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 |
|
binomcxplem.f |
⊢ 𝐹 = ( 𝑗 ∈ ℕ0 ↦ ( 𝐶 C𝑐 𝑗 ) ) |
6 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝐶 ∈ ℂ ) |
7 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℕ0 ) |
8 |
6 7
|
bccp1k |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐶 C𝑐 ( 𝑘 + 1 ) ) = ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐶 − 𝑘 ) / ( 𝑘 + 1 ) ) ) ) |
9 |
5
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 𝐹 = ( 𝑗 ∈ ℕ0 ↦ ( 𝐶 C𝑐 𝑗 ) ) ) |
10 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = ( 𝑘 + 1 ) ) → 𝑗 = ( 𝑘 + 1 ) ) |
11 |
10
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = ( 𝑘 + 1 ) ) → ( 𝐶 C𝑐 𝑗 ) = ( 𝐶 C𝑐 ( 𝑘 + 1 ) ) ) |
12 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
13 |
12
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → 1 ∈ ℕ0 ) |
14 |
7 13
|
nn0addcld |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 + 1 ) ∈ ℕ0 ) |
15 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐶 C𝑐 ( 𝑘 + 1 ) ) ∈ V ) |
16 |
9 11 14 15
|
fvmptd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝐶 C𝑐 ( 𝑘 + 1 ) ) ) |
17 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = 𝑘 ) → 𝑗 = 𝑘 ) |
18 |
17
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) ∧ 𝑗 = 𝑘 ) → ( 𝐶 C𝑐 𝑗 ) = ( 𝐶 C𝑐 𝑘 ) ) |
19 |
|
ovexd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐶 C𝑐 𝑘 ) ∈ V ) |
20 |
9 18 7 19
|
fvmptd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐶 C𝑐 𝑘 ) ) |
21 |
20
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐹 ‘ 𝑘 ) · ( ( 𝐶 − 𝑘 ) / ( 𝑘 + 1 ) ) ) = ( ( 𝐶 C𝑐 𝑘 ) · ( ( 𝐶 − 𝑘 ) / ( 𝑘 + 1 ) ) ) ) |
22 |
8 16 21
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( ( 𝐹 ‘ 𝑘 ) · ( ( 𝐶 − 𝑘 ) / ( 𝑘 + 1 ) ) ) ) |
23 |
22
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( ( 𝐹 ‘ 𝑘 ) · ( ( 𝐶 − 𝑘 ) / ( 𝑘 + 1 ) ) ) ) |
24 |
23
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐹 ‘ 𝑘 ) · ( ( 𝐶 − 𝑘 ) / ( 𝑘 + 1 ) ) ) = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
25 |
6 7
|
bcccl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐶 C𝑐 𝑘 ) ∈ ℂ ) |
26 |
20 25
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ℕ0 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
27 |
26
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
28 |
6
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝐶 ∈ ℂ ) |
29 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℕ0 ) |
30 |
29
|
nn0cnd |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 𝑘 ∈ ℂ ) |
31 |
28 30
|
subcld |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐶 − 𝑘 ) ∈ ℂ ) |
32 |
|
1cnd |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → 1 ∈ ℂ ) |
33 |
30 32
|
addcld |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 + 1 ) ∈ ℂ ) |
34 |
|
nn0p1nn |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝑘 + 1 ) ∈ ℕ ) |
35 |
34
|
nnne0d |
⊢ ( 𝑘 ∈ ℕ0 → ( 𝑘 + 1 ) ≠ 0 ) |
36 |
35
|
adantl |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝑘 + 1 ) ≠ 0 ) |
37 |
31 33 36
|
divcld |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐶 − 𝑘 ) / ( 𝑘 + 1 ) ) ∈ ℂ ) |
38 |
27 37
|
mulcld |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐹 ‘ 𝑘 ) · ( ( 𝐶 − 𝑘 ) / ( 𝑘 + 1 ) ) ) ∈ ℂ ) |
39 |
23 38
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
40 |
20
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐶 C𝑐 𝑘 ) ) |
41 |
|
elfznn0 |
⊢ ( 𝐶 ∈ ( 0 ... ( 𝑘 − 1 ) ) → 𝐶 ∈ ℕ0 ) |
42 |
41
|
con3i |
⊢ ( ¬ 𝐶 ∈ ℕ0 → ¬ 𝐶 ∈ ( 0 ... ( 𝑘 − 1 ) ) ) |
43 |
42
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ¬ 𝐶 ∈ ( 0 ... ( 𝑘 − 1 ) ) ) |
44 |
28 29
|
bcc0 |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐶 C𝑐 𝑘 ) = 0 ↔ 𝐶 ∈ ( 0 ... ( 𝑘 − 1 ) ) ) ) |
45 |
44
|
necon3abid |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐶 C𝑐 𝑘 ) ≠ 0 ↔ ¬ 𝐶 ∈ ( 0 ... ( 𝑘 − 1 ) ) ) ) |
46 |
43 45
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐶 C𝑐 𝑘 ) ≠ 0 ) |
47 |
40 46
|
eqnetrd |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( 𝐹 ‘ 𝑘 ) ≠ 0 ) |
48 |
39 27 37 47
|
divmuld |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) = ( ( 𝐶 − 𝑘 ) / ( 𝑘 + 1 ) ) ↔ ( ( 𝐹 ‘ 𝑘 ) · ( ( 𝐶 − 𝑘 ) / ( 𝑘 + 1 ) ) ) = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
49 |
24 48
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) = ( ( 𝐶 − 𝑘 ) / ( 𝑘 + 1 ) ) ) |
50 |
49
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑘 ∈ ℕ0 ) → ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) = ( abs ‘ ( ( 𝐶 − 𝑘 ) / ( 𝑘 + 1 ) ) ) ) |
51 |
50
|
mpteq2dva |
⊢ ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) → ( 𝑘 ∈ ℕ0 ↦ ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ) = ( 𝑘 ∈ ℕ0 ↦ ( abs ‘ ( ( 𝐶 − 𝑘 ) / ( 𝑘 + 1 ) ) ) ) ) |
52 |
1 2 3 4
|
binomcxplemrat |
⊢ ( 𝜑 → ( 𝑘 ∈ ℕ0 ↦ ( abs ‘ ( ( 𝐶 − 𝑘 ) / ( 𝑘 + 1 ) ) ) ) ⇝ 1 ) |
53 |
52
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) → ( 𝑘 ∈ ℕ0 ↦ ( abs ‘ ( ( 𝐶 − 𝑘 ) / ( 𝑘 + 1 ) ) ) ) ⇝ 1 ) |
54 |
51 53
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) → ( 𝑘 ∈ ℕ0 ↦ ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ) ⇝ 1 ) |