𝑘 ) + 𝑁 ) ∈ ( ℤ≥ ‘ ( 0 + 𝑁 ) ) ) |
59 |
55 57 58
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( ( 𝑀 − 𝑘 ) + 𝑁 ) ∈ ( ℤ≥ ‘ ( 0 + 𝑁 ) ) ) |
60 |
51 59
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( ( 𝑀 + 𝑁 ) − 𝑘 ) ∈ ( ℤ≥ ‘ ( 0 + 𝑁 ) ) ) |
61 |
47
|
addid2d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( 0 + 𝑁 ) = 𝑁 ) |
62 |
61
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( ℤ≥ ‘ ( 0 + 𝑁 ) ) = ( ℤ≥ ‘ 𝑁 ) ) |
63 |
60 62
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( ( 𝑀 + 𝑁 ) − 𝑘 ) ∈ ( ℤ≥ ‘ 𝑁 ) ) |
64 |
|
fzss2 |
⊢ ( ( ( 𝑀 + 𝑁 ) − 𝑘 ) ∈ ( ℤ≥ ‘ 𝑁 ) → ( 0 ... 𝑁 ) ⊆ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ) |
65 |
63 64
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( 0 ... 𝑁 ) ⊆ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ) |
66 |
48 31
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ ℂ ) |
67 |
66
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ ℂ ) |
68 |
|
elfznn0 |
⊢ ( 𝑛 ∈ ( 0 ... 𝑁 ) → 𝑛 ∈ ℕ0 ) |
69 |
68 38
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ∈ ℂ ) |
70 |
69
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ∈ ℂ ) |
71 |
67 70
|
mulcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( 0 ... 𝑁 ) ) → ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) ∈ ℂ ) |
72 |
|
eldifn |
⊢ ( 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) → ¬ 𝑛 ∈ ( 0 ... 𝑁 ) ) |
73 |
72
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ¬ 𝑛 ∈ ( 0 ... 𝑁 ) ) |
74 |
|
eldifi |
⊢ ( 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) → 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ) |
75 |
74 33
|
syl |
⊢ ( 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) → 𝑛 ∈ ℕ0 ) |
76 |
75
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → 𝑛 ∈ ℕ0 ) |
77 |
|
peano2nn0 |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ∈ ℕ0 ) |
78 |
4 77
|
syl |
⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ℕ0 ) |
79 |
78 54
|
eleqtrdi |
⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 0 ) ) |
80 |
|
uzsplit |
⊢ ( ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 0 ) → ( ℤ≥ ‘ 0 ) = ( ( 0 ... ( ( 𝑁 + 1 ) − 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) |
81 |
79 80
|
syl |
⊢ ( 𝜑 → ( ℤ≥ ‘ 0 ) = ( ( 0 ... ( ( 𝑁 + 1 ) − 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) |
82 |
54 81
|
syl5eq |
⊢ ( 𝜑 → ℕ0 = ( ( 0 ... ( ( 𝑁 + 1 ) − 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) |
83 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
84 |
|
pncan |
⊢ ( ( 𝑁 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑁 + 1 ) − 1 ) = 𝑁 ) |
85 |
46 83 84
|
sylancl |
⊢ ( 𝜑 → ( ( 𝑁 + 1 ) − 1 ) = 𝑁 ) |
86 |
85
|
oveq2d |
⊢ ( 𝜑 → ( 0 ... ( ( 𝑁 + 1 ) − 1 ) ) = ( 0 ... 𝑁 ) ) |
87 |
86
|
uneq1d |
⊢ ( 𝜑 → ( ( 0 ... ( ( 𝑁 + 1 ) − 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = ( ( 0 ... 𝑁 ) ∪ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) |
88 |
82 87
|
eqtrd |
⊢ ( 𝜑 → ℕ0 = ( ( 0 ... 𝑁 ) ∪ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) |
89 |
88
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ℕ0 = ( ( 0 ... 𝑁 ) ∪ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) |
90 |
76 89
|
eleqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → 𝑛 ∈ ( ( 0 ... 𝑁 ) ∪ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) |
91 |
|
elun |
⊢ ( 𝑛 ∈ ( ( 0 ... 𝑁 ) ∪ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ↔ ( 𝑛 ∈ ( 0 ... 𝑁 ) ∨ 𝑛 ∈ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) |
92 |
90 91
|
sylib |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( 𝑛 ∈ ( 0 ... 𝑁 ) ∨ 𝑛 ∈ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) |
93 |
92
|
ord |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( ¬ 𝑛 ∈ ( 0 ... 𝑁 ) → 𝑛 ∈ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) |
94 |
73 93
|
mpd |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → 𝑛 ∈ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) |
95 |
6
|
ffund |
⊢ ( 𝜑 → Fun 𝐵 ) |
96 |
|
ssun2 |
⊢ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ⊆ ( ( 0 ... ( ( 𝑁 + 1 ) − 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) |
97 |
96 82
|
sseqtrrid |
⊢ ( 𝜑 → ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ⊆ ℕ0 ) |
98 |
6
|
fdmd |
⊢ ( 𝜑 → dom 𝐵 = ℕ0 ) |
99 |
97 98
|
sseqtrrd |
⊢ ( 𝜑 → ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ⊆ dom 𝐵 ) |
100 |
|
funfvima2 |
⊢ ( ( Fun 𝐵 ∧ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ⊆ dom 𝐵 ) → ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) → ( 𝐵 ‘ 𝑛 ) ∈ ( 𝐵 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) |
101 |
95 99 100
|
syl2anc |
⊢ ( 𝜑 → ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) → ( 𝐵 ‘ 𝑛 ) ∈ ( 𝐵 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) |
102 |
101
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( 𝑛 ∈ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) → ( 𝐵 ‘ 𝑛 ) ∈ ( 𝐵 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) ) |
103 |
94 102
|
mpd |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( 𝐵 ‘ 𝑛 ) ∈ ( 𝐵 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ) |
104 |
8
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( 𝐵 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ) |
105 |
103 104
|
eleqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( 𝐵 ‘ 𝑛 ) ∈ { 0 } ) |
106 |
|
elsni |
⊢ ( ( 𝐵 ‘ 𝑛 ) ∈ { 0 } → ( 𝐵 ‘ 𝑛 ) = 0 ) |
107 |
105 106
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( 𝐵 ‘ 𝑛 ) = 0 ) |
108 |
107
|
oveq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) = ( 0 · ( 𝑧 ↑ 𝑛 ) ) ) |
109 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → 𝑧 ∈ ℂ ) |
110 |
109 75 36
|
syl2an |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( 𝑧 ↑ 𝑛 ) ∈ ℂ ) |
111 |
110
|
mul02d |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( 0 · ( 𝑧 ↑ 𝑛 ) ) = 0 ) |
112 |
108 111
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) = 0 ) |
113 |
112
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) = ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · 0 ) ) |
114 |
66
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ ℂ ) |
115 |
114
|
mul01d |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · 0 ) = 0 ) |
116 |
113 115
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) = 0 ) |
117 |
|
fzfid |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∈ Fin ) |
118 |
65 71 116 117
|
fsumss |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → Σ 𝑛 ∈ ( 0 ... 𝑁 ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) = Σ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) ) |
119 |
118
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... 𝑀 ) Σ 𝑛 ∈ ( 0 ... 𝑁 ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) = Σ 𝑘 ∈ ( 0 ... 𝑀 ) Σ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) ) |
120 |
|
fzfid |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → ( 0 ... 𝑀 ) ∈ Fin ) |
121 |
|
fzfid |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → ( 0 ... 𝑁 ) ∈ Fin ) |
122 |
120 121 66 69
|
fsum2mul |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... 𝑀 ) Σ 𝑛 ∈ ( 0 ... 𝑁 ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) = ( Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · Σ 𝑛 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) ) |
123 |
44 46
|
addcomd |
⊢ ( 𝜑 → ( 𝑀 + 𝑁 ) = ( 𝑁 + 𝑀 ) ) |
124 |
4 54
|
eleqtrdi |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 0 ) ) |
125 |
3
|
nn0zd |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
126 |
|
eluzadd |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 0 ) ∧ 𝑀 ∈ ℤ ) → ( 𝑁 + 𝑀 ) ∈ ( ℤ≥ ‘ ( 0 + 𝑀 ) ) ) |
127 |
124 125 126
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 + 𝑀 ) ∈ ( ℤ≥ ‘ ( 0 + 𝑀 ) ) ) |
128 |
44
|
addid2d |
⊢ ( 𝜑 → ( 0 + 𝑀 ) = 𝑀 ) |
129 |
128
|
fveq2d |
⊢ ( 𝜑 → ( ℤ≥ ‘ ( 0 + 𝑀 ) ) = ( ℤ≥ ‘ 𝑀 ) ) |
130 |
127 129
|
eleqtrd |
⊢ ( 𝜑 → ( 𝑁 + 𝑀 ) ∈ ( ℤ≥ ‘ 𝑀 ) ) |
131 |
123 130
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 𝑀 ) ) |
132 |
|
fzss2 |
⊢ ( ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 𝑀 ) → ( 0 ... 𝑀 ) ⊆ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
133 |
131 132
|
syl |
⊢ ( 𝜑 → ( 0 ... 𝑀 ) ⊆ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
134 |
133
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → ( 0 ... 𝑀 ) ⊆ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
135 |
66
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ ℂ ) |
136 |
39
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ) → ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ∈ ℂ ) |
137 |
135 136
|
mulcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ) → ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) ∈ ℂ ) |
138 |
117 137
|
fsumcl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → Σ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) ∈ ℂ ) |
139 |
|
eldifn |
⊢ ( 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) → ¬ 𝑘 ∈ ( 0 ... 𝑀 ) ) |
140 |
139
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ¬ 𝑘 ∈ ( 0 ... 𝑀 ) ) |
141 |
|
eldifi |
⊢ ( 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) → 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
142 |
141 26
|
syl |
⊢ ( 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) → 𝑘 ∈ ℕ0 ) |
143 |
142
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → 𝑘 ∈ ℕ0 ) |
144 |
|
peano2nn0 |
⊢ ( 𝑀 ∈ ℕ0 → ( 𝑀 + 1 ) ∈ ℕ0 ) |
145 |
3 144
|
syl |
⊢ ( 𝜑 → ( 𝑀 + 1 ) ∈ ℕ0 ) |
146 |
145 54
|
eleqtrdi |
⊢ ( 𝜑 → ( 𝑀 + 1 ) ∈ ( ℤ≥ ‘ 0 ) ) |
147 |
|
uzsplit |
⊢ ( ( 𝑀 + 1 ) ∈ ( ℤ≥ ‘ 0 ) → ( ℤ≥ ‘ 0 ) = ( ( 0 ... ( ( 𝑀 + 1 ) − 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) |
148 |
146 147
|
syl |
⊢ ( 𝜑 → ( ℤ≥ ‘ 0 ) = ( ( 0 ... ( ( 𝑀 + 1 ) − 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) |
149 |
54 148
|
syl5eq |
⊢ ( 𝜑 → ℕ0 = ( ( 0 ... ( ( 𝑀 + 1 ) − 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) |
150 |
|
pncan |
⊢ ( ( 𝑀 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑀 + 1 ) − 1 ) = 𝑀 ) |
151 |
44 83 150
|
sylancl |
⊢ ( 𝜑 → ( ( 𝑀 + 1 ) − 1 ) = 𝑀 ) |
152 |
151
|
oveq2d |
⊢ ( 𝜑 → ( 0 ... ( ( 𝑀 + 1 ) − 1 ) ) = ( 0 ... 𝑀 ) ) |
153 |
152
|
uneq1d |
⊢ ( 𝜑 → ( ( 0 ... ( ( 𝑀 + 1 ) − 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = ( ( 0 ... 𝑀 ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) |
154 |
149 153
|
eqtrd |
⊢ ( 𝜑 → ℕ0 = ( ( 0 ... 𝑀 ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) |
155 |
154
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ℕ0 = ( ( 0 ... 𝑀 ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) |
156 |
143 155
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → 𝑘 ∈ ( ( 0 ... 𝑀 ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) |
157 |
|
elun |
⊢ ( 𝑘 ∈ ( ( 0 ... 𝑀 ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ↔ ( 𝑘 ∈ ( 0 ... 𝑀 ) ∨ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) |
158 |
156 157
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( 𝑘 ∈ ( 0 ... 𝑀 ) ∨ 𝑘 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) |
159 |
158
|
ord |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( ¬ 𝑘 ∈ ( 0 ... 𝑀 ) → 𝑘 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) |
160 |
140 159
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → 𝑘 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) |
161 |
5
|
ffund |
⊢ ( 𝜑 → Fun 𝐴 ) |
162 |
|
ssun2 |
⊢ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ⊆ ( ( 0 ... ( ( 𝑀 + 1 ) − 1 ) ) ∪ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) |
163 |
162 149
|
sseqtrrid |
⊢ ( 𝜑 → ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ⊆ ℕ0 ) |
164 |
5
|
fdmd |
⊢ ( 𝜑 → dom 𝐴 = ℕ0 ) |
165 |
163 164
|
sseqtrrd |
⊢ ( 𝜑 → ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ⊆ dom 𝐴 ) |
166 |
|
funfvima2 |
⊢ ( ( Fun 𝐴 ∧ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ⊆ dom 𝐴 ) → ( 𝑘 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) → ( 𝐴 ‘ 𝑘 ) ∈ ( 𝐴 “ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) ) |
167 |
161 165 166
|
syl2anc |
⊢ ( 𝜑 → ( 𝑘 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) → ( 𝐴 ‘ 𝑘 ) ∈ ( 𝐴 “ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) ) |
168 |
167
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( 𝑘 ∈ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) → ( 𝐴 ‘ 𝑘 ) ∈ ( 𝐴 “ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) ) |
169 |
160 168
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( 𝐴 ‘ 𝑘 ) ∈ ( 𝐴 “ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) ) |
170 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( 𝐴 “ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = { 0 } ) |
171 |
169 170
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( 𝐴 ‘ 𝑘 ) ∈ { 0 } ) |
172 |
|
elsni |
⊢ ( ( 𝐴 ‘ 𝑘 ) ∈ { 0 } → ( 𝐴 ‘ 𝑘 ) = 0 ) |
173 |
171 172
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( 𝐴 ‘ 𝑘 ) = 0 ) |
174 |
173
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = ( 0 · ( 𝑧 ↑ 𝑘 ) ) ) |
175 |
142 30
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( 𝑧 ↑ 𝑘 ) ∈ ℂ ) |
176 |
175
|
mul02d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( 0 · ( 𝑧 ↑ 𝑘 ) ) = 0 ) |
177 |
174 176
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = 0 ) |
178 |
177
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) ∧ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = 0 ) |
179 |
178
|
oveq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) ∧ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ) → ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) = ( 0 · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) ) |
180 |
39
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) ∧ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ) → ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ∈ ℂ ) |
181 |
180
|
mul02d |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) ∧ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ) → ( 0 · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) = 0 ) |
182 |
179 181
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) ∧ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ) → ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) = 0 ) |
183 |
182
|
sumeq2dv |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → Σ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) = Σ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) 0 ) |
184 |
|
fzfid |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∈ Fin ) |
185 |
184
|
olcd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ⊆ ( ℤ≥ ‘ 0 ) ∨ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∈ Fin ) ) |
186 |
|
sumz |
⊢ ( ( ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ⊆ ( ℤ≥ ‘ 0 ) ∨ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ∈ Fin ) → Σ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) 0 = 0 ) |
187 |
185 186
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → Σ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) 0 = 0 ) |
188 |
183 187
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → Σ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) = 0 ) |
189 |
|
fzfid |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → ( 0 ... ( 𝑀 + 𝑁 ) ) ∈ Fin ) |
190 |
134 138 188 189
|
fsumss |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... 𝑀 ) Σ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) = Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) Σ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) ) |
191 |
119 122 190
|
3eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → ( Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · Σ 𝑛 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) = Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) Σ 𝑛 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) − 𝑘 ) ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) ) |
192 |
|
fzfid |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → ( 0 ... 𝑛 ) ∈ Fin ) |
193 |
|
elfznn0 |
⊢ ( 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) → 𝑛 ∈ ℕ0 ) |
194 |
193 37
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → ( 𝑧 ↑ 𝑛 ) ∈ ℂ ) |
195 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → 𝜑 ) |
196 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... 𝑛 ) → 𝑘 ∈ ℕ0 ) |
197 |
5
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ‘ 𝑘 ) ∈ ℂ ) |
198 |
195 196 197
|
syl2an |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( 𝐴 ‘ 𝑘 ) ∈ ℂ ) |
199 |
|
fznn0sub |
⊢ ( 𝑘 ∈ ( 0 ... 𝑛 ) → ( 𝑛 − 𝑘 ) ∈ ℕ0 ) |
200 |
6
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ ( 𝑛 − 𝑘 ) ∈ ℕ0 ) → ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ∈ ℂ ) |
201 |
195 199 200
|
syl2an |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ∈ ℂ ) |
202 |
198 201
|
mulcld |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) ∈ ℂ ) |
203 |
192 194 202
|
fsummulc1 |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → ( Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) · ( 𝑧 ↑ 𝑛 ) ) = Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) · ( 𝑧 ↑ 𝑛 ) ) ) |
204 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → 𝑧 ∈ ℂ ) |
205 |
204 196 29
|
syl2an |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( 𝑧 ↑ 𝑘 ) ∈ ℂ ) |
206 |
|
expcl |
⊢ ( ( 𝑧 ∈ ℂ ∧ ( 𝑛 − 𝑘 ) ∈ ℕ0 ) → ( 𝑧 ↑ ( 𝑛 − 𝑘 ) ) ∈ ℂ ) |
207 |
204 199 206
|
syl2an |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( 𝑧 ↑ ( 𝑛 − 𝑘 ) ) ∈ ℂ ) |
208 |
198 205 201 207
|
mul4d |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) · ( 𝑧 ↑ ( 𝑛 − 𝑘 ) ) ) ) = ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) · ( ( 𝑧 ↑ 𝑘 ) · ( 𝑧 ↑ ( 𝑛 − 𝑘 ) ) ) ) ) |
209 |
204
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → 𝑧 ∈ ℂ ) |
210 |
199
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( 𝑛 − 𝑘 ) ∈ ℕ0 ) |
211 |
196
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → 𝑘 ∈ ℕ0 ) |
212 |
209 210 211
|
expaddd |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( 𝑧 ↑ ( 𝑘 + ( 𝑛 − 𝑘 ) ) ) = ( ( 𝑧 ↑ 𝑘 ) · ( 𝑧 ↑ ( 𝑛 − 𝑘 ) ) ) ) |
213 |
211
|
nn0cnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → 𝑘 ∈ ℂ ) |
214 |
193
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → 𝑛 ∈ ℕ0 ) |
215 |
214
|
nn0cnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → 𝑛 ∈ ℂ ) |
216 |
213 215
|
pncan3d |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( 𝑘 + ( 𝑛 − 𝑘 ) ) = 𝑛 ) |
217 |
216
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( 𝑧 ↑ ( 𝑘 + ( 𝑛 − 𝑘 ) ) ) = ( 𝑧 ↑ 𝑛 ) ) |
218 |
212 217
|
eqtr3d |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( ( 𝑧 ↑ 𝑘 ) · ( 𝑧 ↑ ( 𝑛 − 𝑘 ) ) ) = ( 𝑧 ↑ 𝑛 ) ) |
219 |
218
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) · ( ( 𝑧 ↑ 𝑘 ) · ( 𝑧 ↑ ( 𝑛 − 𝑘 ) ) ) ) = ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) · ( 𝑧 ↑ 𝑛 ) ) ) |
220 |
208 219
|
eqtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) ∧ 𝑘 ∈ ( 0 ... 𝑛 ) ) → ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) · ( 𝑧 ↑ ( 𝑛 − 𝑘 ) ) ) ) = ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) · ( 𝑧 ↑ 𝑛 ) ) ) |
221 |
220
|
sumeq2dv |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) · ( 𝑧 ↑ ( 𝑛 − 𝑘 ) ) ) ) = Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) · ( 𝑧 ↑ 𝑛 ) ) ) |
222 |
203 221
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → ( Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) · ( 𝑧 ↑ 𝑛 ) ) = Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) · ( 𝑧 ↑ ( 𝑛 − 𝑘 ) ) ) ) ) |
223 |
222
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) · ( 𝑧 ↑ 𝑛 ) ) = Σ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · ( ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) · ( 𝑧 ↑ ( 𝑛 − 𝑘 ) ) ) ) ) |
224 |
43 191 223
|
3eqtr4rd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) · ( 𝑧 ↑ 𝑛 ) ) = ( Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · Σ 𝑛 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) ) |
225 |
|
fveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝐵 ‘ 𝑛 ) = ( 𝐵 ‘ 𝑘 ) ) |
226 |
|
oveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝑧 ↑ 𝑛 ) = ( 𝑧 ↑ 𝑘 ) ) |
227 |
225 226
|
oveq12d |
⊢ ( 𝑛 = 𝑘 → ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) = ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
228 |
227
|
cbvsumv |
⊢ Σ 𝑛 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) = Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) |
229 |
228
|
oveq2i |
⊢ ( Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · Σ 𝑛 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑛 ) · ( 𝑧 ↑ 𝑛 ) ) ) = ( Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
230 |
224 229
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) · ( 𝑧 ↑ 𝑛 ) ) = ( Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
231 |
230
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑧 ∈ ℂ ↦ Σ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) · ( 𝑧 ↑ 𝑛 ) ) ) = ( 𝑧 ∈ ℂ ↦ ( Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) · Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) ) |
232 |
17 231
|
eqtr4d |
⊢ ( 𝜑 → ( 𝐹 ∘f · 𝐺 ) = ( 𝑧 ∈ ℂ ↦ Σ 𝑛 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝐵 ‘ ( 𝑛 − 𝑘 ) ) ) · ( 𝑧 ↑ 𝑛 ) ) ) ) |