Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
⊢ ( 𝑚 = 0 → ( 0 ... 𝑚 ) = ( 0 ... 0 ) ) |
2 |
1
|
sumeq1d |
⊢ ( 𝑚 = 0 → Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( 𝑘 C 𝐶 ) = Σ 𝑘 ∈ ( 0 ... 0 ) ( 𝑘 C 𝐶 ) ) |
3 |
|
oveq1 |
⊢ ( 𝑚 = 0 → ( 𝑚 + 1 ) = ( 0 + 1 ) ) |
4 |
|
0p1e1 |
⊢ ( 0 + 1 ) = 1 |
5 |
3 4
|
eqtrdi |
⊢ ( 𝑚 = 0 → ( 𝑚 + 1 ) = 1 ) |
6 |
5
|
oveq1d |
⊢ ( 𝑚 = 0 → ( ( 𝑚 + 1 ) C ( 𝐶 + 1 ) ) = ( 1 C ( 𝐶 + 1 ) ) ) |
7 |
2 6
|
eqeq12d |
⊢ ( 𝑚 = 0 → ( Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( 𝑘 C 𝐶 ) = ( ( 𝑚 + 1 ) C ( 𝐶 + 1 ) ) ↔ Σ 𝑘 ∈ ( 0 ... 0 ) ( 𝑘 C 𝐶 ) = ( 1 C ( 𝐶 + 1 ) ) ) ) |
8 |
7
|
imbi2d |
⊢ ( 𝑚 = 0 → ( ( 𝐶 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( 𝑘 C 𝐶 ) = ( ( 𝑚 + 1 ) C ( 𝐶 + 1 ) ) ) ↔ ( 𝐶 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... 0 ) ( 𝑘 C 𝐶 ) = ( 1 C ( 𝐶 + 1 ) ) ) ) ) |
9 |
|
oveq2 |
⊢ ( 𝑚 = 𝑛 → ( 0 ... 𝑚 ) = ( 0 ... 𝑛 ) ) |
10 |
9
|
sumeq1d |
⊢ ( 𝑚 = 𝑛 → Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( 𝑘 C 𝐶 ) = Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( 𝑘 C 𝐶 ) ) |
11 |
|
oveq1 |
⊢ ( 𝑚 = 𝑛 → ( 𝑚 + 1 ) = ( 𝑛 + 1 ) ) |
12 |
11
|
oveq1d |
⊢ ( 𝑚 = 𝑛 → ( ( 𝑚 + 1 ) C ( 𝐶 + 1 ) ) = ( ( 𝑛 + 1 ) C ( 𝐶 + 1 ) ) ) |
13 |
10 12
|
eqeq12d |
⊢ ( 𝑚 = 𝑛 → ( Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( 𝑘 C 𝐶 ) = ( ( 𝑚 + 1 ) C ( 𝐶 + 1 ) ) ↔ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( 𝑘 C 𝐶 ) = ( ( 𝑛 + 1 ) C ( 𝐶 + 1 ) ) ) ) |
14 |
13
|
imbi2d |
⊢ ( 𝑚 = 𝑛 → ( ( 𝐶 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( 𝑘 C 𝐶 ) = ( ( 𝑚 + 1 ) C ( 𝐶 + 1 ) ) ) ↔ ( 𝐶 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( 𝑘 C 𝐶 ) = ( ( 𝑛 + 1 ) C ( 𝐶 + 1 ) ) ) ) ) |
15 |
|
oveq2 |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( 0 ... 𝑚 ) = ( 0 ... ( 𝑛 + 1 ) ) ) |
16 |
15
|
sumeq1d |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( 𝑘 C 𝐶 ) = Σ 𝑘 ∈ ( 0 ... ( 𝑛 + 1 ) ) ( 𝑘 C 𝐶 ) ) |
17 |
|
oveq1 |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( 𝑚 + 1 ) = ( ( 𝑛 + 1 ) + 1 ) ) |
18 |
17
|
oveq1d |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( ( 𝑚 + 1 ) C ( 𝐶 + 1 ) ) = ( ( ( 𝑛 + 1 ) + 1 ) C ( 𝐶 + 1 ) ) ) |
19 |
16 18
|
eqeq12d |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( 𝑘 C 𝐶 ) = ( ( 𝑚 + 1 ) C ( 𝐶 + 1 ) ) ↔ Σ 𝑘 ∈ ( 0 ... ( 𝑛 + 1 ) ) ( 𝑘 C 𝐶 ) = ( ( ( 𝑛 + 1 ) + 1 ) C ( 𝐶 + 1 ) ) ) ) |
20 |
19
|
imbi2d |
⊢ ( 𝑚 = ( 𝑛 + 1 ) → ( ( 𝐶 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( 𝑘 C 𝐶 ) = ( ( 𝑚 + 1 ) C ( 𝐶 + 1 ) ) ) ↔ ( 𝐶 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... ( 𝑛 + 1 ) ) ( 𝑘 C 𝐶 ) = ( ( ( 𝑛 + 1 ) + 1 ) C ( 𝐶 + 1 ) ) ) ) ) |
21 |
|
oveq2 |
⊢ ( 𝑚 = 𝑁 → ( 0 ... 𝑚 ) = ( 0 ... 𝑁 ) ) |
22 |
21
|
sumeq1d |
⊢ ( 𝑚 = 𝑁 → Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( 𝑘 C 𝐶 ) = Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( 𝑘 C 𝐶 ) ) |
23 |
|
oveq1 |
⊢ ( 𝑚 = 𝑁 → ( 𝑚 + 1 ) = ( 𝑁 + 1 ) ) |
24 |
23
|
oveq1d |
⊢ ( 𝑚 = 𝑁 → ( ( 𝑚 + 1 ) C ( 𝐶 + 1 ) ) = ( ( 𝑁 + 1 ) C ( 𝐶 + 1 ) ) ) |
25 |
22 24
|
eqeq12d |
⊢ ( 𝑚 = 𝑁 → ( Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( 𝑘 C 𝐶 ) = ( ( 𝑚 + 1 ) C ( 𝐶 + 1 ) ) ↔ Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( 𝑘 C 𝐶 ) = ( ( 𝑁 + 1 ) C ( 𝐶 + 1 ) ) ) ) |
26 |
25
|
imbi2d |
⊢ ( 𝑚 = 𝑁 → ( ( 𝐶 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... 𝑚 ) ( 𝑘 C 𝐶 ) = ( ( 𝑚 + 1 ) C ( 𝐶 + 1 ) ) ) ↔ ( 𝐶 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( 𝑘 C 𝐶 ) = ( ( 𝑁 + 1 ) C ( 𝐶 + 1 ) ) ) ) ) |
27 |
|
0z |
⊢ 0 ∈ ℤ |
28 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
29 |
|
nn0z |
⊢ ( 𝐶 ∈ ℕ0 → 𝐶 ∈ ℤ ) |
30 |
|
bccl |
⊢ ( ( 0 ∈ ℕ0 ∧ 𝐶 ∈ ℤ ) → ( 0 C 𝐶 ) ∈ ℕ0 ) |
31 |
28 29 30
|
sylancr |
⊢ ( 𝐶 ∈ ℕ0 → ( 0 C 𝐶 ) ∈ ℕ0 ) |
32 |
31
|
nn0cnd |
⊢ ( 𝐶 ∈ ℕ0 → ( 0 C 𝐶 ) ∈ ℂ ) |
33 |
|
oveq1 |
⊢ ( 𝑘 = 0 → ( 𝑘 C 𝐶 ) = ( 0 C 𝐶 ) ) |
34 |
33
|
fsum1 |
⊢ ( ( 0 ∈ ℤ ∧ ( 0 C 𝐶 ) ∈ ℂ ) → Σ 𝑘 ∈ ( 0 ... 0 ) ( 𝑘 C 𝐶 ) = ( 0 C 𝐶 ) ) |
35 |
27 32 34
|
sylancr |
⊢ ( 𝐶 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... 0 ) ( 𝑘 C 𝐶 ) = ( 0 C 𝐶 ) ) |
36 |
|
elnn0 |
⊢ ( 𝐶 ∈ ℕ0 ↔ ( 𝐶 ∈ ℕ ∨ 𝐶 = 0 ) ) |
37 |
|
1red |
⊢ ( 𝐶 ∈ ℕ → 1 ∈ ℝ ) |
38 |
|
nnrp |
⊢ ( 𝐶 ∈ ℕ → 𝐶 ∈ ℝ+ ) |
39 |
37 38
|
ltaddrp2d |
⊢ ( 𝐶 ∈ ℕ → 1 < ( 𝐶 + 1 ) ) |
40 |
|
peano2nn |
⊢ ( 𝐶 ∈ ℕ → ( 𝐶 + 1 ) ∈ ℕ ) |
41 |
40
|
nnred |
⊢ ( 𝐶 ∈ ℕ → ( 𝐶 + 1 ) ∈ ℝ ) |
42 |
37 41
|
ltnled |
⊢ ( 𝐶 ∈ ℕ → ( 1 < ( 𝐶 + 1 ) ↔ ¬ ( 𝐶 + 1 ) ≤ 1 ) ) |
43 |
39 42
|
mpbid |
⊢ ( 𝐶 ∈ ℕ → ¬ ( 𝐶 + 1 ) ≤ 1 ) |
44 |
|
elfzle2 |
⊢ ( ( 𝐶 + 1 ) ∈ ( 0 ... 1 ) → ( 𝐶 + 1 ) ≤ 1 ) |
45 |
43 44
|
nsyl |
⊢ ( 𝐶 ∈ ℕ → ¬ ( 𝐶 + 1 ) ∈ ( 0 ... 1 ) ) |
46 |
45
|
iffalsed |
⊢ ( 𝐶 ∈ ℕ → if ( ( 𝐶 + 1 ) ∈ ( 0 ... 1 ) , ( ( ! ‘ 1 ) / ( ( ! ‘ ( 1 − ( 𝐶 + 1 ) ) ) · ( ! ‘ ( 𝐶 + 1 ) ) ) ) , 0 ) = 0 ) |
47 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
48 |
40
|
nnzd |
⊢ ( 𝐶 ∈ ℕ → ( 𝐶 + 1 ) ∈ ℤ ) |
49 |
|
bcval |
⊢ ( ( 1 ∈ ℕ0 ∧ ( 𝐶 + 1 ) ∈ ℤ ) → ( 1 C ( 𝐶 + 1 ) ) = if ( ( 𝐶 + 1 ) ∈ ( 0 ... 1 ) , ( ( ! ‘ 1 ) / ( ( ! ‘ ( 1 − ( 𝐶 + 1 ) ) ) · ( ! ‘ ( 𝐶 + 1 ) ) ) ) , 0 ) ) |
50 |
47 48 49
|
sylancr |
⊢ ( 𝐶 ∈ ℕ → ( 1 C ( 𝐶 + 1 ) ) = if ( ( 𝐶 + 1 ) ∈ ( 0 ... 1 ) , ( ( ! ‘ 1 ) / ( ( ! ‘ ( 1 − ( 𝐶 + 1 ) ) ) · ( ! ‘ ( 𝐶 + 1 ) ) ) ) , 0 ) ) |
51 |
|
bc0k |
⊢ ( 𝐶 ∈ ℕ → ( 0 C 𝐶 ) = 0 ) |
52 |
46 50 51
|
3eqtr4rd |
⊢ ( 𝐶 ∈ ℕ → ( 0 C 𝐶 ) = ( 1 C ( 𝐶 + 1 ) ) ) |
53 |
|
bcnn |
⊢ ( 0 ∈ ℕ0 → ( 0 C 0 ) = 1 ) |
54 |
28 53
|
ax-mp |
⊢ ( 0 C 0 ) = 1 |
55 |
|
bcnn |
⊢ ( 1 ∈ ℕ0 → ( 1 C 1 ) = 1 ) |
56 |
47 55
|
ax-mp |
⊢ ( 1 C 1 ) = 1 |
57 |
54 56
|
eqtr4i |
⊢ ( 0 C 0 ) = ( 1 C 1 ) |
58 |
|
oveq2 |
⊢ ( 𝐶 = 0 → ( 0 C 𝐶 ) = ( 0 C 0 ) ) |
59 |
|
oveq1 |
⊢ ( 𝐶 = 0 → ( 𝐶 + 1 ) = ( 0 + 1 ) ) |
60 |
59 4
|
eqtrdi |
⊢ ( 𝐶 = 0 → ( 𝐶 + 1 ) = 1 ) |
61 |
60
|
oveq2d |
⊢ ( 𝐶 = 0 → ( 1 C ( 𝐶 + 1 ) ) = ( 1 C 1 ) ) |
62 |
57 58 61
|
3eqtr4a |
⊢ ( 𝐶 = 0 → ( 0 C 𝐶 ) = ( 1 C ( 𝐶 + 1 ) ) ) |
63 |
52 62
|
jaoi |
⊢ ( ( 𝐶 ∈ ℕ ∨ 𝐶 = 0 ) → ( 0 C 𝐶 ) = ( 1 C ( 𝐶 + 1 ) ) ) |
64 |
36 63
|
sylbi |
⊢ ( 𝐶 ∈ ℕ0 → ( 0 C 𝐶 ) = ( 1 C ( 𝐶 + 1 ) ) ) |
65 |
35 64
|
eqtrd |
⊢ ( 𝐶 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... 0 ) ( 𝑘 C 𝐶 ) = ( 1 C ( 𝐶 + 1 ) ) ) |
66 |
|
elnn0uz |
⊢ ( 𝑛 ∈ ℕ0 ↔ 𝑛 ∈ ( ℤ≥ ‘ 0 ) ) |
67 |
66
|
biimpi |
⊢ ( 𝑛 ∈ ℕ0 → 𝑛 ∈ ( ℤ≥ ‘ 0 ) ) |
68 |
67
|
adantr |
⊢ ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → 𝑛 ∈ ( ℤ≥ ‘ 0 ) ) |
69 |
|
elfznn0 |
⊢ ( 𝑘 ∈ ( 0 ... ( 𝑛 + 1 ) ) → 𝑘 ∈ ℕ0 ) |
70 |
69
|
adantl |
⊢ ( ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... ( 𝑛 + 1 ) ) ) → 𝑘 ∈ ℕ0 ) |
71 |
|
simplr |
⊢ ( ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... ( 𝑛 + 1 ) ) ) → 𝐶 ∈ ℕ0 ) |
72 |
71
|
nn0zd |
⊢ ( ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... ( 𝑛 + 1 ) ) ) → 𝐶 ∈ ℤ ) |
73 |
|
bccl |
⊢ ( ( 𝑘 ∈ ℕ0 ∧ 𝐶 ∈ ℤ ) → ( 𝑘 C 𝐶 ) ∈ ℕ0 ) |
74 |
70 72 73
|
syl2anc |
⊢ ( ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... ( 𝑛 + 1 ) ) ) → ( 𝑘 C 𝐶 ) ∈ ℕ0 ) |
75 |
74
|
nn0cnd |
⊢ ( ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ( 0 ... ( 𝑛 + 1 ) ) ) → ( 𝑘 C 𝐶 ) ∈ ℂ ) |
76 |
|
oveq1 |
⊢ ( 𝑘 = ( 𝑛 + 1 ) → ( 𝑘 C 𝐶 ) = ( ( 𝑛 + 1 ) C 𝐶 ) ) |
77 |
68 75 76
|
fsump1 |
⊢ ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → Σ 𝑘 ∈ ( 0 ... ( 𝑛 + 1 ) ) ( 𝑘 C 𝐶 ) = ( Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( 𝑘 C 𝐶 ) + ( ( 𝑛 + 1 ) C 𝐶 ) ) ) |
78 |
77
|
adantr |
⊢ ( ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( 𝑘 C 𝐶 ) = ( ( 𝑛 + 1 ) C ( 𝐶 + 1 ) ) ) → Σ 𝑘 ∈ ( 0 ... ( 𝑛 + 1 ) ) ( 𝑘 C 𝐶 ) = ( Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( 𝑘 C 𝐶 ) + ( ( 𝑛 + 1 ) C 𝐶 ) ) ) |
79 |
|
id |
⊢ ( Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( 𝑘 C 𝐶 ) = ( ( 𝑛 + 1 ) C ( 𝐶 + 1 ) ) → Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( 𝑘 C 𝐶 ) = ( ( 𝑛 + 1 ) C ( 𝐶 + 1 ) ) ) |
80 |
|
nn0cn |
⊢ ( 𝐶 ∈ ℕ0 → 𝐶 ∈ ℂ ) |
81 |
80
|
adantl |
⊢ ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → 𝐶 ∈ ℂ ) |
82 |
|
1cnd |
⊢ ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → 1 ∈ ℂ ) |
83 |
81 82
|
pncand |
⊢ ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( ( 𝐶 + 1 ) − 1 ) = 𝐶 ) |
84 |
83
|
oveq2d |
⊢ ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( ( 𝑛 + 1 ) C ( ( 𝐶 + 1 ) − 1 ) ) = ( ( 𝑛 + 1 ) C 𝐶 ) ) |
85 |
84
|
eqcomd |
⊢ ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( ( 𝑛 + 1 ) C 𝐶 ) = ( ( 𝑛 + 1 ) C ( ( 𝐶 + 1 ) − 1 ) ) ) |
86 |
79 85
|
oveqan12rd |
⊢ ( ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( 𝑘 C 𝐶 ) = ( ( 𝑛 + 1 ) C ( 𝐶 + 1 ) ) ) → ( Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( 𝑘 C 𝐶 ) + ( ( 𝑛 + 1 ) C 𝐶 ) ) = ( ( ( 𝑛 + 1 ) C ( 𝐶 + 1 ) ) + ( ( 𝑛 + 1 ) C ( ( 𝐶 + 1 ) − 1 ) ) ) ) |
87 |
|
peano2nn0 |
⊢ ( 𝑛 ∈ ℕ0 → ( 𝑛 + 1 ) ∈ ℕ0 ) |
88 |
|
peano2nn0 |
⊢ ( 𝐶 ∈ ℕ0 → ( 𝐶 + 1 ) ∈ ℕ0 ) |
89 |
88
|
nn0zd |
⊢ ( 𝐶 ∈ ℕ0 → ( 𝐶 + 1 ) ∈ ℤ ) |
90 |
|
bcpasc |
⊢ ( ( ( 𝑛 + 1 ) ∈ ℕ0 ∧ ( 𝐶 + 1 ) ∈ ℤ ) → ( ( ( 𝑛 + 1 ) C ( 𝐶 + 1 ) ) + ( ( 𝑛 + 1 ) C ( ( 𝐶 + 1 ) − 1 ) ) ) = ( ( ( 𝑛 + 1 ) + 1 ) C ( 𝐶 + 1 ) ) ) |
91 |
87 89 90
|
syl2an |
⊢ ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( ( ( 𝑛 + 1 ) C ( 𝐶 + 1 ) ) + ( ( 𝑛 + 1 ) C ( ( 𝐶 + 1 ) − 1 ) ) ) = ( ( ( 𝑛 + 1 ) + 1 ) C ( 𝐶 + 1 ) ) ) |
92 |
91
|
adantr |
⊢ ( ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( 𝑘 C 𝐶 ) = ( ( 𝑛 + 1 ) C ( 𝐶 + 1 ) ) ) → ( ( ( 𝑛 + 1 ) C ( 𝐶 + 1 ) ) + ( ( 𝑛 + 1 ) C ( ( 𝐶 + 1 ) − 1 ) ) ) = ( ( ( 𝑛 + 1 ) + 1 ) C ( 𝐶 + 1 ) ) ) |
93 |
78 86 92
|
3eqtrd |
⊢ ( ( ( 𝑛 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( 𝑘 C 𝐶 ) = ( ( 𝑛 + 1 ) C ( 𝐶 + 1 ) ) ) → Σ 𝑘 ∈ ( 0 ... ( 𝑛 + 1 ) ) ( 𝑘 C 𝐶 ) = ( ( ( 𝑛 + 1 ) + 1 ) C ( 𝐶 + 1 ) ) ) |
94 |
93
|
exp31 |
⊢ ( 𝑛 ∈ ℕ0 → ( 𝐶 ∈ ℕ0 → ( Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( 𝑘 C 𝐶 ) = ( ( 𝑛 + 1 ) C ( 𝐶 + 1 ) ) → Σ 𝑘 ∈ ( 0 ... ( 𝑛 + 1 ) ) ( 𝑘 C 𝐶 ) = ( ( ( 𝑛 + 1 ) + 1 ) C ( 𝐶 + 1 ) ) ) ) ) |
95 |
94
|
a2d |
⊢ ( 𝑛 ∈ ℕ0 → ( ( 𝐶 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... 𝑛 ) ( 𝑘 C 𝐶 ) = ( ( 𝑛 + 1 ) C ( 𝐶 + 1 ) ) ) → ( 𝐶 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... ( 𝑛 + 1 ) ) ( 𝑘 C 𝐶 ) = ( ( ( 𝑛 + 1 ) + 1 ) C ( 𝐶 + 1 ) ) ) ) ) |
96 |
8 14 20 26 65 95
|
nn0ind |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝐶 ∈ ℕ0 → Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( 𝑘 C 𝐶 ) = ( ( 𝑁 + 1 ) C ( 𝐶 + 1 ) ) ) ) |
97 |
96
|
imp |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → Σ 𝑘 ∈ ( 0 ... 𝑁 ) ( 𝑘 C 𝐶 ) = ( ( 𝑁 + 1 ) C ( 𝐶 + 1 ) ) ) |