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 |
|
binomcxplem.s |
⊢ 𝑆 = ( 𝑏 ∈ ℂ ↦ ( 𝑘 ∈ ℕ0 ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝑏 ↑ 𝑘 ) ) ) ) |
7 |
|
binomcxplem.r |
⊢ 𝑅 = sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝑆 ‘ 𝑟 ) ) ∈ dom ⇝ } , ℝ* , < ) |
8 |
|
simpl |
⊢ ( ( 𝑏 = 𝑥 ∧ 𝑘 ∈ ℕ0 ) → 𝑏 = 𝑥 ) |
9 |
8
|
oveq1d |
⊢ ( ( 𝑏 = 𝑥 ∧ 𝑘 ∈ ℕ0 ) → ( 𝑏 ↑ 𝑘 ) = ( 𝑥 ↑ 𝑘 ) ) |
10 |
9
|
oveq2d |
⊢ ( ( 𝑏 = 𝑥 ∧ 𝑘 ∈ ℕ0 ) → ( ( 𝐹 ‘ 𝑘 ) · ( 𝑏 ↑ 𝑘 ) ) = ( ( 𝐹 ‘ 𝑘 ) · ( 𝑥 ↑ 𝑘 ) ) ) |
11 |
10
|
mpteq2dva |
⊢ ( 𝑏 = 𝑥 → ( 𝑘 ∈ ℕ0 ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝑏 ↑ 𝑘 ) ) ) = ( 𝑘 ∈ ℕ0 ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝑥 ↑ 𝑘 ) ) ) ) |
12 |
|
fveq2 |
⊢ ( 𝑘 = 𝑦 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑦 ) ) |
13 |
|
oveq2 |
⊢ ( 𝑘 = 𝑦 → ( 𝑥 ↑ 𝑘 ) = ( 𝑥 ↑ 𝑦 ) ) |
14 |
12 13
|
oveq12d |
⊢ ( 𝑘 = 𝑦 → ( ( 𝐹 ‘ 𝑘 ) · ( 𝑥 ↑ 𝑘 ) ) = ( ( 𝐹 ‘ 𝑦 ) · ( 𝑥 ↑ 𝑦 ) ) ) |
15 |
14
|
cbvmptv |
⊢ ( 𝑘 ∈ ℕ0 ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝑥 ↑ 𝑘 ) ) ) = ( 𝑦 ∈ ℕ0 ↦ ( ( 𝐹 ‘ 𝑦 ) · ( 𝑥 ↑ 𝑦 ) ) ) |
16 |
11 15
|
eqtrdi |
⊢ ( 𝑏 = 𝑥 → ( 𝑘 ∈ ℕ0 ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝑏 ↑ 𝑘 ) ) ) = ( 𝑦 ∈ ℕ0 ↦ ( ( 𝐹 ‘ 𝑦 ) · ( 𝑥 ↑ 𝑦 ) ) ) ) |
17 |
16
|
cbvmptv |
⊢ ( 𝑏 ∈ ℂ ↦ ( 𝑘 ∈ ℕ0 ↦ ( ( 𝐹 ‘ 𝑘 ) · ( 𝑏 ↑ 𝑘 ) ) ) ) = ( 𝑥 ∈ ℂ ↦ ( 𝑦 ∈ ℕ0 ↦ ( ( 𝐹 ‘ 𝑦 ) · ( 𝑥 ↑ 𝑦 ) ) ) ) |
18 |
6 17
|
eqtri |
⊢ 𝑆 = ( 𝑥 ∈ ℂ ↦ ( 𝑦 ∈ ℕ0 ↦ ( ( 𝐹 ‘ 𝑦 ) · ( 𝑥 ↑ 𝑦 ) ) ) ) |
19 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑗 ∈ ℕ0 ) → 𝐶 ∈ ℂ ) |
20 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑗 ∈ ℕ0 ) → 𝑗 ∈ ℕ0 ) |
21 |
19 20
|
bcccl |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑗 ∈ ℕ0 ) → ( 𝐶 C𝑐 𝑗 ) ∈ ℂ ) |
22 |
21 5
|
fmptd |
⊢ ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) → 𝐹 : ℕ0 ⟶ ℂ ) |
23 |
|
fvoveq1 |
⊢ ( 𝑘 = 𝑖 → ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝐹 ‘ ( 𝑖 + 1 ) ) ) |
24 |
|
fveq2 |
⊢ ( 𝑘 = 𝑖 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑖 ) ) |
25 |
23 24
|
oveq12d |
⊢ ( 𝑘 = 𝑖 → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) = ( ( 𝐹 ‘ ( 𝑖 + 1 ) ) / ( 𝐹 ‘ 𝑖 ) ) ) |
26 |
25
|
fveq2d |
⊢ ( 𝑘 = 𝑖 → ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) = ( abs ‘ ( ( 𝐹 ‘ ( 𝑖 + 1 ) ) / ( 𝐹 ‘ 𝑖 ) ) ) ) |
27 |
26
|
cbvmptv |
⊢ ( 𝑘 ∈ ℕ0 ↦ ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ) = ( 𝑖 ∈ ℕ0 ↦ ( abs ‘ ( ( 𝐹 ‘ ( 𝑖 + 1 ) ) / ( 𝐹 ‘ 𝑖 ) ) ) ) |
28 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
29 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
30 |
29
|
a1i |
⊢ ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) → 0 ∈ ℕ0 ) |
31 |
5
|
a1i |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑖 ∈ ℕ0 ) → 𝐹 = ( 𝑗 ∈ ℕ0 ↦ ( 𝐶 C𝑐 𝑗 ) ) ) |
32 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑖 ∈ ℕ0 ) ∧ 𝑗 = 𝑖 ) → 𝑗 = 𝑖 ) |
33 |
32
|
oveq2d |
⊢ ( ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑖 ∈ ℕ0 ) ∧ 𝑗 = 𝑖 ) → ( 𝐶 C𝑐 𝑗 ) = ( 𝐶 C𝑐 𝑖 ) ) |
34 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑖 ∈ ℕ0 ) → 𝑖 ∈ ℕ0 ) |
35 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑖 ∈ ℕ0 ) → ( 𝐶 C𝑐 𝑖 ) ∈ V ) |
36 |
31 33 34 35
|
fvmptd |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑖 ∈ ℕ0 ) → ( 𝐹 ‘ 𝑖 ) = ( 𝐶 C𝑐 𝑖 ) ) |
37 |
|
elfznn0 |
⊢ ( 𝐶 ∈ ( 0 ... ( 𝑖 − 1 ) ) → 𝐶 ∈ ℕ0 ) |
38 |
37
|
con3i |
⊢ ( ¬ 𝐶 ∈ ℕ0 → ¬ 𝐶 ∈ ( 0 ... ( 𝑖 − 1 ) ) ) |
39 |
38
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑖 ∈ ℕ0 ) → ¬ 𝐶 ∈ ( 0 ... ( 𝑖 − 1 ) ) ) |
40 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ0 ) → 𝐶 ∈ ℂ ) |
41 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ0 ) → 𝑖 ∈ ℕ0 ) |
42 |
40 41
|
bcc0 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ0 ) → ( ( 𝐶 C𝑐 𝑖 ) = 0 ↔ 𝐶 ∈ ( 0 ... ( 𝑖 − 1 ) ) ) ) |
43 |
42
|
necon3abid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℕ0 ) → ( ( 𝐶 C𝑐 𝑖 ) ≠ 0 ↔ ¬ 𝐶 ∈ ( 0 ... ( 𝑖 − 1 ) ) ) ) |
44 |
43
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑖 ∈ ℕ0 ) → ( ( 𝐶 C𝑐 𝑖 ) ≠ 0 ↔ ¬ 𝐶 ∈ ( 0 ... ( 𝑖 − 1 ) ) ) ) |
45 |
39 44
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑖 ∈ ℕ0 ) → ( 𝐶 C𝑐 𝑖 ) ≠ 0 ) |
46 |
36 45
|
eqnetrd |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) ∧ 𝑖 ∈ ℕ0 ) → ( 𝐹 ‘ 𝑖 ) ≠ 0 ) |
47 |
1 2 3 4 5
|
binomcxplemfrat |
⊢ ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) → ( 𝑘 ∈ ℕ0 ↦ ( abs ‘ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) / ( 𝐹 ‘ 𝑘 ) ) ) ) ⇝ 1 ) |
48 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
49 |
48
|
a1i |
⊢ ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) → 1 ≠ 0 ) |
50 |
18 22 7 27 28 30 46 47 49
|
radcnvrat |
⊢ ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) → 𝑅 = ( 1 / 1 ) ) |
51 |
|
1div1e1 |
⊢ ( 1 / 1 ) = 1 |
52 |
50 51
|
eqtrdi |
⊢ ( ( 𝜑 ∧ ¬ 𝐶 ∈ ℕ0 ) → 𝑅 = 1 ) |