Step |
Hyp |
Ref |
Expression |
1 |
|
pserf.g |
⊢ 𝐺 = ( 𝑥 ∈ ℂ ↦ ( 𝑛 ∈ ℕ0 ↦ ( ( 𝐴 ‘ 𝑛 ) · ( 𝑥 ↑ 𝑛 ) ) ) ) |
2 |
|
pserf.f |
⊢ 𝐹 = ( 𝑦 ∈ 𝑆 ↦ Σ 𝑗 ∈ ℕ0 ( ( 𝐺 ‘ 𝑦 ) ‘ 𝑗 ) ) |
3 |
|
pserf.a |
⊢ ( 𝜑 → 𝐴 : ℕ0 ⟶ ℂ ) |
4 |
|
pserf.r |
⊢ 𝑅 = sup ( { 𝑟 ∈ ℝ ∣ seq 0 ( + , ( 𝐺 ‘ 𝑟 ) ) ∈ dom ⇝ } , ℝ* , < ) |
5 |
|
psercn.s |
⊢ 𝑆 = ( ◡ abs “ ( 0 [,) 𝑅 ) ) |
6 |
|
psercn.m |
⊢ 𝑀 = if ( 𝑅 ∈ ℝ , ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) , ( ( abs ‘ 𝑎 ) + 1 ) ) |
7 |
|
cnvimass |
⊢ ( ◡ abs “ ( 0 [,) 𝑅 ) ) ⊆ dom abs |
8 |
|
absf |
⊢ abs : ℂ ⟶ ℝ |
9 |
8
|
fdmi |
⊢ dom abs = ℂ |
10 |
7 9
|
sseqtri |
⊢ ( ◡ abs “ ( 0 [,) 𝑅 ) ) ⊆ ℂ |
11 |
5 10
|
eqsstri |
⊢ 𝑆 ⊆ ℂ |
12 |
11
|
a1i |
⊢ ( 𝜑 → 𝑆 ⊆ ℂ ) |
13 |
12
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → 𝑎 ∈ ℂ ) |
14 |
13
|
abscld |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( abs ‘ 𝑎 ) ∈ ℝ ) |
15 |
|
readdcl |
⊢ ( ( ( abs ‘ 𝑎 ) ∈ ℝ ∧ 𝑅 ∈ ℝ ) → ( ( abs ‘ 𝑎 ) + 𝑅 ) ∈ ℝ ) |
16 |
14 15
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ 𝑅 ∈ ℝ ) → ( ( abs ‘ 𝑎 ) + 𝑅 ) ∈ ℝ ) |
17 |
16
|
rehalfcld |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ 𝑅 ∈ ℝ ) → ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) ∈ ℝ ) |
18 |
|
peano2re |
⊢ ( ( abs ‘ 𝑎 ) ∈ ℝ → ( ( abs ‘ 𝑎 ) + 1 ) ∈ ℝ ) |
19 |
14 18
|
syl |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( ( abs ‘ 𝑎 ) + 1 ) ∈ ℝ ) |
20 |
19
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ ¬ 𝑅 ∈ ℝ ) → ( ( abs ‘ 𝑎 ) + 1 ) ∈ ℝ ) |
21 |
17 20
|
ifclda |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → if ( 𝑅 ∈ ℝ , ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) , ( ( abs ‘ 𝑎 ) + 1 ) ) ∈ ℝ ) |
22 |
6 21
|
eqeltrid |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → 𝑀 ∈ ℝ ) |
23 |
|
0re |
⊢ 0 ∈ ℝ |
24 |
23
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → 0 ∈ ℝ ) |
25 |
13
|
absge0d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → 0 ≤ ( abs ‘ 𝑎 ) ) |
26 |
|
breq2 |
⊢ ( ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) = if ( 𝑅 ∈ ℝ , ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) , ( ( abs ‘ 𝑎 ) + 1 ) ) → ( ( abs ‘ 𝑎 ) < ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) ↔ ( abs ‘ 𝑎 ) < if ( 𝑅 ∈ ℝ , ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) , ( ( abs ‘ 𝑎 ) + 1 ) ) ) ) |
27 |
|
breq2 |
⊢ ( ( ( abs ‘ 𝑎 ) + 1 ) = if ( 𝑅 ∈ ℝ , ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) , ( ( abs ‘ 𝑎 ) + 1 ) ) → ( ( abs ‘ 𝑎 ) < ( ( abs ‘ 𝑎 ) + 1 ) ↔ ( abs ‘ 𝑎 ) < if ( 𝑅 ∈ ℝ , ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) , ( ( abs ‘ 𝑎 ) + 1 ) ) ) ) |
28 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → 𝑎 ∈ 𝑆 ) |
29 |
28 5
|
eleqtrdi |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → 𝑎 ∈ ( ◡ abs “ ( 0 [,) 𝑅 ) ) ) |
30 |
|
ffn |
⊢ ( abs : ℂ ⟶ ℝ → abs Fn ℂ ) |
31 |
|
elpreima |
⊢ ( abs Fn ℂ → ( 𝑎 ∈ ( ◡ abs “ ( 0 [,) 𝑅 ) ) ↔ ( 𝑎 ∈ ℂ ∧ ( abs ‘ 𝑎 ) ∈ ( 0 [,) 𝑅 ) ) ) ) |
32 |
8 30 31
|
mp2b |
⊢ ( 𝑎 ∈ ( ◡ abs “ ( 0 [,) 𝑅 ) ) ↔ ( 𝑎 ∈ ℂ ∧ ( abs ‘ 𝑎 ) ∈ ( 0 [,) 𝑅 ) ) ) |
33 |
29 32
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( 𝑎 ∈ ℂ ∧ ( abs ‘ 𝑎 ) ∈ ( 0 [,) 𝑅 ) ) ) |
34 |
33
|
simprd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( abs ‘ 𝑎 ) ∈ ( 0 [,) 𝑅 ) ) |
35 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
36 |
1 3 4
|
radcnvcl |
⊢ ( 𝜑 → 𝑅 ∈ ( 0 [,] +∞ ) ) |
37 |
36
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → 𝑅 ∈ ( 0 [,] +∞ ) ) |
38 |
35 37
|
sselid |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → 𝑅 ∈ ℝ* ) |
39 |
|
elico2 |
⊢ ( ( 0 ∈ ℝ ∧ 𝑅 ∈ ℝ* ) → ( ( abs ‘ 𝑎 ) ∈ ( 0 [,) 𝑅 ) ↔ ( ( abs ‘ 𝑎 ) ∈ ℝ ∧ 0 ≤ ( abs ‘ 𝑎 ) ∧ ( abs ‘ 𝑎 ) < 𝑅 ) ) ) |
40 |
23 38 39
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( ( abs ‘ 𝑎 ) ∈ ( 0 [,) 𝑅 ) ↔ ( ( abs ‘ 𝑎 ) ∈ ℝ ∧ 0 ≤ ( abs ‘ 𝑎 ) ∧ ( abs ‘ 𝑎 ) < 𝑅 ) ) ) |
41 |
34 40
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( ( abs ‘ 𝑎 ) ∈ ℝ ∧ 0 ≤ ( abs ‘ 𝑎 ) ∧ ( abs ‘ 𝑎 ) < 𝑅 ) ) |
42 |
41
|
simp3d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( abs ‘ 𝑎 ) < 𝑅 ) |
43 |
42
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ 𝑅 ∈ ℝ ) → ( abs ‘ 𝑎 ) < 𝑅 ) |
44 |
|
avglt1 |
⊢ ( ( ( abs ‘ 𝑎 ) ∈ ℝ ∧ 𝑅 ∈ ℝ ) → ( ( abs ‘ 𝑎 ) < 𝑅 ↔ ( abs ‘ 𝑎 ) < ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) ) ) |
45 |
14 44
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ 𝑅 ∈ ℝ ) → ( ( abs ‘ 𝑎 ) < 𝑅 ↔ ( abs ‘ 𝑎 ) < ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) ) ) |
46 |
43 45
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ 𝑅 ∈ ℝ ) → ( abs ‘ 𝑎 ) < ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) ) |
47 |
14
|
ltp1d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( abs ‘ 𝑎 ) < ( ( abs ‘ 𝑎 ) + 1 ) ) |
48 |
47
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ ¬ 𝑅 ∈ ℝ ) → ( abs ‘ 𝑎 ) < ( ( abs ‘ 𝑎 ) + 1 ) ) |
49 |
26 27 46 48
|
ifbothda |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( abs ‘ 𝑎 ) < if ( 𝑅 ∈ ℝ , ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) , ( ( abs ‘ 𝑎 ) + 1 ) ) ) |
50 |
49 6
|
breqtrrdi |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( abs ‘ 𝑎 ) < 𝑀 ) |
51 |
24 14 22 25 50
|
lelttrd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → 0 < 𝑀 ) |
52 |
22 51
|
elrpd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → 𝑀 ∈ ℝ+ ) |
53 |
|
breq1 |
⊢ ( ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) = if ( 𝑅 ∈ ℝ , ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) , ( ( abs ‘ 𝑎 ) + 1 ) ) → ( ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) < 𝑅 ↔ if ( 𝑅 ∈ ℝ , ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) , ( ( abs ‘ 𝑎 ) + 1 ) ) < 𝑅 ) ) |
54 |
|
breq1 |
⊢ ( ( ( abs ‘ 𝑎 ) + 1 ) = if ( 𝑅 ∈ ℝ , ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) , ( ( abs ‘ 𝑎 ) + 1 ) ) → ( ( ( abs ‘ 𝑎 ) + 1 ) < 𝑅 ↔ if ( 𝑅 ∈ ℝ , ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) , ( ( abs ‘ 𝑎 ) + 1 ) ) < 𝑅 ) ) |
55 |
|
avglt2 |
⊢ ( ( ( abs ‘ 𝑎 ) ∈ ℝ ∧ 𝑅 ∈ ℝ ) → ( ( abs ‘ 𝑎 ) < 𝑅 ↔ ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) < 𝑅 ) ) |
56 |
14 55
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ 𝑅 ∈ ℝ ) → ( ( abs ‘ 𝑎 ) < 𝑅 ↔ ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) < 𝑅 ) ) |
57 |
43 56
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ 𝑅 ∈ ℝ ) → ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) < 𝑅 ) |
58 |
19
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( ( abs ‘ 𝑎 ) + 1 ) ∈ ℝ* ) |
59 |
38 58
|
xrlenltd |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( 𝑅 ≤ ( ( abs ‘ 𝑎 ) + 1 ) ↔ ¬ ( ( abs ‘ 𝑎 ) + 1 ) < 𝑅 ) ) |
60 |
|
0xr |
⊢ 0 ∈ ℝ* |
61 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
62 |
|
elicc1 |
⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( 𝑅 ∈ ( 0 [,] +∞ ) ↔ ( 𝑅 ∈ ℝ* ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞ ) ) ) |
63 |
60 61 62
|
mp2an |
⊢ ( 𝑅 ∈ ( 0 [,] +∞ ) ↔ ( 𝑅 ∈ ℝ* ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞ ) ) |
64 |
36 63
|
sylib |
⊢ ( 𝜑 → ( 𝑅 ∈ ℝ* ∧ 0 ≤ 𝑅 ∧ 𝑅 ≤ +∞ ) ) |
65 |
64
|
simp2d |
⊢ ( 𝜑 → 0 ≤ 𝑅 ) |
66 |
65
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → 0 ≤ 𝑅 ) |
67 |
|
ge0gtmnf |
⊢ ( ( 𝑅 ∈ ℝ* ∧ 0 ≤ 𝑅 ) → -∞ < 𝑅 ) |
68 |
38 66 67
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → -∞ < 𝑅 ) |
69 |
|
xrre |
⊢ ( ( ( 𝑅 ∈ ℝ* ∧ ( ( abs ‘ 𝑎 ) + 1 ) ∈ ℝ ) ∧ ( -∞ < 𝑅 ∧ 𝑅 ≤ ( ( abs ‘ 𝑎 ) + 1 ) ) ) → 𝑅 ∈ ℝ ) |
70 |
69
|
expr |
⊢ ( ( ( 𝑅 ∈ ℝ* ∧ ( ( abs ‘ 𝑎 ) + 1 ) ∈ ℝ ) ∧ -∞ < 𝑅 ) → ( 𝑅 ≤ ( ( abs ‘ 𝑎 ) + 1 ) → 𝑅 ∈ ℝ ) ) |
71 |
38 19 68 70
|
syl21anc |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( 𝑅 ≤ ( ( abs ‘ 𝑎 ) + 1 ) → 𝑅 ∈ ℝ ) ) |
72 |
59 71
|
sylbird |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( ¬ ( ( abs ‘ 𝑎 ) + 1 ) < 𝑅 → 𝑅 ∈ ℝ ) ) |
73 |
72
|
con1d |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( ¬ 𝑅 ∈ ℝ → ( ( abs ‘ 𝑎 ) + 1 ) < 𝑅 ) ) |
74 |
73
|
imp |
⊢ ( ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) ∧ ¬ 𝑅 ∈ ℝ ) → ( ( abs ‘ 𝑎 ) + 1 ) < 𝑅 ) |
75 |
53 54 57 74
|
ifbothda |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → if ( 𝑅 ∈ ℝ , ( ( ( abs ‘ 𝑎 ) + 𝑅 ) / 2 ) , ( ( abs ‘ 𝑎 ) + 1 ) ) < 𝑅 ) |
76 |
6 75
|
eqbrtrid |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → 𝑀 < 𝑅 ) |
77 |
52 50 76
|
3jca |
⊢ ( ( 𝜑 ∧ 𝑎 ∈ 𝑆 ) → ( 𝑀 ∈ ℝ+ ∧ ( abs ‘ 𝑎 ) < 𝑀 ∧ 𝑀 < 𝑅 ) ) |