Step |
Hyp |
Ref |
Expression |
1 |
|
fzsum2sub.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ0 ) |
2 |
|
fzsum2sub.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
3 |
|
fzsum2sub.1 |
⊢ ( 𝑖 = ( 𝑘 − 𝑗 ) → 𝐴 = 𝐵 ) |
4 |
|
fzsum2sub.2 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ℤ≥ ‘ - 𝑗 ) ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝐴 ∈ ℂ ) |
5 |
|
fzsum2sub.3 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) ) → 𝐵 = 0 ) |
6 |
|
fzsum2sub.4 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( 0 ..^ 𝑗 ) ) → 𝐵 = 0 ) |
7 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑗 ∈ ( 1 ... 𝑁 ) ) |
8 |
7
|
elfzelzd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑗 ∈ ℤ ) |
9 |
|
0zd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 0 ∈ ℤ ) |
10 |
1
|
nn0zd |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
11 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑀 ∈ ℤ ) |
12 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑖 ∈ ( 0 ... 𝑀 ) ) → 𝜑 ) |
13 |
|
fz1ssnn |
⊢ ( 1 ... 𝑁 ) ⊆ ℕ |
14 |
|
nnssnn0 |
⊢ ℕ ⊆ ℕ0 |
15 |
13 14
|
sstri |
⊢ ( 1 ... 𝑁 ) ⊆ ℕ0 |
16 |
15 7
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑗 ∈ ℕ0 ) |
17 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
18 |
16 17
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑗 ∈ ( ℤ≥ ‘ 0 ) ) |
19 |
|
neg0 |
⊢ - 0 = 0 |
20 |
|
uzneg |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 0 ) → - 0 ∈ ( ℤ≥ ‘ - 𝑗 ) ) |
21 |
19 20
|
eqeltrrid |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 0 ) → 0 ∈ ( ℤ≥ ‘ - 𝑗 ) ) |
22 |
|
fzss1 |
⊢ ( 0 ∈ ( ℤ≥ ‘ - 𝑗 ) → ( 0 ... 𝑀 ) ⊆ ( - 𝑗 ... 𝑀 ) ) |
23 |
18 21 22
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 0 ... 𝑀 ) ⊆ ( - 𝑗 ... 𝑀 ) ) |
24 |
|
fzssuz |
⊢ ( - 𝑗 ... 𝑀 ) ⊆ ( ℤ≥ ‘ - 𝑗 ) |
25 |
23 24
|
sstrdi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 0 ... 𝑀 ) ⊆ ( ℤ≥ ‘ - 𝑗 ) ) |
26 |
25
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑖 ∈ ( 0 ... 𝑀 ) ) → 𝑖 ∈ ( ℤ≥ ‘ - 𝑗 ) ) |
27 |
7
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑖 ∈ ( 0 ... 𝑀 ) ) → 𝑗 ∈ ( 1 ... 𝑁 ) ) |
28 |
12 26 27 4
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑖 ∈ ( 0 ... 𝑀 ) ) → 𝐴 ∈ ℂ ) |
29 |
8 9 11 28 3
|
fsumshft |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → Σ 𝑖 ∈ ( 0 ... 𝑀 ) 𝐴 = Σ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) 𝐵 ) |
30 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑀 ∈ ℕ0 ) |
31 |
13 7
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑗 ∈ ℕ ) |
32 |
31
|
nnnn0d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑗 ∈ ℕ0 ) |
33 |
30 32
|
nn0addcld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑀 + 𝑗 ) ∈ ℕ0 ) |
34 |
33
|
nn0red |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑀 + 𝑗 ) ∈ ℝ ) |
35 |
34
|
ltp1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑀 + 𝑗 ) < ( ( 𝑀 + 𝑗 ) + 1 ) ) |
36 |
|
fzdisj |
⊢ ( ( 𝑀 + 𝑗 ) < ( ( 𝑀 + 𝑗 ) + 1 ) → ( ( 𝑗 ... ( 𝑀 + 𝑗 ) ) ∩ ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) ) = ∅ ) |
37 |
35 36
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( ( 𝑗 ... ( 𝑀 + 𝑗 ) ) ∩ ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) ) = ∅ ) |
38 |
2
|
nn0zd |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
39 |
10 38
|
zaddcld |
⊢ ( 𝜑 → ( 𝑀 + 𝑁 ) ∈ ℤ ) |
40 |
39
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑀 + 𝑁 ) ∈ ℤ ) |
41 |
33
|
nn0zd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑀 + 𝑗 ) ∈ ℤ ) |
42 |
31
|
nnred |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑗 ∈ ℝ ) |
43 |
|
nn0addge2 |
⊢ ( ( 𝑗 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) → 𝑗 ≤ ( 𝑀 + 𝑗 ) ) |
44 |
42 30 43
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑗 ≤ ( 𝑀 + 𝑗 ) ) |
45 |
2
|
nn0red |
⊢ ( 𝜑 → 𝑁 ∈ ℝ ) |
46 |
45
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑁 ∈ ℝ ) |
47 |
30
|
nn0red |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑀 ∈ ℝ ) |
48 |
|
elfzle2 |
⊢ ( 𝑗 ∈ ( 1 ... 𝑁 ) → 𝑗 ≤ 𝑁 ) |
49 |
48
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑗 ≤ 𝑁 ) |
50 |
42 46 47 49
|
leadd2dd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑀 + 𝑗 ) ≤ ( 𝑀 + 𝑁 ) ) |
51 |
8 40 41 44 50
|
elfzd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑀 + 𝑗 ) ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ) |
52 |
|
fzsplit |
⊢ ( ( 𝑀 + 𝑗 ) ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) → ( 𝑗 ... ( 𝑀 + 𝑁 ) ) = ( ( 𝑗 ... ( 𝑀 + 𝑗 ) ) ∪ ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) ) ) |
53 |
51 52
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑗 ... ( 𝑀 + 𝑁 ) ) = ( ( 𝑗 ... ( 𝑀 + 𝑗 ) ) ∪ ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) ) ) |
54 |
|
fzfid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ∈ Fin ) |
55 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ) → 𝜑 ) |
56 |
7
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ) → 𝑗 ∈ ( 1 ... 𝑁 ) ) |
57 |
15 56
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ) → 𝑗 ∈ ℕ0 ) |
58 |
|
fz2ssnn0 |
⊢ ( 𝑗 ∈ ℕ0 → ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ⊆ ℕ0 ) |
59 |
57 58
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ) → ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ⊆ ℕ0 ) |
60 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ) → 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ) |
61 |
59 60
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ) → 𝑘 ∈ ℕ0 ) |
62 |
3
|
eleq1d |
⊢ ( 𝑖 = ( 𝑘 − 𝑗 ) → ( 𝐴 ∈ ℂ ↔ 𝐵 ∈ ℂ ) ) |
63 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( ℤ≥ ‘ - 𝑗 ) ) ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝜑 ) |
64 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( ℤ≥ ‘ - 𝑗 ) ) ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑖 ∈ ( ℤ≥ ‘ - 𝑗 ) ) |
65 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( ℤ≥ ‘ - 𝑗 ) ) ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑗 ∈ ( 1 ... 𝑁 ) ) |
66 |
63 64 65 4
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( ℤ≥ ‘ - 𝑗 ) ) ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝐴 ∈ ℂ ) |
67 |
66
|
an32s |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑖 ∈ ( ℤ≥ ‘ - 𝑗 ) ) → 𝐴 ∈ ℂ ) |
68 |
67
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ∀ 𝑖 ∈ ( ℤ≥ ‘ - 𝑗 ) 𝐴 ∈ ℂ ) |
69 |
68
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ℕ0 ) → ∀ 𝑖 ∈ ( ℤ≥ ‘ - 𝑗 ) 𝐴 ∈ ℂ ) |
70 |
|
nnsscn |
⊢ ℕ ⊆ ℂ |
71 |
13 70
|
sstri |
⊢ ( 1 ... 𝑁 ) ⊆ ℂ |
72 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝑗 ∈ ( 1 ... 𝑁 ) ) |
73 |
71 72
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝑗 ∈ ℂ ) |
74 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℕ0 ) |
75 |
74
|
nn0cnd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℂ ) |
76 |
73 75
|
negsubdi2d |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ℕ0 ) → - ( 𝑗 − 𝑘 ) = ( 𝑘 − 𝑗 ) ) |
77 |
72
|
elfzelzd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝑗 ∈ ℤ ) |
78 |
|
eluzmn |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑘 ∈ ℕ0 ) → 𝑗 ∈ ( ℤ≥ ‘ ( 𝑗 − 𝑘 ) ) ) |
79 |
77 74 78
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝑗 ∈ ( ℤ≥ ‘ ( 𝑗 − 𝑘 ) ) ) |
80 |
|
uzneg |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑗 − 𝑘 ) ) → - ( 𝑗 − 𝑘 ) ∈ ( ℤ≥ ‘ - 𝑗 ) ) |
81 |
79 80
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ℕ0 ) → - ( 𝑗 − 𝑘 ) ∈ ( ℤ≥ ‘ - 𝑗 ) ) |
82 |
76 81
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 − 𝑗 ) ∈ ( ℤ≥ ‘ - 𝑗 ) ) |
83 |
62 69 82
|
rspcdva |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ℕ0 ) → 𝐵 ∈ ℂ ) |
84 |
55 56 61 83
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ) → 𝐵 ∈ ℂ ) |
85 |
37 53 54 84
|
fsumsplit |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → Σ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) 𝐵 = ( Σ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑗 ) ) 𝐵 + Σ 𝑘 ∈ ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) 𝐵 ) ) |
86 |
8
|
zcnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑗 ∈ ℂ ) |
87 |
86
|
addid2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 0 + 𝑗 ) = 𝑗 ) |
88 |
87
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) = ( 𝑗 ... ( 𝑀 + 𝑗 ) ) ) |
89 |
88
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑗 ... ( 𝑀 + 𝑗 ) ) = ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) ) |
90 |
89
|
sumeq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → Σ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑗 ) ) 𝐵 = Σ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) 𝐵 ) |
91 |
5
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → Σ 𝑘 ∈ ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) 𝐵 = Σ 𝑘 ∈ ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) 0 ) |
92 |
|
fzfi |
⊢ ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) ∈ Fin |
93 |
|
sumz |
⊢ ( ( ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) ⊆ ( ℤ≥ ‘ 0 ) ∨ ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) ∈ Fin ) → Σ 𝑘 ∈ ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) 0 = 0 ) |
94 |
93
|
olcs |
⊢ ( ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) ∈ Fin → Σ 𝑘 ∈ ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) 0 = 0 ) |
95 |
92 94
|
ax-mp |
⊢ Σ 𝑘 ∈ ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) 0 = 0 |
96 |
91 95
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → Σ 𝑘 ∈ ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) 𝐵 = 0 ) |
97 |
90 96
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( Σ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑗 ) ) 𝐵 + Σ 𝑘 ∈ ( ( ( 𝑀 + 𝑗 ) + 1 ) ... ( 𝑀 + 𝑁 ) ) 𝐵 ) = ( Σ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) 𝐵 + 0 ) ) |
98 |
|
fzfid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) ∈ Fin ) |
99 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) ) → 𝜑 ) |
100 |
7
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) ) → 𝑗 ∈ ( 1 ... 𝑁 ) ) |
101 |
|
elfzuz3 |
⊢ ( 𝑗 ∈ ( 1 ... 𝑁 ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
102 |
101
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑁 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
103 |
|
eluzadd |
⊢ ( ( 𝑁 ∈ ( ℤ≥ ‘ 𝑗 ) ∧ 𝑀 ∈ ℤ ) → ( 𝑁 + 𝑀 ) ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑀 ) ) ) |
104 |
102 11 103
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑁 + 𝑀 ) ∈ ( ℤ≥ ‘ ( 𝑗 + 𝑀 ) ) ) |
105 |
2
|
nn0cnd |
⊢ ( 𝜑 → 𝑁 ∈ ℂ ) |
106 |
105
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑁 ∈ ℂ ) |
107 |
|
zsscn |
⊢ ℤ ⊆ ℂ |
108 |
107 11
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑀 ∈ ℂ ) |
109 |
106 108
|
addcomd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑁 + 𝑀 ) = ( 𝑀 + 𝑁 ) ) |
110 |
86 108
|
addcomd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑗 + 𝑀 ) = ( 𝑀 + 𝑗 ) ) |
111 |
110
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( ℤ≥ ‘ ( 𝑗 + 𝑀 ) ) = ( ℤ≥ ‘ ( 𝑀 + 𝑗 ) ) ) |
112 |
104 109 111
|
3eltr3d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ ( 𝑀 + 𝑗 ) ) ) |
113 |
112
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) ) → ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ ( 𝑀 + 𝑗 ) ) ) |
114 |
|
fzss2 |
⊢ ( ( 𝑀 + 𝑁 ) ∈ ( ℤ≥ ‘ ( 𝑀 + 𝑗 ) ) → ( 𝑗 ... ( 𝑀 + 𝑗 ) ) ⊆ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ) |
115 |
113 114
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) ) → ( 𝑗 ... ( 𝑀 + 𝑗 ) ) ⊆ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ) |
116 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) ) → 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) ) |
117 |
88
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) ) → ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) = ( 𝑗 ... ( 𝑀 + 𝑗 ) ) ) |
118 |
116 117
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) ) → 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑗 ) ) ) |
119 |
115 118
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) ) → 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ) |
120 |
99 100 119 61
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) ) → 𝑘 ∈ ℕ0 ) |
121 |
99 100 120 83
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) ) → 𝐵 ∈ ℂ ) |
122 |
98 121
|
fsumcl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → Σ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) 𝐵 ∈ ℂ ) |
123 |
122
|
addid1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( Σ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) 𝐵 + 0 ) = Σ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) 𝐵 ) |
124 |
85 97 123
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → Σ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) 𝐵 = Σ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) 𝐵 ) |
125 |
|
fzval3 |
⊢ ( ( 𝑀 + 𝑁 ) ∈ ℤ → ( 𝑗 ... ( 𝑀 + 𝑁 ) ) = ( 𝑗 ..^ ( ( 𝑀 + 𝑁 ) + 1 ) ) ) |
126 |
40 125
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑗 ... ( 𝑀 + 𝑁 ) ) = ( 𝑗 ..^ ( ( 𝑀 + 𝑁 ) + 1 ) ) ) |
127 |
126
|
ineq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( ( 0 ..^ 𝑗 ) ∩ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ) = ( ( 0 ..^ 𝑗 ) ∩ ( 𝑗 ..^ ( ( 𝑀 + 𝑁 ) + 1 ) ) ) ) |
128 |
|
fzodisj |
⊢ ( ( 0 ..^ 𝑗 ) ∩ ( 𝑗 ..^ ( ( 𝑀 + 𝑁 ) + 1 ) ) ) = ∅ |
129 |
127 128
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( ( 0 ..^ 𝑗 ) ∩ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ) = ∅ ) |
130 |
40
|
peano2zd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( ( 𝑀 + 𝑁 ) + 1 ) ∈ ℤ ) |
131 |
32
|
nn0ge0d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 0 ≤ 𝑗 ) |
132 |
130
|
zred |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( ( 𝑀 + 𝑁 ) + 1 ) ∈ ℝ ) |
133 |
40
|
zred |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑀 + 𝑁 ) ∈ ℝ ) |
134 |
|
nn0addge2 |
⊢ ( ( 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℕ0 ) → 𝑁 ≤ ( 𝑀 + 𝑁 ) ) |
135 |
45 1 134
|
syl2anc |
⊢ ( 𝜑 → 𝑁 ≤ ( 𝑀 + 𝑁 ) ) |
136 |
135
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑁 ≤ ( 𝑀 + 𝑁 ) ) |
137 |
133
|
lep1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 𝑀 + 𝑁 ) ≤ ( ( 𝑀 + 𝑁 ) + 1 ) ) |
138 |
46 133 132 136 137
|
letrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑁 ≤ ( ( 𝑀 + 𝑁 ) + 1 ) ) |
139 |
42 46 132 49 138
|
letrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑗 ≤ ( ( 𝑀 + 𝑁 ) + 1 ) ) |
140 |
9 130 8 131 139
|
elfzd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → 𝑗 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) + 1 ) ) ) |
141 |
|
fzosplit |
⊢ ( 𝑗 ∈ ( 0 ... ( ( 𝑀 + 𝑁 ) + 1 ) ) → ( 0 ..^ ( ( 𝑀 + 𝑁 ) + 1 ) ) = ( ( 0 ..^ 𝑗 ) ∪ ( 𝑗 ..^ ( ( 𝑀 + 𝑁 ) + 1 ) ) ) ) |
142 |
140 141
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 0 ..^ ( ( 𝑀 + 𝑁 ) + 1 ) ) = ( ( 0 ..^ 𝑗 ) ∪ ( 𝑗 ..^ ( ( 𝑀 + 𝑁 ) + 1 ) ) ) ) |
143 |
|
fzval3 |
⊢ ( ( 𝑀 + 𝑁 ) ∈ ℤ → ( 0 ... ( 𝑀 + 𝑁 ) ) = ( 0 ..^ ( ( 𝑀 + 𝑁 ) + 1 ) ) ) |
144 |
40 143
|
syl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 0 ... ( 𝑀 + 𝑁 ) ) = ( 0 ..^ ( ( 𝑀 + 𝑁 ) + 1 ) ) ) |
145 |
126
|
uneq2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( ( 0 ..^ 𝑗 ) ∪ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ) = ( ( 0 ..^ 𝑗 ) ∪ ( 𝑗 ..^ ( ( 𝑀 + 𝑁 ) + 1 ) ) ) ) |
146 |
142 144 145
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 0 ... ( 𝑀 + 𝑁 ) ) = ( ( 0 ..^ 𝑗 ) ∪ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) ) ) |
147 |
|
fzfid |
⊢ ( 𝜑 → ( 0 ... ( 𝑀 + 𝑁 ) ) ∈ Fin ) |
148 |
147
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 0 ... ( 𝑀 + 𝑁 ) ) ∈ Fin ) |
149 |
|
simpl |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ) → 𝜑 ) |
150 |
7
|
adantrl |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ) → 𝑗 ∈ ( 1 ... 𝑁 ) ) |
151 |
|
fz0ssnn0 |
⊢ ( 0 ... ( 𝑀 + 𝑁 ) ) ⊆ ℕ0 |
152 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ) → 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) |
153 |
151 152
|
sselid |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ) → 𝑘 ∈ ℕ0 ) |
154 |
149 150 153 83
|
syl21anc |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ) → 𝐵 ∈ ℂ ) |
155 |
154
|
anass1rs |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ∧ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) ) → 𝐵 ∈ ℂ ) |
156 |
129 146 148 155
|
fsumsplit |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) 𝐵 = ( Σ 𝑘 ∈ ( 0 ..^ 𝑗 ) 𝐵 + Σ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) 𝐵 ) ) |
157 |
6
|
sumeq2dv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → Σ 𝑘 ∈ ( 0 ..^ 𝑗 ) 𝐵 = Σ 𝑘 ∈ ( 0 ..^ 𝑗 ) 0 ) |
158 |
|
fzofi |
⊢ ( 0 ..^ 𝑗 ) ∈ Fin |
159 |
|
sumz |
⊢ ( ( ( 0 ..^ 𝑗 ) ⊆ ( ℤ≥ ‘ 0 ) ∨ ( 0 ..^ 𝑗 ) ∈ Fin ) → Σ 𝑘 ∈ ( 0 ..^ 𝑗 ) 0 = 0 ) |
160 |
159
|
olcs |
⊢ ( ( 0 ..^ 𝑗 ) ∈ Fin → Σ 𝑘 ∈ ( 0 ..^ 𝑗 ) 0 = 0 ) |
161 |
158 160
|
ax-mp |
⊢ Σ 𝑘 ∈ ( 0 ..^ 𝑗 ) 0 = 0 |
162 |
157 161
|
eqtrdi |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → Σ 𝑘 ∈ ( 0 ..^ 𝑗 ) 𝐵 = 0 ) |
163 |
162
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( Σ 𝑘 ∈ ( 0 ..^ 𝑗 ) 𝐵 + Σ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) 𝐵 ) = ( 0 + Σ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) 𝐵 ) ) |
164 |
54 84
|
fsumcl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → Σ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) 𝐵 ∈ ℂ ) |
165 |
164
|
addid2d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → ( 0 + Σ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) 𝐵 ) = Σ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) 𝐵 ) |
166 |
156 163 165
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → Σ 𝑘 ∈ ( 𝑗 ... ( 𝑀 + 𝑁 ) ) 𝐵 = Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) 𝐵 ) |
167 |
124 166
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → Σ 𝑘 ∈ ( ( 0 + 𝑗 ) ... ( 𝑀 + 𝑗 ) ) 𝐵 = Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) 𝐵 ) |
168 |
29 167
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) → Σ 𝑖 ∈ ( 0 ... 𝑀 ) 𝐴 = Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) 𝐵 ) |
169 |
168
|
sumeq2dv |
⊢ ( 𝜑 → Σ 𝑗 ∈ ( 1 ... 𝑁 ) Σ 𝑖 ∈ ( 0 ... 𝑀 ) 𝐴 = Σ 𝑗 ∈ ( 1 ... 𝑁 ) Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) 𝐵 ) |
170 |
|
fzfid |
⊢ ( 𝜑 → ( 0 ... 𝑀 ) ∈ Fin ) |
171 |
|
fzfid |
⊢ ( 𝜑 → ( 1 ... 𝑁 ) ∈ Fin ) |
172 |
28
|
anasss |
⊢ ( ( 𝜑 ∧ ( 𝑗 ∈ ( 1 ... 𝑁 ) ∧ 𝑖 ∈ ( 0 ... 𝑀 ) ) ) → 𝐴 ∈ ℂ ) |
173 |
172
|
ancom2s |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ ( 0 ... 𝑀 ) ∧ 𝑗 ∈ ( 1 ... 𝑁 ) ) ) → 𝐴 ∈ ℂ ) |
174 |
170 171 173
|
fsumcom |
⊢ ( 𝜑 → Σ 𝑖 ∈ ( 0 ... 𝑀 ) Σ 𝑗 ∈ ( 1 ... 𝑁 ) 𝐴 = Σ 𝑗 ∈ ( 1 ... 𝑁 ) Σ 𝑖 ∈ ( 0 ... 𝑀 ) 𝐴 ) |
175 |
147 171 154
|
fsumcom |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) Σ 𝑗 ∈ ( 1 ... 𝑁 ) 𝐵 = Σ 𝑗 ∈ ( 1 ... 𝑁 ) Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) 𝐵 ) |
176 |
169 174 175
|
3eqtr4d |
⊢ ( 𝜑 → Σ 𝑖 ∈ ( 0 ... 𝑀 ) Σ 𝑗 ∈ ( 1 ... 𝑁 ) 𝐴 = Σ 𝑘 ∈ ( 0 ... ( 𝑀 + 𝑁 ) ) Σ 𝑗 ∈ ( 1 ... 𝑁 ) 𝐵 ) |