Step |
Hyp |
Ref |
Expression |
1 |
|
coeeu.1 |
⊢ ( 𝜑 → 𝐹 ∈ ( Poly ‘ 𝑆 ) ) |
2 |
|
coeeu.2 |
⊢ ( 𝜑 → 𝐴 ∈ ( ℂ ↑m ℕ0 ) ) |
3 |
|
coeeu.3 |
⊢ ( 𝜑 → 𝐵 ∈ ( ℂ ↑m ℕ0 ) ) |
4 |
|
coeeu.4 |
⊢ ( 𝜑 → 𝑀 ∈ ℕ0 ) |
5 |
|
coeeu.5 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
6 |
|
coeeu.6 |
⊢ ( 𝜑 → ( 𝐴 “ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = { 0 } ) |
7 |
|
coeeu.7 |
⊢ ( 𝜑 → ( 𝐵 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ) |
8 |
|
coeeu.8 |
⊢ ( 𝜑 → 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
9 |
|
coeeu.9 |
⊢ ( 𝜑 → 𝐹 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
10 |
|
ssidd |
⊢ ( 𝜑 → ℂ ⊆ ℂ ) |
11 |
4 5
|
nn0addcld |
⊢ ( 𝜑 → ( 𝑀 + 𝑁 ) ∈ ℕ0 ) |
12 |
|
subcl |
⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 − 𝑦 ) ∈ ℂ ) |
13 |
12
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( 𝑥 − 𝑦 ) ∈ ℂ ) |
14 |
|
cnex |
⊢ ℂ ∈ V |
15 |
|
nn0ex |
⊢ ℕ0 ∈ V |
16 |
14 15
|
elmap |
⊢ ( 𝐴 ∈ ( ℂ ↑m ℕ0 ) ↔ 𝐴 : ℕ0 ⟶ ℂ ) |
17 |
2 16
|
sylib |
⊢ ( 𝜑 → 𝐴 : ℕ0 ⟶ ℂ ) |
18 |
14 15
|
elmap |
⊢ ( 𝐵 ∈ ( ℂ ↑m ℕ0 ) ↔ 𝐵 : ℕ0 ⟶ ℂ ) |
19 |
3 18
|
sylib |
⊢ ( 𝜑 → 𝐵 : ℕ0 ⟶ ℂ ) |
20 |
15
|
a1i |
⊢ ( 𝜑 → ℕ0 ∈ V ) |
21 |
|
inidm |
⊢ ( ℕ0 ∩ ℕ0 ) = ℕ0 |
22 |
13 17 19 20 20 21
|
off |
⊢ ( 𝜑 → ( 𝐴 ∘f − 𝐵 ) : ℕ0 ⟶ ℂ ) |
23 |
14 15
|
elmap |
⊢ ( ( 𝐴 ∘f − 𝐵 ) ∈ ( ℂ ↑m ℕ0 ) ↔ ( 𝐴 ∘f − 𝐵 ) : ℕ0 ⟶ ℂ ) |
24 |
22 23
|
sylibr |
⊢ ( 𝜑 → ( 𝐴 ∘f − 𝐵 ) ∈ ( ℂ ↑m ℕ0 ) ) |
25 |
|
0cn |
⊢ 0 ∈ ℂ |
26 |
|
snssi |
⊢ ( 0 ∈ ℂ → { 0 } ⊆ ℂ ) |
27 |
25 26
|
ax-mp |
⊢ { 0 } ⊆ ℂ |
28 |
|
ssequn2 |
⊢ ( { 0 } ⊆ ℂ ↔ ( ℂ ∪ { 0 } ) = ℂ ) |
29 |
27 28
|
mpbi |
⊢ ( ℂ ∪ { 0 } ) = ℂ |
30 |
29
|
oveq1i |
⊢ ( ( ℂ ∪ { 0 } ) ↑m ℕ0 ) = ( ℂ ↑m ℕ0 ) |
31 |
24 30
|
eleqtrrdi |
⊢ ( 𝜑 → ( 𝐴 ∘f − 𝐵 ) ∈ ( ( ℂ ∪ { 0 } ) ↑m ℕ0 ) ) |
32 |
11
|
nn0red |
⊢ ( 𝜑 → ( 𝑀 + 𝑁 ) ∈ ℝ ) |
33 |
|
nn0re |
⊢ ( 𝑘 ∈ ℕ0 → 𝑘 ∈ ℝ ) |
34 |
|
ltnle |
⊢ ( ( ( 𝑀 + 𝑁 ) ∈ ℝ ∧ 𝑘 ∈ ℝ ) → ( ( 𝑀 + 𝑁 ) < 𝑘 ↔ ¬ 𝑘 ≤ ( 𝑀 + 𝑁 ) ) ) |
35 |
32 33 34
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑀 + 𝑁 ) < 𝑘 ↔ ¬ 𝑘 ≤ ( 𝑀 + 𝑁 ) ) ) |
36 |
17
|
ffnd |
⊢ ( 𝜑 → 𝐴 Fn ℕ0 ) |
37 |
19
|
ffnd |
⊢ ( 𝜑 → 𝐵 Fn ℕ0 ) |
38 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑘 ) ) |
39 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐵 ‘ 𝑘 ) = ( 𝐵 ‘ 𝑘 ) ) |
40 |
36 37 20 20 21 38 39
|
ofval |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) = ( ( 𝐴 ‘ 𝑘 ) − ( 𝐵 ‘ 𝑘 ) ) ) |
41 |
40
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) = ( ( 𝐴 ‘ 𝑘 ) − ( 𝐵 ‘ 𝑘 ) ) ) |
42 |
4
|
nn0red |
⊢ ( 𝜑 → 𝑀 ∈ ℝ ) |
43 |
42
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → 𝑀 ∈ ℝ ) |
44 |
32
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → ( 𝑀 + 𝑁 ) ∈ ℝ ) |
45 |
33
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℝ ) |
46 |
45
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → 𝑘 ∈ ℝ ) |
47 |
4
|
nn0cnd |
⊢ ( 𝜑 → 𝑀 ∈ ℂ ) |
48 |
5
|
nn0cnd |
⊢ ( 𝜑 → 𝑁 ∈ ℂ ) |
49 |
47 48
|
addcomd |
⊢ ( 𝜑 → ( 𝑀 + 𝑁 ) = ( 𝑁 + 𝑀 ) ) |
50 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
51 |
5 50
|
eleqtrdi |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 0 ) ) |
52 |
4
|
nn0zd |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
53 |
|
eluzadd |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 0 ) ∧ 𝑀 ∈ ℤ ) → ( 𝑁 + 𝑀 ) ∈ ( ℤ≥ ‘ ( 0 + 𝑀 ) ) ) |
54 |
51 52 53
|
syl2anc |
⊢ ( 𝜑 → ( 𝑁 + 𝑀 ) ∈ ( ℤ≥ ‘ ( 0 + 𝑀 ) ) ) |
55 |
49 54
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ ( 0 + 𝑀 ) ) ) |
56 |
47
|
addid2d |
⊢ ( 𝜑 → ( 0 + 𝑀 ) = 𝑀 ) |
57 |
56
|
fveq2d |
⊢ ( 𝜑 → ( ℤ≥ ‘ ( 0 + 𝑀 ) ) = ( ℤ≥ ‘ 𝑀 ) ) |
58 |
55 57
|
eleqtrd |
⊢ ( 𝜑 → ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 𝑀 ) ) |
59 |
|
eluzle |
⊢ ( ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ≤ ( 𝑀 + 𝑁 ) ) |
60 |
58 59
|
syl |
⊢ ( 𝜑 → 𝑀 ≤ ( 𝑀 + 𝑁 ) ) |
61 |
60
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → 𝑀 ≤ ( 𝑀 + 𝑁 ) ) |
62 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → ( 𝑀 + 𝑁 ) < 𝑘 ) |
63 |
43 44 46 61 62
|
lelttrd |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → 𝑀 < 𝑘 ) |
64 |
43 46
|
ltnled |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → ( 𝑀 < 𝑘 ↔ ¬ 𝑘 ≤ 𝑀 ) ) |
65 |
63 64
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → ¬ 𝑘 ≤ 𝑀 ) |
66 |
|
plyco0 |
⊢ ( ( 𝑀 ∈ ℕ0 ∧ 𝐴 : ℕ0 ⟶ ℂ ) → ( ( 𝐴 “ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = { 0 } ↔ ∀ 𝑘 ∈ ℕ0 ( ( 𝐴 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑀 ) ) ) |
67 |
4 17 66
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐴 “ ( ℤ≥ ‘ ( 𝑀 + 1 ) ) ) = { 0 } ↔ ∀ 𝑘 ∈ ℕ0 ( ( 𝐴 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑀 ) ) ) |
68 |
6 67
|
mpbid |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ℕ0 ( ( 𝐴 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑀 ) ) |
69 |
68
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑀 ) ) |
70 |
69
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → ( ( 𝐴 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑀 ) ) |
71 |
70
|
necon1bd |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → ( ¬ 𝑘 ≤ 𝑀 → ( 𝐴 ‘ 𝑘 ) = 0 ) ) |
72 |
65 71
|
mpd |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → ( 𝐴 ‘ 𝑘 ) = 0 ) |
73 |
5
|
nn0red |
⊢ ( 𝜑 → 𝑁 ∈ ℝ ) |
74 |
73
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → 𝑁 ∈ ℝ ) |
75 |
4 50
|
eleqtrdi |
⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ 0 ) ) |
76 |
5
|
nn0zd |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
77 |
|
eluzadd |
⊢ ( ( 𝑀 ∈ ( ℤ≥ ‘ 0 ) ∧ 𝑁 ∈ ℤ ) → ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ ( 0 + 𝑁 ) ) ) |
78 |
75 76 77
|
syl2anc |
⊢ ( 𝜑 → ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ ( 0 + 𝑁 ) ) ) |
79 |
48
|
addid2d |
⊢ ( 𝜑 → ( 0 + 𝑁 ) = 𝑁 ) |
80 |
79
|
fveq2d |
⊢ ( 𝜑 → ( ℤ≥ ‘ ( 0 + 𝑁 ) ) = ( ℤ≥ ‘ 𝑁 ) ) |
81 |
78 80
|
eleqtrd |
⊢ ( 𝜑 → ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 𝑁 ) ) |
82 |
|
eluzle |
⊢ ( ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 𝑁 ) → 𝑁 ≤ ( 𝑀 + 𝑁 ) ) |
83 |
81 82
|
syl |
⊢ ( 𝜑 → 𝑁 ≤ ( 𝑀 + 𝑁 ) ) |
84 |
83
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → 𝑁 ≤ ( 𝑀 + 𝑁 ) ) |
85 |
74 44 46 84 62
|
lelttrd |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → 𝑁 < 𝑘 ) |
86 |
74 46
|
ltnled |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → ( 𝑁 < 𝑘 ↔ ¬ 𝑘 ≤ 𝑁 ) ) |
87 |
85 86
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → ¬ 𝑘 ≤ 𝑁 ) |
88 |
|
plyco0 |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐵 : ℕ0 ⟶ ℂ ) → ( ( 𝐵 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ↔ ∀ 𝑘 ∈ ℕ0 ( ( 𝐵 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑁 ) ) ) |
89 |
5 19 88
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐵 “ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = { 0 } ↔ ∀ 𝑘 ∈ ℕ0 ( ( 𝐵 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑁 ) ) ) |
90 |
7 89
|
mpbid |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ℕ0 ( ( 𝐵 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑁 ) ) |
91 |
90
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐵 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑁 ) ) |
92 |
91
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → ( ( 𝐵 ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ 𝑁 ) ) |
93 |
92
|
necon1bd |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → ( ¬ 𝑘 ≤ 𝑁 → ( 𝐵 ‘ 𝑘 ) = 0 ) ) |
94 |
87 93
|
mpd |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → ( 𝐵 ‘ 𝑘 ) = 0 ) |
95 |
72 94
|
oveq12d |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → ( ( 𝐴 ‘ 𝑘 ) − ( 𝐵 ‘ 𝑘 ) ) = ( 0 − 0 ) ) |
96 |
|
0m0e0 |
⊢ ( 0 − 0 ) = 0 |
97 |
95 96
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → ( ( 𝐴 ‘ 𝑘 ) − ( 𝐵 ‘ 𝑘 ) ) = 0 ) |
98 |
41 97
|
eqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℕ0 ∧ ( 𝑀 + 𝑁 ) < 𝑘 ) ) → ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) = 0 ) |
99 |
98
|
expr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝑀 + 𝑁 ) < 𝑘 → ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) = 0 ) ) |
100 |
35 99
|
sylbird |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ¬ 𝑘 ≤ ( 𝑀 + 𝑁 ) → ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) = 0 ) ) |
101 |
100
|
necon1ad |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ ( 𝑀 + 𝑁 ) ) ) |
102 |
101
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ℕ0 ( ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ ( 𝑀 + 𝑁 ) ) ) |
103 |
|
plyco0 |
⊢ ( ( ( 𝑀 + 𝑁 ) ∈ ℕ0 ∧ ( 𝐴 ∘f − 𝐵 ) : ℕ0 ⟶ ℂ ) → ( ( ( 𝐴 ∘f − 𝐵 ) “ ( ℤ≥ ‘ ( ( 𝑀 + 𝑁 ) + 1 ) ) ) = { 0 } ↔ ∀ 𝑘 ∈ ℕ0 ( ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ ( 𝑀 + 𝑁 ) ) ) ) |
104 |
11 22 103
|
syl2anc |
⊢ ( 𝜑 → ( ( ( 𝐴 ∘f − 𝐵 ) “ ( ℤ≥ ‘ ( ( 𝑀 + 𝑁 ) + 1 ) ) ) = { 0 } ↔ ∀ 𝑘 ∈ ℕ0 ( ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) ≠ 0 → 𝑘 ≤ ( 𝑀 + 𝑁 ) ) ) ) |
105 |
102 104
|
mpbird |
⊢ ( 𝜑 → ( ( 𝐴 ∘f − 𝐵 ) “ ( ℤ≥ ‘ ( ( 𝑀 + 𝑁 ) + 1 ) ) ) = { 0 } ) |
106 |
|
df-0p |
⊢ 0𝑝 = ( ℂ × { 0 } ) |
107 |
|
fconstmpt |
⊢ ( ℂ × { 0 } ) = ( 𝑧 ∈ ℂ ↦ 0 ) |
108 |
106 107
|
eqtri |
⊢ 0𝑝 = ( 𝑧 ∈ ℂ ↦ 0 ) |
109 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) → 𝑘 ∈ ℕ0 ) |
110 |
40
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) = ( ( 𝐴 ‘ 𝑘 ) − ( 𝐵 ‘ 𝑘 ) ) ) |
111 |
110
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = ( ( ( 𝐴 ‘ 𝑘 ) − ( 𝐵 ‘ 𝑘 ) ) · ( 𝑧 ↑ 𝑘 ) ) ) |
112 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → 𝐴 : ℕ0 ⟶ ℂ ) |
113 |
112
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐴 ‘ 𝑘 ) ∈ ℂ ) |
114 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → 𝐵 : ℕ0 ⟶ ℂ ) |
115 |
114
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐵 ‘ 𝑘 ) ∈ ℂ ) |
116 |
|
expcl |
⊢ ( ( 𝑧 ∈ ℂ ∧ 𝑘 ∈ ℕ0 ) → ( 𝑧 ↑ 𝑘 ) ∈ ℂ ) |
117 |
116
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑧 ↑ 𝑘 ) ∈ ℂ ) |
118 |
113 115 117
|
subdird |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝐴 ‘ 𝑘 ) − ( 𝐵 ‘ 𝑘 ) ) · ( 𝑧 ↑ 𝑘 ) ) = ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) − ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
119 |
111 118
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) − ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
120 |
109 119
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → ( ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) − ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
121 |
120
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) − ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
122 |
|
fzfid |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → ( 0 ... ( 𝑀 + 𝑁 ) ) ∈ Fin ) |
123 |
113 117
|
mulcld |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ ℂ ) |
124 |
109 123
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ ℂ ) |
125 |
115 117
|
mulcld |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ ℂ ) |
126 |
109 125
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ ℂ ) |
127 |
122 124 126
|
fsumsub |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) − ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) = ( Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) − Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
128 |
122 124
|
fsumcl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ ℂ ) |
129 |
8 9
|
eqtr3d |
⊢ ( 𝜑 → ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
130 |
129
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ‘ 𝑧 ) = ( ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ‘ 𝑧 ) ) |
131 |
130
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → ( ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ‘ 𝑧 ) = ( ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ‘ 𝑧 ) ) |
132 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → 𝑧 ∈ ℂ ) |
133 |
|
sumex |
⊢ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ V |
134 |
|
eqid |
⊢ ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
135 |
134
|
fvmpt2 |
⊢ ( ( 𝑧 ∈ ℂ ∧ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ V ) → ( ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ‘ 𝑧 ) = Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
136 |
132 133 135
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → ( ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ‘ 𝑧 ) = Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
137 |
|
fzss2 |
⊢ ( ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 𝑀 ) → ( 0 ... 𝑀 ) ⊆ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
138 |
58 137
|
syl |
⊢ ( 𝜑 → ( 0 ... 𝑀 ) ⊆ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
139 |
138
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → ( 0 ... 𝑀 ) ⊆ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
140 |
139
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
141 |
140 124
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑀 ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ ℂ ) |
142 |
|
eldifn |
⊢ ( 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) → ¬ 𝑘 ∈ ( 0 ... 𝑀 ) ) |
143 |
142
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ¬ 𝑘 ∈ ( 0 ... 𝑀 ) ) |
144 |
|
eldifi |
⊢ ( 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) → 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
145 |
144 109
|
syl |
⊢ ( 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) → 𝑘 ∈ ℕ0 ) |
146 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℕ0 ) |
147 |
146 50
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ( ℤ≥ ‘ 0 ) ) |
148 |
52
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑀 ∈ ℤ ) |
149 |
|
elfz5 |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 0 ) ∧ 𝑀 ∈ ℤ ) → ( 𝑘 ∈ ( 0 ... 𝑀 ) ↔ 𝑘 ≤ 𝑀 ) ) |
150 |
147 148 149
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 ∈ ( 0 ... 𝑀 ) ↔ 𝑘 ≤ 𝑀 ) ) |
151 |
69 150
|
sylibrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ‘ 𝑘 ) ≠ 0 → 𝑘 ∈ ( 0 ... 𝑀 ) ) ) |
152 |
151
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐴 ‘ 𝑘 ) ≠ 0 → 𝑘 ∈ ( 0 ... 𝑀 ) ) ) |
153 |
152
|
necon1bd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( ¬ 𝑘 ∈ ( 0 ... 𝑀 ) → ( 𝐴 ‘ 𝑘 ) = 0 ) ) |
154 |
145 153
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( ¬ 𝑘 ∈ ( 0 ... 𝑀 ) → ( 𝐴 ‘ 𝑘 ) = 0 ) ) |
155 |
143 154
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( 𝐴 ‘ 𝑘 ) = 0 ) |
156 |
155
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = ( 0 · ( 𝑧 ↑ 𝑘 ) ) ) |
157 |
132 145 116
|
syl2an |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( 𝑧 ↑ 𝑘 ) ∈ ℂ ) |
158 |
157
|
mul02d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( 0 · ( 𝑧 ↑ 𝑘 ) ) = 0 ) |
159 |
156 158
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑀 ) ) ) → ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = 0 ) |
160 |
139 141 159 122
|
fsumss |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
161 |
136 160
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → ( ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑀 ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ‘ 𝑧 ) = Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
162 |
|
sumex |
⊢ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ V |
163 |
|
eqid |
⊢ ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
164 |
163
|
fvmpt2 |
⊢ ( ( 𝑧 ∈ ℂ ∧ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ V ) → ( ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ‘ 𝑧 ) = Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
165 |
132 162 164
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → ( ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ‘ 𝑧 ) = Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
166 |
|
fzss2 |
⊢ ( ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ 𝑁 ) → ( 0 ... 𝑁 ) ⊆ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
167 |
81 166
|
syl |
⊢ ( 𝜑 → ( 0 ... 𝑁 ) ⊆ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
168 |
167
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → ( 0 ... 𝑁 ) ⊆ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
169 |
168
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
170 |
169 126
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ∈ ℂ ) |
171 |
|
eldifn |
⊢ ( 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑁 ) ) → ¬ 𝑘 ∈ ( 0 ... 𝑁 ) ) |
172 |
171
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ¬ 𝑘 ∈ ( 0 ... 𝑁 ) ) |
173 |
|
eldifi |
⊢ ( 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑁 ) ) → 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
174 |
173 109
|
syl |
⊢ ( 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑁 ) ) → 𝑘 ∈ ℕ0 ) |
175 |
76
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑁 ∈ ℤ ) |
176 |
|
elfz5 |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 0 ) ∧ 𝑁 ∈ ℤ ) → ( 𝑘 ∈ ( 0 ... 𝑁 ) ↔ 𝑘 ≤ 𝑁 ) ) |
177 |
147 175 176
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 ∈ ( 0 ... 𝑁 ) ↔ 𝑘 ≤ 𝑁 ) ) |
178 |
91 177
|
sylibrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐵 ‘ 𝑘 ) ≠ 0 → 𝑘 ∈ ( 0 ... 𝑁 ) ) ) |
179 |
178
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐵 ‘ 𝑘 ) ≠ 0 → 𝑘 ∈ ( 0 ... 𝑁 ) ) ) |
180 |
179
|
necon1bd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ℕ0 ) → ( ¬ 𝑘 ∈ ( 0 ... 𝑁 ) → ( 𝐵 ‘ 𝑘 ) = 0 ) ) |
181 |
174 180
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( ¬ 𝑘 ∈ ( 0 ... 𝑁 ) → ( 𝐵 ‘ 𝑘 ) = 0 ) ) |
182 |
172 181
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( 𝐵 ‘ 𝑘 ) = 0 ) |
183 |
182
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = ( 0 · ( 𝑧 ↑ 𝑘 ) ) ) |
184 |
132 174 116
|
syl2an |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( 𝑧 ↑ 𝑘 ) ∈ ℂ ) |
185 |
184
|
mul02d |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( 0 · ( 𝑧 ↑ 𝑘 ) ) = 0 ) |
186 |
183 185
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) ∧ 𝑘 ∈ ( ( 0 ... ( 𝑀 + 𝑁 ) ) ∖ ( 0 ... 𝑁 ) ) ) → ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = 0 ) |
187 |
168 170 186 122
|
fsumss |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
188 |
165 187
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → ( ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ‘ 𝑧 ) = Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
189 |
131 161 188
|
3eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) = Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
190 |
128 189
|
subeq0bd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → ( Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐴 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) − Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( 𝐵 ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) = 0 ) |
191 |
121 127 190
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℂ ) → 0 = Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) |
192 |
191
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑧 ∈ ℂ ↦ 0 ) = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
193 |
108 192
|
syl5eq |
⊢ ( 𝜑 → 0𝑝 = ( 𝑧 ∈ ℂ ↦ Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ( ( ( 𝐴 ∘f − 𝐵 ) ‘ 𝑘 ) · ( 𝑧 ↑ 𝑘 ) ) ) ) |
194 |
10 11 31 105 193
|
plyeq0 |
⊢ ( 𝜑 → ( 𝐴 ∘f − 𝐵 ) = ( ℕ0 × { 0 } ) ) |
195 |
|
ofsubeq0 |
⊢ ( ( ℕ0 ∈ V ∧ 𝐴 : ℕ0 ⟶ ℂ ∧ 𝐵 : ℕ0 ⟶ ℂ ) → ( ( 𝐴 ∘f − 𝐵 ) = ( ℕ0 × { 0 } ) ↔ 𝐴 = 𝐵 ) ) |
196 |
15 17 19 195
|
mp3an2i |
⊢ ( 𝜑 → ( ( 𝐴 ∘f − 𝐵 ) = ( ℕ0 × { 0 } ) ↔ 𝐴 = 𝐵 ) ) |
197 |
194 196
|
mpbid |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |