Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem48.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
fourierdlem48.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
fourierdlem48.altb |
⊢ ( 𝜑 → 𝐴 < 𝐵 ) |
4 |
|
fourierdlem48.p |
⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) |
5 |
|
fourierdlem48.t |
⊢ 𝑇 = ( 𝐵 − 𝐴 ) |
6 |
|
fourierdlem48.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
7 |
|
fourierdlem48.q |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ) |
8 |
|
fourierdlem48.f |
⊢ ( 𝜑 → 𝐹 : 𝐷 ⟶ ℝ ) |
9 |
|
fourierdlem48.dper |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
10 |
|
fourierdlem48.per |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
11 |
|
fourierdlem48.cn |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
12 |
|
fourierdlem48.r |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑅 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
13 |
|
fourierdlem48.x |
⊢ ( 𝜑 → 𝑋 ∈ ℝ ) |
14 |
|
fourierdlem48.z |
⊢ 𝑍 = ( 𝑥 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) |
15 |
|
fourierdlem48.e |
⊢ 𝐸 = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) ) |
16 |
|
fourierdlem48.ch |
⊢ ( 𝜒 ↔ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑘 ∈ ℤ ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) |
17 |
|
simpl |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → 𝜑 ) |
18 |
|
0zd |
⊢ ( 𝜑 → 0 ∈ ℤ ) |
19 |
6
|
nnzd |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
20 |
6
|
nngt0d |
⊢ ( 𝜑 → 0 < 𝑀 ) |
21 |
|
fzolb |
⊢ ( 0 ∈ ( 0 ..^ 𝑀 ) ↔ ( 0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀 ) ) |
22 |
18 19 20 21
|
syl3anbrc |
⊢ ( 𝜑 → 0 ∈ ( 0 ..^ 𝑀 ) ) |
23 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → 0 ∈ ( 0 ..^ 𝑀 ) ) |
24 |
2 13
|
resubcld |
⊢ ( 𝜑 → ( 𝐵 − 𝑋 ) ∈ ℝ ) |
25 |
2 1
|
resubcld |
⊢ ( 𝜑 → ( 𝐵 − 𝐴 ) ∈ ℝ ) |
26 |
5 25
|
eqeltrid |
⊢ ( 𝜑 → 𝑇 ∈ ℝ ) |
27 |
1 2
|
posdifd |
⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ 0 < ( 𝐵 − 𝐴 ) ) ) |
28 |
3 27
|
mpbid |
⊢ ( 𝜑 → 0 < ( 𝐵 − 𝐴 ) ) |
29 |
28 5
|
breqtrrdi |
⊢ ( 𝜑 → 0 < 𝑇 ) |
30 |
29
|
gt0ne0d |
⊢ ( 𝜑 → 𝑇 ≠ 0 ) |
31 |
24 26 30
|
redivcld |
⊢ ( 𝜑 → ( ( 𝐵 − 𝑋 ) / 𝑇 ) ∈ ℝ ) |
32 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( ( 𝐵 − 𝑋 ) / 𝑇 ) ∈ ℝ ) |
33 |
32
|
flcld |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) |
34 |
|
1zzd |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → 1 ∈ ℤ ) |
35 |
33 34
|
zsubcld |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) − 1 ) ∈ ℤ ) |
36 |
|
id |
⊢ ( ( 𝐸 ‘ 𝑋 ) = 𝐵 → ( 𝐸 ‘ 𝑋 ) = 𝐵 ) |
37 |
5
|
a1i |
⊢ ( ( 𝐸 ‘ 𝑋 ) = 𝐵 → 𝑇 = ( 𝐵 − 𝐴 ) ) |
38 |
36 37
|
oveq12d |
⊢ ( ( 𝐸 ‘ 𝑋 ) = 𝐵 → ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝐵 − ( 𝐵 − 𝐴 ) ) ) |
39 |
2
|
recnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
40 |
1
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
41 |
39 40
|
nncand |
⊢ ( 𝜑 → ( 𝐵 − ( 𝐵 − 𝐴 ) ) = 𝐴 ) |
42 |
38 41
|
sylan9eqr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = 𝐴 ) |
43 |
4
|
fourierdlem2 |
⊢ ( 𝑀 ∈ ℕ → ( 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
44 |
6 43
|
syl |
⊢ ( 𝜑 → ( 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
45 |
7 44
|
mpbid |
⊢ ( 𝜑 → ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
46 |
45
|
simpld |
⊢ ( 𝜑 → 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ) |
47 |
|
elmapi |
⊢ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
48 |
46 47
|
syl |
⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
49 |
6
|
nnnn0d |
⊢ ( 𝜑 → 𝑀 ∈ ℕ0 ) |
50 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
51 |
49 50
|
eleqtrdi |
⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ 0 ) ) |
52 |
|
eluzfz1 |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 0 ) → 0 ∈ ( 0 ... 𝑀 ) ) |
53 |
51 52
|
syl |
⊢ ( 𝜑 → 0 ∈ ( 0 ... 𝑀 ) ) |
54 |
48 53
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) ∈ ℝ ) |
55 |
54
|
rexrd |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) ∈ ℝ* ) |
56 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
57 |
|
0le1 |
⊢ 0 ≤ 1 |
58 |
57
|
a1i |
⊢ ( 𝜑 → 0 ≤ 1 ) |
59 |
6
|
nnge1d |
⊢ ( 𝜑 → 1 ≤ 𝑀 ) |
60 |
18 19 56 58 59
|
elfzd |
⊢ ( 𝜑 → 1 ∈ ( 0 ... 𝑀 ) ) |
61 |
48 60
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑄 ‘ 1 ) ∈ ℝ ) |
62 |
61
|
rexrd |
⊢ ( 𝜑 → ( 𝑄 ‘ 1 ) ∈ ℝ* ) |
63 |
1
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
64 |
45
|
simprd |
⊢ ( 𝜑 → ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
65 |
64
|
simplld |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) = 𝐴 ) |
66 |
1
|
leidd |
⊢ ( 𝜑 → 𝐴 ≤ 𝐴 ) |
67 |
65 66
|
eqbrtrd |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) ≤ 𝐴 ) |
68 |
65
|
eqcomd |
⊢ ( 𝜑 → 𝐴 = ( 𝑄 ‘ 0 ) ) |
69 |
|
0re |
⊢ 0 ∈ ℝ |
70 |
|
eleq1 |
⊢ ( 𝑖 = 0 → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↔ 0 ∈ ( 0 ..^ 𝑀 ) ) ) |
71 |
70
|
anbi2d |
⊢ ( 𝑖 = 0 → ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ↔ ( 𝜑 ∧ 0 ∈ ( 0 ..^ 𝑀 ) ) ) ) |
72 |
|
fveq2 |
⊢ ( 𝑖 = 0 → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 0 ) ) |
73 |
|
oveq1 |
⊢ ( 𝑖 = 0 → ( 𝑖 + 1 ) = ( 0 + 1 ) ) |
74 |
73
|
fveq2d |
⊢ ( 𝑖 = 0 → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ ( 0 + 1 ) ) ) |
75 |
72 74
|
breq12d |
⊢ ( 𝑖 = 0 → ( ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ↔ ( 𝑄 ‘ 0 ) < ( 𝑄 ‘ ( 0 + 1 ) ) ) ) |
76 |
71 75
|
imbi12d |
⊢ ( 𝑖 = 0 → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( ( 𝜑 ∧ 0 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 0 ) < ( 𝑄 ‘ ( 0 + 1 ) ) ) ) ) |
77 |
45
|
simprrd |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
78 |
77
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
79 |
76 78
|
vtoclg |
⊢ ( 0 ∈ ℝ → ( ( 𝜑 ∧ 0 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 0 ) < ( 𝑄 ‘ ( 0 + 1 ) ) ) ) |
80 |
69 79
|
ax-mp |
⊢ ( ( 𝜑 ∧ 0 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 0 ) < ( 𝑄 ‘ ( 0 + 1 ) ) ) |
81 |
22 80
|
mpdan |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) < ( 𝑄 ‘ ( 0 + 1 ) ) ) |
82 |
|
1e0p1 |
⊢ 1 = ( 0 + 1 ) |
83 |
82
|
fveq2i |
⊢ ( 𝑄 ‘ 1 ) = ( 𝑄 ‘ ( 0 + 1 ) ) |
84 |
81 83
|
breqtrrdi |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) < ( 𝑄 ‘ 1 ) ) |
85 |
68 84
|
eqbrtrd |
⊢ ( 𝜑 → 𝐴 < ( 𝑄 ‘ 1 ) ) |
86 |
55 62 63 67 85
|
elicod |
⊢ ( 𝜑 → 𝐴 ∈ ( ( 𝑄 ‘ 0 ) [,) ( 𝑄 ‘ 1 ) ) ) |
87 |
83
|
oveq2i |
⊢ ( ( 𝑄 ‘ 0 ) [,) ( 𝑄 ‘ 1 ) ) = ( ( 𝑄 ‘ 0 ) [,) ( 𝑄 ‘ ( 0 + 1 ) ) ) |
88 |
86 87
|
eleqtrdi |
⊢ ( 𝜑 → 𝐴 ∈ ( ( 𝑄 ‘ 0 ) [,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ) |
89 |
88
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → 𝐴 ∈ ( ( 𝑄 ‘ 0 ) [,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ) |
90 |
42 89
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 0 ) [,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ) |
91 |
15
|
a1i |
⊢ ( 𝜑 → 𝐸 = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) ) ) |
92 |
|
id |
⊢ ( 𝑥 = 𝑋 → 𝑥 = 𝑋 ) |
93 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝑍 ‘ 𝑥 ) = ( 𝑍 ‘ 𝑋 ) ) |
94 |
92 93
|
oveq12d |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) = ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) |
95 |
94
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) = ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) |
96 |
14
|
a1i |
⊢ ( 𝜑 → 𝑍 = ( 𝑥 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
97 |
|
oveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝐵 − 𝑥 ) = ( 𝐵 − 𝑋 ) ) |
98 |
97
|
oveq1d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝐵 − 𝑥 ) / 𝑇 ) = ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) |
99 |
98
|
fveq2d |
⊢ ( 𝑥 = 𝑋 → ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) = ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ) |
100 |
99
|
oveq1d |
⊢ ( 𝑥 = 𝑋 → ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
101 |
100
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
102 |
31
|
flcld |
⊢ ( 𝜑 → ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) |
103 |
102
|
zred |
⊢ ( 𝜑 → ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℝ ) |
104 |
103 26
|
remulcld |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ∈ ℝ ) |
105 |
96 101 13 104
|
fvmptd |
⊢ ( 𝜑 → ( 𝑍 ‘ 𝑋 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
106 |
105 104
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑍 ‘ 𝑋 ) ∈ ℝ ) |
107 |
13 106
|
readdcld |
⊢ ( 𝜑 → ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ) |
108 |
91 95 13 107
|
fvmptd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) |
109 |
105
|
oveq2d |
⊢ ( 𝜑 → ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) = ( 𝑋 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
110 |
108 109
|
eqtrd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
111 |
110
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( ( 𝑋 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) − 𝑇 ) ) |
112 |
13
|
recnd |
⊢ ( 𝜑 → 𝑋 ∈ ℂ ) |
113 |
104
|
recnd |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ∈ ℂ ) |
114 |
26
|
recnd |
⊢ ( 𝜑 → 𝑇 ∈ ℂ ) |
115 |
112 113 114
|
addsubassd |
⊢ ( 𝜑 → ( ( 𝑋 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) − 𝑇 ) = ( 𝑋 + ( ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) − 𝑇 ) ) ) |
116 |
102
|
zcnd |
⊢ ( 𝜑 → ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℂ ) |
117 |
116 114
|
mulsubfacd |
⊢ ( 𝜑 → ( ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) − 𝑇 ) = ( ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) − 1 ) · 𝑇 ) ) |
118 |
117
|
oveq2d |
⊢ ( 𝜑 → ( 𝑋 + ( ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) − 𝑇 ) ) = ( 𝑋 + ( ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) − 1 ) · 𝑇 ) ) ) |
119 |
111 115 118
|
3eqtrd |
⊢ ( 𝜑 → ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) − 1 ) · 𝑇 ) ) ) |
120 |
119
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) − 1 ) · 𝑇 ) ) ) |
121 |
|
oveq1 |
⊢ ( 𝑘 = ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) − 1 ) → ( 𝑘 · 𝑇 ) = ( ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) − 1 ) · 𝑇 ) ) |
122 |
121
|
oveq2d |
⊢ ( 𝑘 = ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) − 1 ) → ( 𝑋 + ( 𝑘 · 𝑇 ) ) = ( 𝑋 + ( ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) − 1 ) · 𝑇 ) ) ) |
123 |
122
|
eqeq2d |
⊢ ( 𝑘 = ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) − 1 ) → ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ↔ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) − 1 ) · 𝑇 ) ) ) ) |
124 |
123
|
anbi2d |
⊢ ( 𝑘 = ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) − 1 ) → ( ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 0 ) [,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ↔ ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 0 ) [,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) − 1 ) · 𝑇 ) ) ) ) ) |
125 |
124
|
rspcev |
⊢ ( ( ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) − 1 ) ∈ ℤ ∧ ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 0 ) [,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) − 1 ) · 𝑇 ) ) ) ) → ∃ 𝑘 ∈ ℤ ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 0 ) [,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) |
126 |
35 90 120 125
|
syl12anc |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ∃ 𝑘 ∈ ℤ ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 0 ) [,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) |
127 |
72 74
|
oveq12d |
⊢ ( 𝑖 = 0 → ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑄 ‘ 0 ) [,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ) |
128 |
127
|
eleq2d |
⊢ ( 𝑖 = 0 → ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 0 ) [,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ) ) |
129 |
128
|
anbi1d |
⊢ ( 𝑖 = 0 → ( ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ↔ ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 0 ) [,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) ) |
130 |
129
|
rexbidv |
⊢ ( 𝑖 = 0 → ( ∃ 𝑘 ∈ ℤ ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ↔ ∃ 𝑘 ∈ ℤ ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 0 ) [,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) ) |
131 |
130
|
rspcev |
⊢ ( ( 0 ∈ ( 0 ..^ 𝑀 ) ∧ ∃ 𝑘 ∈ ℤ ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 0 ) [,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) |
132 |
23 126 131
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) |
133 |
|
ovex |
⊢ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ V |
134 |
|
eleq1 |
⊢ ( 𝑦 = ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) → ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
135 |
|
eqeq1 |
⊢ ( 𝑦 = ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) → ( 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ↔ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) |
136 |
134 135
|
anbi12d |
⊢ ( 𝑦 = ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) → ( ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ↔ ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) ) |
137 |
136
|
2rexbidv |
⊢ ( 𝑦 = ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ↔ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) ) |
138 |
137
|
anbi2d |
⊢ ( 𝑦 = ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) → ( ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) ↔ ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) ) ) |
139 |
138
|
imbi1d |
⊢ ( 𝑦 = ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) → ( ( ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) ↔ ( ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) ) ) |
140 |
|
simpr |
⊢ ( ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) |
141 |
|
nfv |
⊢ Ⅎ 𝑖 𝜑 |
142 |
|
nfre1 |
⊢ Ⅎ 𝑖 ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) |
143 |
141 142
|
nfan |
⊢ Ⅎ 𝑖 ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) |
144 |
|
nfv |
⊢ Ⅎ 𝑘 𝜑 |
145 |
|
nfcv |
⊢ Ⅎ 𝑘 ( 0 ..^ 𝑀 ) |
146 |
|
nfre1 |
⊢ Ⅎ 𝑘 ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) |
147 |
145 146
|
nfrex |
⊢ Ⅎ 𝑘 ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) |
148 |
144 147
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) |
149 |
|
simp1 |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑘 ∈ ℤ ) ∧ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → 𝜑 ) |
150 |
|
simp2l |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑘 ∈ ℤ ) ∧ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → 𝑖 ∈ ( 0 ..^ 𝑀 ) ) |
151 |
|
simp3l |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑘 ∈ ℤ ) ∧ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
152 |
149 150 151
|
jca31 |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑘 ∈ ℤ ) ∧ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
153 |
|
simp2r |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑘 ∈ ℤ ) ∧ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → 𝑘 ∈ ℤ ) |
154 |
|
simp3r |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑘 ∈ ℤ ) ∧ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) |
155 |
16
|
biimpi |
⊢ ( 𝜒 → ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑘 ∈ ℤ ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) |
156 |
155
|
simplld |
⊢ ( 𝜒 → ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
157 |
156
|
simplld |
⊢ ( 𝜒 → 𝜑 ) |
158 |
|
frel |
⊢ ( 𝐹 : 𝐷 ⟶ ℝ → Rel 𝐹 ) |
159 |
157 8 158
|
3syl |
⊢ ( 𝜒 → Rel 𝐹 ) |
160 |
|
resindm |
⊢ ( Rel 𝐹 → ( 𝐹 ↾ ( ( 𝑋 (,) +∞ ) ∩ dom 𝐹 ) ) = ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) ) |
161 |
160
|
eqcomd |
⊢ ( Rel 𝐹 → ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) = ( 𝐹 ↾ ( ( 𝑋 (,) +∞ ) ∩ dom 𝐹 ) ) ) |
162 |
159 161
|
syl |
⊢ ( 𝜒 → ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) = ( 𝐹 ↾ ( ( 𝑋 (,) +∞ ) ∩ dom 𝐹 ) ) ) |
163 |
|
fdm |
⊢ ( 𝐹 : 𝐷 ⟶ ℝ → dom 𝐹 = 𝐷 ) |
164 |
157 8 163
|
3syl |
⊢ ( 𝜒 → dom 𝐹 = 𝐷 ) |
165 |
164
|
ineq2d |
⊢ ( 𝜒 → ( ( 𝑋 (,) +∞ ) ∩ dom 𝐹 ) = ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ) |
166 |
165
|
reseq2d |
⊢ ( 𝜒 → ( 𝐹 ↾ ( ( 𝑋 (,) +∞ ) ∩ dom 𝐹 ) ) = ( 𝐹 ↾ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ) ) |
167 |
162 166
|
eqtrd |
⊢ ( 𝜒 → ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) = ( 𝐹 ↾ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ) ) |
168 |
167
|
oveq1d |
⊢ ( 𝜒 → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) = ( ( 𝐹 ↾ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ) limℂ 𝑋 ) ) |
169 |
157 8
|
syl |
⊢ ( 𝜒 → 𝐹 : 𝐷 ⟶ ℝ ) |
170 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
171 |
170
|
a1i |
⊢ ( 𝜒 → ℝ ⊆ ℂ ) |
172 |
169 171
|
fssd |
⊢ ( 𝜒 → 𝐹 : 𝐷 ⟶ ℂ ) |
173 |
|
inss2 |
⊢ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ⊆ 𝐷 |
174 |
173
|
a1i |
⊢ ( 𝜒 → ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ⊆ 𝐷 ) |
175 |
172 174
|
fssresd |
⊢ ( 𝜒 → ( 𝐹 ↾ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ) : ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ⟶ ℂ ) |
176 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
177 |
176
|
a1i |
⊢ ( 𝜒 → +∞ ∈ ℝ* ) |
178 |
156
|
simplrd |
⊢ ( 𝜒 → 𝑖 ∈ ( 0 ..^ 𝑀 ) ) |
179 |
48
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
180 |
|
fzofzp1 |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
181 |
180
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
182 |
179 181
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
183 |
157 178 182
|
syl2anc |
⊢ ( 𝜒 → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
184 |
155
|
simplrd |
⊢ ( 𝜒 → 𝑘 ∈ ℤ ) |
185 |
184
|
zred |
⊢ ( 𝜒 → 𝑘 ∈ ℝ ) |
186 |
157 26
|
syl |
⊢ ( 𝜒 → 𝑇 ∈ ℝ ) |
187 |
185 186
|
remulcld |
⊢ ( 𝜒 → ( 𝑘 · 𝑇 ) ∈ ℝ ) |
188 |
183 187
|
resubcld |
⊢ ( 𝜒 → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ∈ ℝ ) |
189 |
188
|
rexrd |
⊢ ( 𝜒 → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ∈ ℝ* ) |
190 |
188
|
ltpnfd |
⊢ ( 𝜒 → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) < +∞ ) |
191 |
189 177 190
|
xrltled |
⊢ ( 𝜒 → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ≤ +∞ ) |
192 |
|
iooss2 |
⊢ ( ( +∞ ∈ ℝ* ∧ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ≤ +∞ ) → ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ⊆ ( 𝑋 (,) +∞ ) ) |
193 |
177 191 192
|
syl2anc |
⊢ ( 𝜒 → ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ⊆ ( 𝑋 (,) +∞ ) ) |
194 |
184
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑘 ∈ ℤ ) |
195 |
194
|
zcnd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑘 ∈ ℂ ) |
196 |
186
|
recnd |
⊢ ( 𝜒 → 𝑇 ∈ ℂ ) |
197 |
196
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑇 ∈ ℂ ) |
198 |
195 197
|
mulneg1d |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( - 𝑘 · 𝑇 ) = - ( 𝑘 · 𝑇 ) ) |
199 |
198
|
oveq2d |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( - 𝑘 · 𝑇 ) ) = ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + - ( 𝑘 · 𝑇 ) ) ) |
200 |
|
elioore |
⊢ ( 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) → 𝑤 ∈ ℝ ) |
201 |
200
|
recnd |
⊢ ( 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) → 𝑤 ∈ ℂ ) |
202 |
201
|
adantl |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑤 ∈ ℂ ) |
203 |
195 197
|
mulcld |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑘 · 𝑇 ) ∈ ℂ ) |
204 |
202 203
|
addcld |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ ℂ ) |
205 |
204 203
|
negsubd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + - ( 𝑘 · 𝑇 ) ) = ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) − ( 𝑘 · 𝑇 ) ) ) |
206 |
202 203
|
pncand |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) − ( 𝑘 · 𝑇 ) ) = 𝑤 ) |
207 |
199 205 206
|
3eqtrrd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑤 = ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( - 𝑘 · 𝑇 ) ) ) |
208 |
157
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝜑 ) |
209 |
156
|
simpld |
⊢ ( 𝜒 → ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ) |
210 |
|
cncff |
⊢ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ ) |
211 |
|
fdm |
⊢ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) : ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⟶ ℂ → dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
212 |
11 210 211
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
213 |
|
ssdmres |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ dom 𝐹 ↔ dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
214 |
212 213
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ dom 𝐹 ) |
215 |
8 163
|
syl |
⊢ ( 𝜑 → dom 𝐹 = 𝐷 ) |
216 |
215
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → dom 𝐹 = 𝐷 ) |
217 |
214 216
|
sseqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ 𝐷 ) |
218 |
209 217
|
syl |
⊢ ( 𝜒 → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ 𝐷 ) |
219 |
218
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ 𝐷 ) |
220 |
|
elfzofz |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
221 |
220
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
222 |
179 221
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
223 |
157 178 222
|
syl2anc |
⊢ ( 𝜒 → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
224 |
223
|
rexrd |
⊢ ( 𝜒 → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
225 |
224
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
226 |
183
|
rexrd |
⊢ ( 𝜒 → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
227 |
226
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
228 |
200
|
adantl |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑤 ∈ ℝ ) |
229 |
194
|
zred |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑘 ∈ ℝ ) |
230 |
208 26
|
syl |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑇 ∈ ℝ ) |
231 |
229 230
|
remulcld |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑘 · 𝑇 ) ∈ ℝ ) |
232 |
228 231
|
readdcld |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ ℝ ) |
233 |
223
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
234 |
157 13
|
syl |
⊢ ( 𝜒 → 𝑋 ∈ ℝ ) |
235 |
234 187
|
readdcld |
⊢ ( 𝜒 → ( 𝑋 + ( 𝑘 · 𝑇 ) ) ∈ ℝ ) |
236 |
235
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑋 + ( 𝑘 · 𝑇 ) ) ∈ ℝ ) |
237 |
16
|
simprbi |
⊢ ( 𝜒 → 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) |
238 |
237
|
eqcomd |
⊢ ( 𝜒 → ( 𝑋 + ( 𝑘 · 𝑇 ) ) = 𝑦 ) |
239 |
156
|
simprd |
⊢ ( 𝜒 → 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
240 |
238 239
|
eqeltrd |
⊢ ( 𝜒 → ( 𝑋 + ( 𝑘 · 𝑇 ) ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
241 |
|
icogelb |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ ( 𝑋 + ( 𝑘 · 𝑇 ) ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) |
242 |
224 226 240 241
|
syl3anc |
⊢ ( 𝜒 → ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) |
243 |
242
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ≤ ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) |
244 |
208 13
|
syl |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑋 ∈ ℝ ) |
245 |
244
|
rexrd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑋 ∈ ℝ* ) |
246 |
183
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
247 |
246 231
|
resubcld |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ∈ ℝ ) |
248 |
247
|
rexrd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ∈ ℝ* ) |
249 |
|
simpr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) |
250 |
|
ioogtlb |
⊢ ( ( 𝑋 ∈ ℝ* ∧ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ∈ ℝ* ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑋 < 𝑤 ) |
251 |
245 248 249 250
|
syl3anc |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑋 < 𝑤 ) |
252 |
244 228 231 251
|
ltadd1dd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑋 + ( 𝑘 · 𝑇 ) ) < ( 𝑤 + ( 𝑘 · 𝑇 ) ) ) |
253 |
233 236 232 243 252
|
lelttrd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑤 + ( 𝑘 · 𝑇 ) ) ) |
254 |
|
iooltub |
⊢ ( ( 𝑋 ∈ ℝ* ∧ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ∈ ℝ* ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑤 < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) |
255 |
245 248 249 254
|
syl3anc |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑤 < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) |
256 |
228 247 231 255
|
ltadd1dd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑤 + ( 𝑘 · 𝑇 ) ) < ( ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) + ( 𝑘 · 𝑇 ) ) ) |
257 |
183
|
recnd |
⊢ ( 𝜒 → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℂ ) |
258 |
187
|
recnd |
⊢ ( 𝜒 → ( 𝑘 · 𝑇 ) ∈ ℂ ) |
259 |
257 258
|
npcand |
⊢ ( 𝜒 → ( ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) + ( 𝑘 · 𝑇 ) ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
260 |
259
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) + ( 𝑘 · 𝑇 ) ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
261 |
256 260
|
breqtrd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑤 + ( 𝑘 · 𝑇 ) ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
262 |
225 227 232 253 261
|
eliood |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
263 |
219 262
|
sseldd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
264 |
194
|
znegcld |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → - 𝑘 ∈ ℤ ) |
265 |
|
ovex |
⊢ ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ V |
266 |
|
eleq1 |
⊢ ( 𝑥 = ( 𝑤 + ( 𝑘 · 𝑇 ) ) → ( 𝑥 ∈ 𝐷 ↔ ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ) |
267 |
266
|
3anbi2d |
⊢ ( 𝑥 = ( 𝑤 + ( 𝑘 · 𝑇 ) ) → ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ - 𝑘 ∈ ℤ ) ↔ ( 𝜑 ∧ ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ∧ - 𝑘 ∈ ℤ ) ) ) |
268 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝑤 + ( 𝑘 · 𝑇 ) ) → ( 𝑥 + ( - 𝑘 · 𝑇 ) ) = ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( - 𝑘 · 𝑇 ) ) ) |
269 |
268
|
eleq1d |
⊢ ( 𝑥 = ( 𝑤 + ( 𝑘 · 𝑇 ) ) → ( ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ∈ 𝐷 ↔ ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( - 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ) |
270 |
267 269
|
imbi12d |
⊢ ( 𝑥 = ( 𝑤 + ( 𝑘 · 𝑇 ) ) → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ - 𝑘 ∈ ℤ ) → ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ↔ ( ( 𝜑 ∧ ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ∧ - 𝑘 ∈ ℤ ) → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( - 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ) ) |
271 |
|
negex |
⊢ - 𝑘 ∈ V |
272 |
|
eleq1 |
⊢ ( 𝑗 = - 𝑘 → ( 𝑗 ∈ ℤ ↔ - 𝑘 ∈ ℤ ) ) |
273 |
272
|
3anbi3d |
⊢ ( 𝑗 = - 𝑘 → ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑗 ∈ ℤ ) ↔ ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ - 𝑘 ∈ ℤ ) ) ) |
274 |
|
oveq1 |
⊢ ( 𝑗 = - 𝑘 → ( 𝑗 · 𝑇 ) = ( - 𝑘 · 𝑇 ) ) |
275 |
274
|
oveq2d |
⊢ ( 𝑗 = - 𝑘 → ( 𝑥 + ( 𝑗 · 𝑇 ) ) = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ) |
276 |
275
|
eleq1d |
⊢ ( 𝑗 = - 𝑘 → ( ( 𝑥 + ( 𝑗 · 𝑇 ) ) ∈ 𝐷 ↔ ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ) |
277 |
273 276
|
imbi12d |
⊢ ( 𝑗 = - 𝑘 → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑗 ∈ ℤ ) → ( 𝑥 + ( 𝑗 · 𝑇 ) ) ∈ 𝐷 ) ↔ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ - 𝑘 ∈ ℤ ) → ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ) ) |
278 |
|
eleq1 |
⊢ ( 𝑘 = 𝑗 → ( 𝑘 ∈ ℤ ↔ 𝑗 ∈ ℤ ) ) |
279 |
278
|
3anbi3d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) ↔ ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑗 ∈ ℤ ) ) ) |
280 |
|
oveq1 |
⊢ ( 𝑘 = 𝑗 → ( 𝑘 · 𝑇 ) = ( 𝑗 · 𝑇 ) ) |
281 |
280
|
oveq2d |
⊢ ( 𝑘 = 𝑗 → ( 𝑥 + ( 𝑘 · 𝑇 ) ) = ( 𝑥 + ( 𝑗 · 𝑇 ) ) ) |
282 |
281
|
eleq1d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ↔ ( 𝑥 + ( 𝑗 · 𝑇 ) ) ∈ 𝐷 ) ) |
283 |
279 282
|
imbi12d |
⊢ ( 𝑘 = 𝑗 → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ↔ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑗 ∈ ℤ ) → ( 𝑥 + ( 𝑗 · 𝑇 ) ) ∈ 𝐷 ) ) ) |
284 |
283 9
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑗 ∈ ℤ ) → ( 𝑥 + ( 𝑗 · 𝑇 ) ) ∈ 𝐷 ) |
285 |
271 277 284
|
vtocl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ - 𝑘 ∈ ℤ ) → ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
286 |
265 270 285
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ∧ - 𝑘 ∈ ℤ ) → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( - 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
287 |
208 263 264 286
|
syl3anc |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( - 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
288 |
207 287
|
eqeltrd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑤 ∈ 𝐷 ) |
289 |
288
|
ralrimiva |
⊢ ( 𝜒 → ∀ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑤 ∈ 𝐷 ) |
290 |
|
dfss3 |
⊢ ( ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ⊆ 𝐷 ↔ ∀ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑤 ∈ 𝐷 ) |
291 |
289 290
|
sylibr |
⊢ ( 𝜒 → ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ⊆ 𝐷 ) |
292 |
193 291
|
ssind |
⊢ ( 𝜒 → ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ⊆ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ) |
293 |
|
ioosscn |
⊢ ( 𝑋 (,) +∞ ) ⊆ ℂ |
294 |
|
ssinss1 |
⊢ ( ( 𝑋 (,) +∞ ) ⊆ ℂ → ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ⊆ ℂ ) |
295 |
293 294
|
mp1i |
⊢ ( 𝜒 → ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ⊆ ℂ ) |
296 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
297 |
|
eqid |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) = ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
298 |
234
|
rexrd |
⊢ ( 𝜒 → 𝑋 ∈ ℝ* ) |
299 |
234
|
leidd |
⊢ ( 𝜒 → 𝑋 ≤ 𝑋 ) |
300 |
237
|
oveq1d |
⊢ ( 𝜒 → ( 𝑦 − ( 𝑘 · 𝑇 ) ) = ( ( 𝑋 + ( 𝑘 · 𝑇 ) ) − ( 𝑘 · 𝑇 ) ) ) |
301 |
234
|
recnd |
⊢ ( 𝜒 → 𝑋 ∈ ℂ ) |
302 |
301 258
|
pncand |
⊢ ( 𝜒 → ( ( 𝑋 + ( 𝑘 · 𝑇 ) ) − ( 𝑘 · 𝑇 ) ) = 𝑋 ) |
303 |
300 302
|
eqtr2d |
⊢ ( 𝜒 → 𝑋 = ( 𝑦 − ( 𝑘 · 𝑇 ) ) ) |
304 |
|
icossre |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) → ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ ) |
305 |
223 226 304
|
syl2anc |
⊢ ( 𝜒 → ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℝ ) |
306 |
305 239
|
sseldd |
⊢ ( 𝜒 → 𝑦 ∈ ℝ ) |
307 |
|
icoltub |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑦 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
308 |
224 226 239 307
|
syl3anc |
⊢ ( 𝜒 → 𝑦 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
309 |
306 183 187 308
|
ltsub1dd |
⊢ ( 𝜒 → ( 𝑦 − ( 𝑘 · 𝑇 ) ) < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) |
310 |
303 309
|
eqbrtrd |
⊢ ( 𝜒 → 𝑋 < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) |
311 |
298 189 298 299 310
|
elicod |
⊢ ( 𝜒 → 𝑋 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) |
312 |
|
snunioo1 |
⊢ ( ( 𝑋 ∈ ℝ* ∧ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ∈ ℝ* ∧ 𝑋 < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) → ( ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∪ { 𝑋 } ) = ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) |
313 |
298 189 310 312
|
syl3anc |
⊢ ( 𝜒 → ( ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∪ { 𝑋 } ) = ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) |
314 |
313
|
fveq2d |
⊢ ( 𝜒 → ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) ‘ ( ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∪ { 𝑋 } ) ) = ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) ‘ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ) |
315 |
296
|
cnfldtop |
⊢ ( TopOpen ‘ ℂfld ) ∈ Top |
316 |
|
ovex |
⊢ ( 𝑋 (,) +∞ ) ∈ V |
317 |
316
|
inex1 |
⊢ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∈ V |
318 |
|
snex |
⊢ { 𝑋 } ∈ V |
319 |
317 318
|
unex |
⊢ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ∈ V |
320 |
|
resttop |
⊢ ( ( ( TopOpen ‘ ℂfld ) ∈ Top ∧ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ∈ V ) → ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∈ Top ) |
321 |
315 319 320
|
mp2an |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∈ Top |
322 |
321
|
a1i |
⊢ ( 𝜒 → ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∈ Top ) |
323 |
|
retop |
⊢ ( topGen ‘ ran (,) ) ∈ Top |
324 |
323
|
a1i |
⊢ ( 𝜒 → ( topGen ‘ ran (,) ) ∈ Top ) |
325 |
319
|
a1i |
⊢ ( 𝜒 → ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ∈ V ) |
326 |
|
iooretop |
⊢ ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∈ ( topGen ‘ ran (,) ) |
327 |
326
|
a1i |
⊢ ( 𝜒 → ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∈ ( topGen ‘ ran (,) ) ) |
328 |
|
elrestr |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ Top ∧ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ∈ V ∧ ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∈ ( topGen ‘ ran (,) ) ) → ( ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∩ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∈ ( ( topGen ‘ ran (,) ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) |
329 |
324 325 327 328
|
syl3anc |
⊢ ( 𝜒 → ( ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∩ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∈ ( ( topGen ‘ ran (,) ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) |
330 |
|
mnfxr |
⊢ -∞ ∈ ℝ* |
331 |
330
|
a1i |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → -∞ ∈ ℝ* ) |
332 |
189
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ∈ ℝ* ) |
333 |
|
icossre |
⊢ ( ( 𝑋 ∈ ℝ ∧ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ∈ ℝ* ) → ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ⊆ ℝ ) |
334 |
234 189 333
|
syl2anc |
⊢ ( 𝜒 → ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ⊆ ℝ ) |
335 |
334
|
sselda |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑥 ∈ ℝ ) |
336 |
335
|
mnfltd |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → -∞ < 𝑥 ) |
337 |
298
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑋 ∈ ℝ* ) |
338 |
|
simpr |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) |
339 |
|
icoltub |
⊢ ( ( 𝑋 ∈ ℝ* ∧ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ∈ ℝ* ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑥 < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) |
340 |
337 332 338 339
|
syl3anc |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑥 < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) |
341 |
331 332 335 336 340
|
eliood |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑥 ∈ ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) |
342 |
|
vsnid |
⊢ 𝑥 ∈ { 𝑥 } |
343 |
342
|
a1i |
⊢ ( 𝑥 = 𝑋 → 𝑥 ∈ { 𝑥 } ) |
344 |
|
sneq |
⊢ ( 𝑥 = 𝑋 → { 𝑥 } = { 𝑋 } ) |
345 |
343 344
|
eleqtrd |
⊢ ( 𝑥 = 𝑋 → 𝑥 ∈ { 𝑋 } ) |
346 |
|
elun2 |
⊢ ( 𝑥 ∈ { 𝑋 } → 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
347 |
345 346
|
syl |
⊢ ( 𝑥 = 𝑋 → 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
348 |
347
|
adantl |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ 𝑥 = 𝑋 ) → 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
349 |
298
|
ad2antrr |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑋 ∈ ℝ* ) |
350 |
176
|
a1i |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ ¬ 𝑥 = 𝑋 ) → +∞ ∈ ℝ* ) |
351 |
335
|
adantr |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 ∈ ℝ ) |
352 |
234
|
ad2antrr |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑋 ∈ ℝ ) |
353 |
|
icogelb |
⊢ ( ( 𝑋 ∈ ℝ* ∧ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ∈ ℝ* ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑋 ≤ 𝑥 ) |
354 |
337 332 338 353
|
syl3anc |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑋 ≤ 𝑥 ) |
355 |
354
|
adantr |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑋 ≤ 𝑥 ) |
356 |
|
neqne |
⊢ ( ¬ 𝑥 = 𝑋 → 𝑥 ≠ 𝑋 ) |
357 |
356
|
adantl |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 ≠ 𝑋 ) |
358 |
352 351 355 357
|
leneltd |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑋 < 𝑥 ) |
359 |
351
|
ltpnfd |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 < +∞ ) |
360 |
349 350 351 358 359
|
eliood |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 ∈ ( 𝑋 (,) +∞ ) ) |
361 |
184
|
zcnd |
⊢ ( 𝜒 → 𝑘 ∈ ℂ ) |
362 |
361 196
|
mulneg1d |
⊢ ( 𝜒 → ( - 𝑘 · 𝑇 ) = - ( 𝑘 · 𝑇 ) ) |
363 |
362
|
oveq2d |
⊢ ( 𝜒 → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( - 𝑘 · 𝑇 ) ) = ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + - ( 𝑘 · 𝑇 ) ) ) |
364 |
363
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( - 𝑘 · 𝑇 ) ) = ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + - ( 𝑘 · 𝑇 ) ) ) |
365 |
|
ioosscn |
⊢ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ⊆ ℂ |
366 |
365
|
sseli |
⊢ ( 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) → 𝑤 ∈ ℂ ) |
367 |
366
|
adantl |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑤 ∈ ℂ ) |
368 |
258
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑘 · 𝑇 ) ∈ ℂ ) |
369 |
367 368
|
addcld |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ ℂ ) |
370 |
369 368
|
negsubd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + - ( 𝑘 · 𝑇 ) ) = ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) − ( 𝑘 · 𝑇 ) ) ) |
371 |
367 368
|
pncand |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) − ( 𝑘 · 𝑇 ) ) = 𝑤 ) |
372 |
364 370 371
|
3eqtrrd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑤 = ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( - 𝑘 · 𝑇 ) ) ) |
373 |
187
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑘 · 𝑇 ) ∈ ℝ ) |
374 |
228 373
|
readdcld |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ ℝ ) |
375 |
225 227 374 253 261
|
eliood |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
376 |
219 375
|
sseldd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
377 |
272
|
3anbi3d |
⊢ ( 𝑗 = - 𝑘 → ( ( 𝜑 ∧ ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ∧ 𝑗 ∈ ℤ ) ↔ ( 𝜑 ∧ ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ∧ - 𝑘 ∈ ℤ ) ) ) |
378 |
274
|
oveq2d |
⊢ ( 𝑗 = - 𝑘 → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( 𝑗 · 𝑇 ) ) = ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( - 𝑘 · 𝑇 ) ) ) |
379 |
378
|
eleq1d |
⊢ ( 𝑗 = - 𝑘 → ( ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( 𝑗 · 𝑇 ) ) ∈ 𝐷 ↔ ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( - 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ) |
380 |
377 379
|
imbi12d |
⊢ ( 𝑗 = - 𝑘 → ( ( ( 𝜑 ∧ ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ∧ 𝑗 ∈ ℤ ) → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( 𝑗 · 𝑇 ) ) ∈ 𝐷 ) ↔ ( ( 𝜑 ∧ ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ∧ - 𝑘 ∈ ℤ ) → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( - 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ) ) |
381 |
266
|
3anbi2d |
⊢ ( 𝑥 = ( 𝑤 + ( 𝑘 · 𝑇 ) ) → ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑗 ∈ ℤ ) ↔ ( 𝜑 ∧ ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ∧ 𝑗 ∈ ℤ ) ) ) |
382 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝑤 + ( 𝑘 · 𝑇 ) ) → ( 𝑥 + ( 𝑗 · 𝑇 ) ) = ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( 𝑗 · 𝑇 ) ) ) |
383 |
382
|
eleq1d |
⊢ ( 𝑥 = ( 𝑤 + ( 𝑘 · 𝑇 ) ) → ( ( 𝑥 + ( 𝑗 · 𝑇 ) ) ∈ 𝐷 ↔ ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( 𝑗 · 𝑇 ) ) ∈ 𝐷 ) ) |
384 |
381 383
|
imbi12d |
⊢ ( 𝑥 = ( 𝑤 + ( 𝑘 · 𝑇 ) ) → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑗 ∈ ℤ ) → ( 𝑥 + ( 𝑗 · 𝑇 ) ) ∈ 𝐷 ) ↔ ( ( 𝜑 ∧ ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ∧ 𝑗 ∈ ℤ ) → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( 𝑗 · 𝑇 ) ) ∈ 𝐷 ) ) ) |
385 |
265 384 284
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ∧ 𝑗 ∈ ℤ ) → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( 𝑗 · 𝑇 ) ) ∈ 𝐷 ) |
386 |
271 380 385
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ∧ - 𝑘 ∈ ℤ ) → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( - 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
387 |
208 376 264 386
|
syl3anc |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) + ( - 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
388 |
372 387
|
eqeltrd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑤 ∈ 𝐷 ) |
389 |
388
|
ralrimiva |
⊢ ( 𝜒 → ∀ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑤 ∈ 𝐷 ) |
390 |
389 290
|
sylibr |
⊢ ( 𝜒 → ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ⊆ 𝐷 ) |
391 |
390
|
ad2antrr |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ ¬ 𝑥 = 𝑋 ) → ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ⊆ 𝐷 ) |
392 |
189
|
ad2antrr |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ ¬ 𝑥 = 𝑋 ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ∈ ℝ* ) |
393 |
340
|
adantr |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) |
394 |
349 392 351 358 393
|
eliood |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) |
395 |
391 394
|
sseldd |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 ∈ 𝐷 ) |
396 |
360 395
|
elind |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 ∈ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ) |
397 |
|
elun1 |
⊢ ( 𝑥 ∈ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) → 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
398 |
396 397
|
syl |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
399 |
348 398
|
pm2.61dan |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
400 |
341 399
|
elind |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑥 ∈ ( ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∩ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) |
401 |
298
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∩ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → 𝑋 ∈ ℝ* ) |
402 |
189
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∩ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ∈ ℝ* ) |
403 |
|
elinel1 |
⊢ ( 𝑥 ∈ ( ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∩ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) → 𝑥 ∈ ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) |
404 |
|
elioore |
⊢ ( 𝑥 ∈ ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) → 𝑥 ∈ ℝ ) |
405 |
403 404
|
syl |
⊢ ( 𝑥 ∈ ( ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∩ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) → 𝑥 ∈ ℝ ) |
406 |
405
|
rexrd |
⊢ ( 𝑥 ∈ ( ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∩ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) → 𝑥 ∈ ℝ* ) |
407 |
406
|
adantl |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∩ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → 𝑥 ∈ ℝ* ) |
408 |
|
elinel2 |
⊢ ( 𝑥 ∈ ( ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∩ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) → 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
409 |
234
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑥 = 𝑋 ) → 𝑋 ∈ ℝ ) |
410 |
92
|
eqcomd |
⊢ ( 𝑥 = 𝑋 → 𝑋 = 𝑥 ) |
411 |
410
|
adantl |
⊢ ( ( 𝜒 ∧ 𝑥 = 𝑋 ) → 𝑋 = 𝑥 ) |
412 |
409 411
|
eqled |
⊢ ( ( 𝜒 ∧ 𝑥 = 𝑋 ) → 𝑋 ≤ 𝑥 ) |
413 |
412
|
adantlr |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∧ 𝑥 = 𝑋 ) → 𝑋 ≤ 𝑥 ) |
414 |
|
simpll |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝜒 ) |
415 |
|
simplr |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
416 |
|
id |
⊢ ( ¬ 𝑥 = 𝑋 → ¬ 𝑥 = 𝑋 ) |
417 |
|
velsn |
⊢ ( 𝑥 ∈ { 𝑋 } ↔ 𝑥 = 𝑋 ) |
418 |
416 417
|
sylnibr |
⊢ ( ¬ 𝑥 = 𝑋 → ¬ 𝑥 ∈ { 𝑋 } ) |
419 |
418
|
adantl |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∧ ¬ 𝑥 = 𝑋 ) → ¬ 𝑥 ∈ { 𝑋 } ) |
420 |
|
elunnel2 |
⊢ ( ( 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ∧ ¬ 𝑥 ∈ { 𝑋 } ) → 𝑥 ∈ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ) |
421 |
415 419 420
|
syl2anc |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 ∈ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ) |
422 |
|
elinel1 |
⊢ ( 𝑥 ∈ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) → 𝑥 ∈ ( 𝑋 (,) +∞ ) ) |
423 |
421 422
|
syl |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 ∈ ( 𝑋 (,) +∞ ) ) |
424 |
234
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) +∞ ) ) → 𝑋 ∈ ℝ ) |
425 |
|
elioore |
⊢ ( 𝑥 ∈ ( 𝑋 (,) +∞ ) → 𝑥 ∈ ℝ ) |
426 |
425
|
adantl |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) +∞ ) ) → 𝑥 ∈ ℝ ) |
427 |
298
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) +∞ ) ) → 𝑋 ∈ ℝ* ) |
428 |
176
|
a1i |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) +∞ ) ) → +∞ ∈ ℝ* ) |
429 |
|
simpr |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) +∞ ) ) → 𝑥 ∈ ( 𝑋 (,) +∞ ) ) |
430 |
|
ioogtlb |
⊢ ( ( 𝑋 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑥 ∈ ( 𝑋 (,) +∞ ) ) → 𝑋 < 𝑥 ) |
431 |
427 428 429 430
|
syl3anc |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) +∞ ) ) → 𝑋 < 𝑥 ) |
432 |
424 426 431
|
ltled |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) +∞ ) ) → 𝑋 ≤ 𝑥 ) |
433 |
414 423 432
|
syl2anc |
⊢ ( ( ( 𝜒 ∧ 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑋 ≤ 𝑥 ) |
434 |
413 433
|
pm2.61dan |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) → 𝑋 ≤ 𝑥 ) |
435 |
408 434
|
sylan2 |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∩ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → 𝑋 ≤ 𝑥 ) |
436 |
330
|
a1i |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → -∞ ∈ ℝ* ) |
437 |
189
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ∈ ℝ* ) |
438 |
|
simpr |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑥 ∈ ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) |
439 |
|
iooltub |
⊢ ( ( -∞ ∈ ℝ* ∧ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ∈ ℝ* ∧ 𝑥 ∈ ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑥 < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) |
440 |
436 437 438 439
|
syl3anc |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑥 < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) |
441 |
403 440
|
sylan2 |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∩ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → 𝑥 < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) |
442 |
401 402 407 435 441
|
elicod |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∩ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) |
443 |
400 442
|
impbida |
⊢ ( 𝜒 → ( 𝑥 ∈ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ↔ 𝑥 ∈ ( ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∩ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) ) |
444 |
443
|
eqrdv |
⊢ ( 𝜒 → ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) = ( ( -∞ (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∩ ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) |
445 |
|
ioossre |
⊢ ( 𝑋 (,) +∞ ) ⊆ ℝ |
446 |
|
ssinss1 |
⊢ ( ( 𝑋 (,) +∞ ) ⊆ ℝ → ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ⊆ ℝ ) |
447 |
445 446
|
mp1i |
⊢ ( 𝜒 → ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ⊆ ℝ ) |
448 |
234
|
snssd |
⊢ ( 𝜒 → { 𝑋 } ⊆ ℝ ) |
449 |
447 448
|
unssd |
⊢ ( 𝜒 → ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ⊆ ℝ ) |
450 |
|
eqid |
⊢ ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) ) |
451 |
296 450
|
rerest |
⊢ ( ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ⊆ ℝ → ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) = ( ( topGen ‘ ran (,) ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) |
452 |
449 451
|
syl |
⊢ ( 𝜒 → ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) = ( ( topGen ‘ ran (,) ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) |
453 |
329 444 452
|
3eltr4d |
⊢ ( 𝜒 → ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∈ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) |
454 |
|
isopn3i |
⊢ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∈ Top ∧ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∈ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) ‘ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) = ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) |
455 |
322 453 454
|
syl2anc |
⊢ ( 𝜒 → ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) ‘ ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) = ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) |
456 |
314 455
|
eqtr2d |
⊢ ( 𝜒 → ( 𝑋 [,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) = ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) ‘ ( ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∪ { 𝑋 } ) ) ) |
457 |
311 456
|
eleqtrd |
⊢ ( 𝜒 → 𝑋 ∈ ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) ‘ ( ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∪ { 𝑋 } ) ) ) |
458 |
175 292 295 296 297 457
|
limcres |
⊢ ( 𝜒 → ( ( ( 𝐹 ↾ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ) ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) = ( ( 𝐹 ↾ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ) limℂ 𝑋 ) ) |
459 |
292
|
resabs1d |
⊢ ( 𝜒 → ( ( 𝐹 ↾ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ) ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) = ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ) |
460 |
459
|
oveq1d |
⊢ ( 𝜒 → ( ( ( 𝐹 ↾ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ) ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) = ( ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) ) |
461 |
170
|
a1i |
⊢ ( 𝜑 → ℝ ⊆ ℂ ) |
462 |
8 461
|
fssd |
⊢ ( 𝜑 → 𝐹 : 𝐷 ⟶ ℂ ) |
463 |
215
|
feq2d |
⊢ ( 𝜑 → ( 𝐹 : dom 𝐹 ⟶ ℂ ↔ 𝐹 : 𝐷 ⟶ ℂ ) ) |
464 |
462 463
|
mpbird |
⊢ ( 𝜑 → 𝐹 : dom 𝐹 ⟶ ℂ ) |
465 |
157 464
|
syl |
⊢ ( 𝜒 → 𝐹 : dom 𝐹 ⟶ ℂ ) |
466 |
465
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) ) → 𝐹 : dom 𝐹 ⟶ ℂ ) |
467 |
365
|
a1i |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) ) → ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ⊆ ℂ ) |
468 |
390 164
|
sseqtrrd |
⊢ ( 𝜒 → ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ⊆ dom 𝐹 ) |
469 |
468
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) ) → ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ⊆ dom 𝐹 ) |
470 |
258
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) ) → ( 𝑘 · 𝑇 ) ∈ ℂ ) |
471 |
|
eqid |
⊢ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } = { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } |
472 |
|
eqeq1 |
⊢ ( 𝑧 = 𝑤 → ( 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) ↔ 𝑤 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) ) |
473 |
472
|
rexbidv |
⊢ ( 𝑧 = 𝑤 → ( ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) ↔ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑤 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) ) |
474 |
473
|
elrab |
⊢ ( 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } ↔ ( 𝑤 ∈ ℂ ∧ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑤 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) ) |
475 |
474
|
simprbi |
⊢ ( 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } → ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑤 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) |
476 |
475
|
adantl |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } ) → ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑤 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) |
477 |
|
nfv |
⊢ Ⅎ 𝑥 𝜒 |
478 |
|
nfre1 |
⊢ Ⅎ 𝑥 ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) |
479 |
|
nfcv |
⊢ Ⅎ 𝑥 ℂ |
480 |
478 479
|
nfrabw |
⊢ Ⅎ 𝑥 { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } |
481 |
480
|
nfcri |
⊢ Ⅎ 𝑥 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } |
482 |
477 481
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜒 ∧ 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } ) |
483 |
|
nfv |
⊢ Ⅎ 𝑥 𝑤 ∈ 𝐷 |
484 |
|
simp3 |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∧ 𝑤 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) → 𝑤 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) |
485 |
|
eleq1 |
⊢ ( 𝑤 = 𝑥 → ( 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ↔ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ) |
486 |
485
|
anbi2d |
⊢ ( 𝑤 = 𝑥 → ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ↔ ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ) ) |
487 |
|
oveq1 |
⊢ ( 𝑤 = 𝑥 → ( 𝑤 + ( 𝑘 · 𝑇 ) ) = ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) |
488 |
487
|
eleq1d |
⊢ ( 𝑤 = 𝑥 → ( ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ↔ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ) |
489 |
486 488
|
imbi12d |
⊢ ( 𝑤 = 𝑥 → ( ( ( 𝜒 ∧ 𝑤 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑤 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ↔ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ) ) |
490 |
489 263
|
chvarvv |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
491 |
490
|
3adant3 |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∧ 𝑤 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) → ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
492 |
484 491
|
eqeltrd |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ∧ 𝑤 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) → 𝑤 ∈ 𝐷 ) |
493 |
492
|
3exp |
⊢ ( 𝜒 → ( 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) → ( 𝑤 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) → 𝑤 ∈ 𝐷 ) ) ) |
494 |
493
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } ) → ( 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) → ( 𝑤 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) → 𝑤 ∈ 𝐷 ) ) ) |
495 |
482 483 494
|
rexlimd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } ) → ( ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑤 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) → 𝑤 ∈ 𝐷 ) ) |
496 |
476 495
|
mpd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } ) → 𝑤 ∈ 𝐷 ) |
497 |
496
|
ralrimiva |
⊢ ( 𝜒 → ∀ 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } 𝑤 ∈ 𝐷 ) |
498 |
|
dfss3 |
⊢ ( { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } ⊆ 𝐷 ↔ ∀ 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } 𝑤 ∈ 𝐷 ) |
499 |
497 498
|
sylibr |
⊢ ( 𝜒 → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } ⊆ 𝐷 ) |
500 |
499 164
|
sseqtrrd |
⊢ ( 𝜒 → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } ⊆ dom 𝐹 ) |
501 |
500
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) ) → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } ⊆ dom 𝐹 ) |
502 |
157
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝜑 ) |
503 |
390
|
sselda |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑥 ∈ 𝐷 ) |
504 |
184
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → 𝑘 ∈ ℤ ) |
505 |
502 503 504 10
|
syl3anc |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
506 |
505
|
adantlr |
⊢ ( ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) ) ∧ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
507 |
|
simpr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) ) → 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) ) |
508 |
466 467 469 470 471 501 506 507
|
limcperiod |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) ) → 𝑤 ∈ ( ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } ) limℂ ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) |
509 |
259
|
eqcomd |
⊢ ( 𝜒 → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) + ( 𝑘 · 𝑇 ) ) ) |
510 |
237 509
|
oveq12d |
⊢ ( 𝜒 → ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑋 + ( 𝑘 · 𝑇 ) ) (,) ( ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) + ( 𝑘 · 𝑇 ) ) ) ) |
511 |
234 188 187
|
iooshift |
⊢ ( 𝜒 → ( ( 𝑋 + ( 𝑘 · 𝑇 ) ) (,) ( ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) + ( 𝑘 · 𝑇 ) ) ) = { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } ) |
512 |
510 511
|
eqtr2d |
⊢ ( 𝜒 → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } = ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
513 |
512
|
reseq2d |
⊢ ( 𝜒 → ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } ) = ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
514 |
513 238
|
oveq12d |
⊢ ( 𝜒 → ( ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } ) limℂ ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) = ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) |
515 |
514
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) ) → ( ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) 𝑧 = ( 𝑥 + ( 𝑘 · 𝑇 ) ) } ) limℂ ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) = ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) |
516 |
508 515
|
eleqtrd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) ) → 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) |
517 |
465
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) → 𝐹 : dom 𝐹 ⟶ ℂ ) |
518 |
|
ioosscn |
⊢ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ |
519 |
518
|
a1i |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) → ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ℂ ) |
520 |
|
icogelb |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ≤ 𝑦 ) |
521 |
224 226 239 520
|
syl3anc |
⊢ ( 𝜒 → ( 𝑄 ‘ 𝑖 ) ≤ 𝑦 ) |
522 |
|
iooss1 |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ 𝑖 ) ≤ 𝑦 ) → ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
523 |
224 521 522
|
syl2anc |
⊢ ( 𝜒 → ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
524 |
523 218
|
sstrd |
⊢ ( 𝜒 → ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ 𝐷 ) |
525 |
524 164
|
sseqtrrd |
⊢ ( 𝜒 → ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ dom 𝐹 ) |
526 |
525
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) → ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ dom 𝐹 ) |
527 |
361
|
negcld |
⊢ ( 𝜒 → - 𝑘 ∈ ℂ ) |
528 |
527 196
|
mulcld |
⊢ ( 𝜒 → ( - 𝑘 · 𝑇 ) ∈ ℂ ) |
529 |
528
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) → ( - 𝑘 · 𝑇 ) ∈ ℂ ) |
530 |
|
eqid |
⊢ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } = { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } |
531 |
|
eqeq1 |
⊢ ( 𝑧 = 𝑤 → ( 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ↔ 𝑤 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ) ) |
532 |
531
|
rexbidv |
⊢ ( 𝑧 = 𝑤 → ( ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ↔ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ) ) |
533 |
532
|
elrab |
⊢ ( 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } ↔ ( 𝑤 ∈ ℂ ∧ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ) ) |
534 |
533
|
simprbi |
⊢ ( 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } → ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ) |
535 |
534
|
adantl |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } ) → ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ) |
536 |
|
nfre1 |
⊢ Ⅎ 𝑥 ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) |
537 |
536 479
|
nfrabw |
⊢ Ⅎ 𝑥 { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } |
538 |
537
|
nfcri |
⊢ Ⅎ 𝑥 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } |
539 |
477 538
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜒 ∧ 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } ) |
540 |
|
simp3 |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑤 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ) → 𝑤 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ) |
541 |
157
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝜑 ) |
542 |
524
|
sselda |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ 𝐷 ) |
543 |
184
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑘 ∈ ℤ ) |
544 |
543
|
znegcld |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → - 𝑘 ∈ ℤ ) |
545 |
541 542 544 285
|
syl3anc |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
546 |
545
|
3adant3 |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑤 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ) → ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
547 |
540 546
|
eqeltrd |
⊢ ( ( 𝜒 ∧ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑤 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ) → 𝑤 ∈ 𝐷 ) |
548 |
547
|
3exp |
⊢ ( 𝜒 → ( 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑤 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) → 𝑤 ∈ 𝐷 ) ) ) |
549 |
548
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } ) → ( 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝑤 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) → 𝑤 ∈ 𝐷 ) ) ) |
550 |
539 483 549
|
rexlimd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } ) → ( ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑤 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) → 𝑤 ∈ 𝐷 ) ) |
551 |
535 550
|
mpd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } ) → 𝑤 ∈ 𝐷 ) |
552 |
551
|
ralrimiva |
⊢ ( 𝜒 → ∀ 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } 𝑤 ∈ 𝐷 ) |
553 |
|
dfss3 |
⊢ ( { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } ⊆ 𝐷 ↔ ∀ 𝑤 ∈ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } 𝑤 ∈ 𝐷 ) |
554 |
552 553
|
sylibr |
⊢ ( 𝜒 → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } ⊆ 𝐷 ) |
555 |
554 164
|
sseqtrrd |
⊢ ( 𝜒 → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } ⊆ dom 𝐹 ) |
556 |
555
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } ⊆ dom 𝐹 ) |
557 |
157
|
ad2antrr |
⊢ ( ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) ∧ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝜑 ) |
558 |
542
|
adantlr |
⊢ ( ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) ∧ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑥 ∈ 𝐷 ) |
559 |
544
|
adantlr |
⊢ ( ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) ∧ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → - 𝑘 ∈ ℤ ) |
560 |
275
|
fveq2d |
⊢ ( 𝑗 = - 𝑘 → ( 𝐹 ‘ ( 𝑥 + ( 𝑗 · 𝑇 ) ) ) = ( 𝐹 ‘ ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ) ) |
561 |
560
|
eqeq1d |
⊢ ( 𝑗 = - 𝑘 → ( ( 𝐹 ‘ ( 𝑥 + ( 𝑗 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) ) |
562 |
273 561
|
imbi12d |
⊢ ( 𝑗 = - 𝑘 → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑗 ∈ ℤ ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑗 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) ↔ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ - 𝑘 ∈ ℤ ) → ( 𝐹 ‘ ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) |
563 |
281
|
fveq2d |
⊢ ( 𝑘 = 𝑗 → ( 𝐹 ‘ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ ( 𝑥 + ( 𝑗 · 𝑇 ) ) ) ) |
564 |
563
|
eqeq1d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝐹 ‘ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ ( 𝑥 + ( 𝑗 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) ) |
565 |
279 564
|
imbi12d |
⊢ ( 𝑘 = 𝑗 → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) ↔ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑗 ∈ ℤ ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑗 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) |
566 |
565 10
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑗 ∈ ℤ ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑗 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
567 |
271 562 566
|
vtocl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ - 𝑘 ∈ ℤ ) → ( 𝐹 ‘ ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
568 |
557 558 559 567
|
syl3anc |
⊢ ( ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) ∧ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ‘ ( 𝑥 + ( - 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
569 |
|
simpr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) → 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) |
570 |
517 519 526 529 530 556 568 569
|
limcperiod |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) → 𝑤 ∈ ( ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } ) limℂ ( 𝑦 + ( - 𝑘 · 𝑇 ) ) ) ) |
571 |
362
|
oveq2d |
⊢ ( 𝜒 → ( 𝑦 + ( - 𝑘 · 𝑇 ) ) = ( 𝑦 + - ( 𝑘 · 𝑇 ) ) ) |
572 |
306
|
recnd |
⊢ ( 𝜒 → 𝑦 ∈ ℂ ) |
573 |
572 258
|
negsubd |
⊢ ( 𝜒 → ( 𝑦 + - ( 𝑘 · 𝑇 ) ) = ( 𝑦 − ( 𝑘 · 𝑇 ) ) ) |
574 |
303
|
eqcomd |
⊢ ( 𝜒 → ( 𝑦 − ( 𝑘 · 𝑇 ) ) = 𝑋 ) |
575 |
571 573 574
|
3eqtrd |
⊢ ( 𝜒 → ( 𝑦 + ( - 𝑘 · 𝑇 ) ) = 𝑋 ) |
576 |
575
|
eqcomd |
⊢ ( 𝜒 → 𝑋 = ( 𝑦 + ( - 𝑘 · 𝑇 ) ) ) |
577 |
362
|
oveq2d |
⊢ ( 𝜒 → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + ( - 𝑘 · 𝑇 ) ) = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + - ( 𝑘 · 𝑇 ) ) ) |
578 |
257 258
|
negsubd |
⊢ ( 𝜒 → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + - ( 𝑘 · 𝑇 ) ) = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) |
579 |
577 578
|
eqtr2d |
⊢ ( 𝜒 → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + ( - 𝑘 · 𝑇 ) ) ) |
580 |
576 579
|
oveq12d |
⊢ ( 𝜒 → ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) = ( ( 𝑦 + ( - 𝑘 · 𝑇 ) ) (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + ( - 𝑘 · 𝑇 ) ) ) ) |
581 |
185
|
renegcld |
⊢ ( 𝜒 → - 𝑘 ∈ ℝ ) |
582 |
581 186
|
remulcld |
⊢ ( 𝜒 → ( - 𝑘 · 𝑇 ) ∈ ℝ ) |
583 |
306 183 582
|
iooshift |
⊢ ( 𝜒 → ( ( 𝑦 + ( - 𝑘 · 𝑇 ) ) (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) + ( - 𝑘 · 𝑇 ) ) ) = { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } ) |
584 |
580 583
|
eqtr2d |
⊢ ( 𝜒 → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } = ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) |
585 |
584
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } = ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) |
586 |
585
|
reseq2d |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) → ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } ) = ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) ) |
587 |
575
|
adantr |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) → ( 𝑦 + ( - 𝑘 · 𝑇 ) ) = 𝑋 ) |
588 |
586 587
|
oveq12d |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) → ( ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) 𝑧 = ( 𝑥 + ( - 𝑘 · 𝑇 ) ) } ) limℂ ( 𝑦 + ( - 𝑘 · 𝑇 ) ) ) = ( ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) ) |
589 |
570 588
|
eleqtrd |
⊢ ( ( 𝜒 ∧ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) → 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) ) |
590 |
516 589
|
impbida |
⊢ ( 𝜒 → ( 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) ↔ 𝑤 ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) ) |
591 |
590
|
eqrdv |
⊢ ( 𝜒 → ( ( 𝐹 ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) = ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) |
592 |
460 591
|
eqtrd |
⊢ ( 𝜒 → ( ( ( 𝐹 ↾ ( ( 𝑋 (,) +∞ ) ∩ 𝐷 ) ) ↾ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑘 · 𝑇 ) ) ) ) limℂ 𝑋 ) = ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) |
593 |
168 458 592
|
3eqtr2d |
⊢ ( 𝜒 → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) = ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) |
594 |
157 178 78
|
syl2anc |
⊢ ( 𝜒 → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
595 |
157 178 11
|
syl2anc |
⊢ ( 𝜒 → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
596 |
157 178 12
|
syl2anc |
⊢ ( 𝜒 → 𝑅 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ 𝑖 ) ) ) |
597 |
|
eqid |
⊢ if ( 𝑦 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑦 ) ) = if ( 𝑦 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑦 ) ) |
598 |
|
eqid |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( TopOpen ‘ ℂfld ) ↾t ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
599 |
223 183 594 595 596 306 183 308 523 597 598
|
fourierdlem32 |
⊢ ( 𝜒 → if ( 𝑦 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑦 ) ) ∈ ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) |
600 |
523
|
resabs1d |
⊢ ( 𝜒 → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
601 |
600
|
oveq1d |
⊢ ( 𝜒 → ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) = ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) |
602 |
599 601
|
eleqtrd |
⊢ ( 𝜒 → if ( 𝑦 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑦 ) ) ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ) |
603 |
|
ne0i |
⊢ ( if ( 𝑦 = ( 𝑄 ‘ 𝑖 ) , 𝑅 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ 𝑦 ) ) ∈ ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) → ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ≠ ∅ ) |
604 |
602 603
|
syl |
⊢ ( 𝜒 → ( ( 𝐹 ↾ ( 𝑦 (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ 𝑦 ) ≠ ∅ ) |
605 |
593 604
|
eqnetrd |
⊢ ( 𝜒 → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) |
606 |
16 605
|
sylbir |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑘 ∈ ℤ ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) |
607 |
152 153 154 606
|
syl21anc |
⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑘 ∈ ℤ ) ∧ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) |
608 |
607
|
3exp |
⊢ ( 𝜑 → ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑘 ∈ ℤ ) → ( ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) ) ) |
609 |
608
|
adantr |
⊢ ( ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑘 ∈ ℤ ) → ( ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) ) ) |
610 |
143 148 609
|
rexlim2d |
⊢ ( ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) ) |
611 |
140 610
|
mpd |
⊢ ( ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) |
612 |
133 139 611
|
vtocl |
⊢ ( ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) |
613 |
17 132 612
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) |
614 |
|
iocssre |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) → ( 𝐴 (,] 𝐵 ) ⊆ ℝ ) |
615 |
63 2 614
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 (,] 𝐵 ) ⊆ ℝ ) |
616 |
|
ovex |
⊢ ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ∈ V |
617 |
14
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ ℝ ∧ ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ∈ V ) → ( 𝑍 ‘ 𝑥 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) |
618 |
616 617
|
mpan2 |
⊢ ( 𝑥 ∈ ℝ → ( 𝑍 ‘ 𝑥 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) |
619 |
618
|
oveq2d |
⊢ ( 𝑥 ∈ ℝ → ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) = ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
620 |
619
|
mpteq2ia |
⊢ ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
621 |
15 620
|
eqtri |
⊢ 𝐸 = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
622 |
1 2 3 5 621
|
fourierdlem4 |
⊢ ( 𝜑 → 𝐸 : ℝ ⟶ ( 𝐴 (,] 𝐵 ) ) |
623 |
622 13
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) |
624 |
615 623
|
sseldd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
625 |
624
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
626 |
|
simpl |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) → 𝜑 ) |
627 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) |
628 |
|
ffn |
⊢ ( 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ → 𝑄 Fn ( 0 ... 𝑀 ) ) |
629 |
48 628
|
syl |
⊢ ( 𝜑 → 𝑄 Fn ( 0 ... 𝑀 ) ) |
630 |
629
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → 𝑄 Fn ( 0 ... 𝑀 ) ) |
631 |
|
fvelrnb |
⊢ ( 𝑄 Fn ( 0 ... 𝑀 ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ) |
632 |
630 631
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ) |
633 |
627 632
|
mpbid |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) |
634 |
|
1zzd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 1 ∈ ℤ ) |
635 |
|
elfzelz |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ∈ ℤ ) |
636 |
635
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑗 ∈ ℤ ) |
637 |
636
|
zred |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑗 ∈ ℝ ) |
638 |
|
elfzle1 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 0 ≤ 𝑗 ) |
639 |
638
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 0 ≤ 𝑗 ) |
640 |
|
id |
⊢ ( ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) → ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) |
641 |
640
|
eqcomd |
⊢ ( ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ 𝑗 ) ) |
642 |
641
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑗 = 0 ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ 𝑗 ) ) |
643 |
|
fveq2 |
⊢ ( 𝑗 = 0 → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ 0 ) ) |
644 |
643
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑗 = 0 ) → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ 0 ) ) |
645 |
45
|
simprld |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ) |
646 |
645
|
simpld |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) = 𝐴 ) |
647 |
646
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑗 = 0 ) → ( 𝑄 ‘ 0 ) = 𝐴 ) |
648 |
642 644 647
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑗 = 0 ) → ( 𝐸 ‘ 𝑋 ) = 𝐴 ) |
649 |
648
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑗 = 0 ) → ( 𝐸 ‘ 𝑋 ) = 𝐴 ) |
650 |
649
|
adantllr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑗 = 0 ) → ( 𝐸 ‘ 𝑋 ) = 𝐴 ) |
651 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐴 ∈ ℝ ) |
652 |
63
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐴 ∈ ℝ* ) |
653 |
2
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
654 |
653
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐵 ∈ ℝ* ) |
655 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) |
656 |
|
iocgtlb |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐴 < ( 𝐸 ‘ 𝑋 ) ) |
657 |
652 654 655 656
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐴 < ( 𝐸 ‘ 𝑋 ) ) |
658 |
651 657
|
gtned |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → ( 𝐸 ‘ 𝑋 ) ≠ 𝐴 ) |
659 |
658
|
neneqd |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → ¬ ( 𝐸 ‘ 𝑋 ) = 𝐴 ) |
660 |
659
|
ad3antrrr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑗 = 0 ) → ¬ ( 𝐸 ‘ 𝑋 ) = 𝐴 ) |
661 |
650 660
|
pm2.65da |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ¬ 𝑗 = 0 ) |
662 |
661
|
neqned |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑗 ≠ 0 ) |
663 |
637 639 662
|
ne0gt0d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 0 < 𝑗 ) |
664 |
|
0zd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 0 ∈ ℤ ) |
665 |
|
zltp1le |
⊢ ( ( 0 ∈ ℤ ∧ 𝑗 ∈ ℤ ) → ( 0 < 𝑗 ↔ ( 0 + 1 ) ≤ 𝑗 ) ) |
666 |
664 636 665
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 0 < 𝑗 ↔ ( 0 + 1 ) ≤ 𝑗 ) ) |
667 |
663 666
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 0 + 1 ) ≤ 𝑗 ) |
668 |
82 667
|
eqbrtrid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 1 ≤ 𝑗 ) |
669 |
|
eluz2 |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 1 ) ↔ ( 1 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 1 ≤ 𝑗 ) ) |
670 |
634 636 668 669
|
syl3anbrc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑗 ∈ ( ℤ≥ ‘ 1 ) ) |
671 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
672 |
670 671
|
eleqtrrdi |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑗 ∈ ℕ ) |
673 |
|
nnm1nn0 |
⊢ ( 𝑗 ∈ ℕ → ( 𝑗 − 1 ) ∈ ℕ0 ) |
674 |
672 673
|
syl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ∈ ℕ0 ) |
675 |
674 50
|
eleqtrdi |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ∈ ( ℤ≥ ‘ 0 ) ) |
676 |
19
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑀 ∈ ℤ ) |
677 |
|
peano2zm |
⊢ ( 𝑗 ∈ ℤ → ( 𝑗 − 1 ) ∈ ℤ ) |
678 |
635 677
|
syl |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 − 1 ) ∈ ℤ ) |
679 |
678
|
zred |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 − 1 ) ∈ ℝ ) |
680 |
635
|
zred |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ∈ ℝ ) |
681 |
|
elfzel2 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑀 ∈ ℤ ) |
682 |
681
|
zred |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑀 ∈ ℝ ) |
683 |
680
|
ltm1d |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 − 1 ) < 𝑗 ) |
684 |
|
elfzle2 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ≤ 𝑀 ) |
685 |
679 680 682 683 684
|
ltletrd |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 − 1 ) < 𝑀 ) |
686 |
685
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) < 𝑀 ) |
687 |
|
elfzo2 |
⊢ ( ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ↔ ( ( 𝑗 − 1 ) ∈ ( ℤ≥ ‘ 0 ) ∧ 𝑀 ∈ ℤ ∧ ( 𝑗 − 1 ) < 𝑀 ) ) |
688 |
675 676 686 687
|
syl3anbrc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ) |
689 |
48
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
690 |
636 677
|
syl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ∈ ℤ ) |
691 |
674
|
nn0ge0d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 0 ≤ ( 𝑗 − 1 ) ) |
692 |
679 682 685
|
ltled |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 − 1 ) ≤ 𝑀 ) |
693 |
692
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ≤ 𝑀 ) |
694 |
664 676 690 691 693
|
elfzd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ∈ ( 0 ... 𝑀 ) ) |
695 |
689 694
|
ffvelrnd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) ∈ ℝ ) |
696 |
695
|
rexrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) ∈ ℝ* ) |
697 |
48
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ ) |
698 |
697
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ* ) |
699 |
698
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ* ) |
700 |
699
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ* ) |
701 |
615
|
sselda |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
702 |
701
|
rexrd |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
703 |
702
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
704 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝜑 ) |
705 |
|
ovex |
⊢ ( 𝑗 − 1 ) ∈ V |
706 |
|
eleq1 |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↔ ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ) ) |
707 |
706
|
anbi2d |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ↔ ( 𝜑 ∧ ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ) ) ) |
708 |
|
fveq2 |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ ( 𝑗 − 1 ) ) ) |
709 |
|
oveq1 |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( 𝑖 + 1 ) = ( ( 𝑗 − 1 ) + 1 ) ) |
710 |
709
|
fveq2d |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) |
711 |
708 710
|
breq12d |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ↔ ( 𝑄 ‘ ( 𝑗 − 1 ) ) < ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) |
712 |
707 711
|
imbi12d |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( ( 𝜑 ∧ ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) < ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) ) |
713 |
705 712 78
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) < ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) |
714 |
704 688 713
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) < ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) |
715 |
635
|
zcnd |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ∈ ℂ ) |
716 |
|
1cnd |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 1 ∈ ℂ ) |
717 |
715 716
|
npcand |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 𝑗 − 1 ) + 1 ) = 𝑗 ) |
718 |
717
|
eqcomd |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 = ( ( 𝑗 − 1 ) + 1 ) ) |
719 |
718
|
fveq2d |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) |
720 |
719
|
eqcomd |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) = ( 𝑄 ‘ 𝑗 ) ) |
721 |
720
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) = ( 𝑄 ‘ 𝑗 ) ) |
722 |
714 721
|
breqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) < ( 𝑄 ‘ 𝑗 ) ) |
723 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) |
724 |
722 723
|
breqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) < ( 𝐸 ‘ 𝑋 ) ) |
725 |
624
|
leidd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
726 |
725
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
727 |
641
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ 𝑗 ) ) |
728 |
726 727
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ 𝑗 ) ) |
729 |
728
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ 𝑗 ) ) |
730 |
696 700 703 724 729
|
eliocd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ 𝑗 ) ) ) |
731 |
719
|
oveq2d |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ 𝑗 ) ) = ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) |
732 |
731
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ 𝑗 ) ) = ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) |
733 |
730 732
|
eleqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) |
734 |
708 710
|
oveq12d |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) |
735 |
734
|
eleq2d |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) ) |
736 |
735
|
rspcev |
⊢ ( ( ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
737 |
688 733 736
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
738 |
737
|
ex |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
739 |
738
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
740 |
739
|
rexlimdva |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
741 |
633 740
|
mpd |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
742 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → 𝑀 ∈ ℕ ) |
743 |
48
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
744 |
|
iocssicc |
⊢ ( 𝐴 (,] 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) |
745 |
646
|
eqcomd |
⊢ ( 𝜑 → 𝐴 = ( 𝑄 ‘ 0 ) ) |
746 |
645
|
simprd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) = 𝐵 ) |
747 |
746
|
eqcomd |
⊢ ( 𝜑 → 𝐵 = ( 𝑄 ‘ 𝑀 ) ) |
748 |
745 747
|
oveq12d |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) = ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) |
749 |
744 748
|
sseqtrid |
⊢ ( 𝜑 → ( 𝐴 (,] 𝐵 ) ⊆ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) |
750 |
749
|
sselda |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) |
751 |
750
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) |
752 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) |
753 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝑄 ‘ 𝑘 ) = ( 𝑄 ‘ 𝑗 ) ) |
754 |
753
|
breq1d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝑄 ‘ 𝑘 ) < ( 𝐸 ‘ 𝑋 ) ↔ ( 𝑄 ‘ 𝑗 ) < ( 𝐸 ‘ 𝑋 ) ) ) |
755 |
754
|
cbvrabv |
⊢ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < ( 𝐸 ‘ 𝑋 ) } = { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) < ( 𝐸 ‘ 𝑋 ) } |
756 |
755
|
supeq1i |
⊢ sup ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < ( 𝐸 ‘ 𝑋 ) } , ℝ , < ) = sup ( { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) < ( 𝐸 ‘ 𝑋 ) } , ℝ , < ) |
757 |
742 743 751 752 756
|
fourierdlem25 |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
758 |
|
ioossioc |
⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
759 |
758
|
sseli |
⊢ ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
760 |
759
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
761 |
760
|
reximdva |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
762 |
757 761
|
mpd |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
763 |
741 762
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
764 |
623 763
|
mpdan |
⊢ ( 𝜑 → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
765 |
|
fveq2 |
⊢ ( 𝑖 = 𝑗 → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 𝑗 ) ) |
766 |
|
oveq1 |
⊢ ( 𝑖 = 𝑗 → ( 𝑖 + 1 ) = ( 𝑗 + 1 ) ) |
767 |
766
|
fveq2d |
⊢ ( 𝑖 = 𝑗 → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
768 |
765 767
|
oveq12d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) |
769 |
768
|
eleq2d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ) |
770 |
769
|
cbvrexvw |
⊢ ( ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ∃ 𝑗 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) |
771 |
764 770
|
sylib |
⊢ ( 𝜑 → ∃ 𝑗 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) |
772 |
771
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) → ∃ 𝑗 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) |
773 |
|
elfzonn0 |
⊢ ( 𝑗 ∈ ( 0 ..^ 𝑀 ) → 𝑗 ∈ ℕ0 ) |
774 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
775 |
774
|
a1i |
⊢ ( 𝑗 ∈ ( 0 ..^ 𝑀 ) → 1 ∈ ℕ0 ) |
776 |
773 775
|
nn0addcld |
⊢ ( 𝑗 ∈ ( 0 ..^ 𝑀 ) → ( 𝑗 + 1 ) ∈ ℕ0 ) |
777 |
776 50
|
eleqtrdi |
⊢ ( 𝑗 ∈ ( 0 ..^ 𝑀 ) → ( 𝑗 + 1 ) ∈ ( ℤ≥ ‘ 0 ) ) |
778 |
777
|
adantr |
⊢ ( ( 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑗 + 1 ) ∈ ( ℤ≥ ‘ 0 ) ) |
779 |
778
|
3ad2antl2 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑗 + 1 ) ∈ ( ℤ≥ ‘ 0 ) ) |
780 |
19
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → 𝑀 ∈ ℤ ) |
781 |
780
|
3ad2antl1 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → 𝑀 ∈ ℤ ) |
782 |
773
|
nn0red |
⊢ ( 𝑗 ∈ ( 0 ..^ 𝑀 ) → 𝑗 ∈ ℝ ) |
783 |
782
|
adantr |
⊢ ( ( 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → 𝑗 ∈ ℝ ) |
784 |
783
|
3ad2antl2 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → 𝑗 ∈ ℝ ) |
785 |
|
1red |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → 1 ∈ ℝ ) |
786 |
784 785
|
readdcld |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑗 + 1 ) ∈ ℝ ) |
787 |
781
|
zred |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → 𝑀 ∈ ℝ ) |
788 |
|
elfzop1le2 |
⊢ ( 𝑗 ∈ ( 0 ..^ 𝑀 ) → ( 𝑗 + 1 ) ≤ 𝑀 ) |
789 |
788
|
adantr |
⊢ ( ( 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑗 + 1 ) ≤ 𝑀 ) |
790 |
789
|
3ad2antl2 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑗 + 1 ) ≤ 𝑀 ) |
791 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ∧ 𝑀 = ( 𝑗 + 1 ) ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
792 |
|
fveq2 |
⊢ ( 𝑀 = ( 𝑗 + 1 ) → ( 𝑄 ‘ 𝑀 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
793 |
792
|
eqcomd |
⊢ ( 𝑀 = ( 𝑗 + 1 ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) = ( 𝑄 ‘ 𝑀 ) ) |
794 |
793
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ∧ 𝑀 = ( 𝑗 + 1 ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) = ( 𝑄 ‘ 𝑀 ) ) |
795 |
746
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ∧ 𝑀 = ( 𝑗 + 1 ) ) → ( 𝑄 ‘ 𝑀 ) = 𝐵 ) |
796 |
791 794 795
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ∧ 𝑀 = ( 𝑗 + 1 ) ) → ( 𝐸 ‘ 𝑋 ) = 𝐵 ) |
797 |
796
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ∧ 𝑀 = ( 𝑗 + 1 ) ) → ( 𝐸 ‘ 𝑋 ) = 𝐵 ) |
798 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ∧ 𝑀 = ( 𝑗 + 1 ) ) → ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) |
799 |
798
|
neneqd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ∧ 𝑀 = ( 𝑗 + 1 ) ) → ¬ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) |
800 |
797 799
|
pm2.65da |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ¬ 𝑀 = ( 𝑗 + 1 ) ) |
801 |
800
|
neqned |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → 𝑀 ≠ ( 𝑗 + 1 ) ) |
802 |
801
|
3ad2antl1 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → 𝑀 ≠ ( 𝑗 + 1 ) ) |
803 |
786 787 790 802
|
leneltd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑗 + 1 ) < 𝑀 ) |
804 |
|
elfzo2 |
⊢ ( ( 𝑗 + 1 ) ∈ ( 0 ..^ 𝑀 ) ↔ ( ( 𝑗 + 1 ) ∈ ( ℤ≥ ‘ 0 ) ∧ 𝑀 ∈ ℤ ∧ ( 𝑗 + 1 ) < 𝑀 ) ) |
805 |
779 781 803 804
|
syl3anbrc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑗 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) |
806 |
48
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
807 |
|
fzofzp1 |
⊢ ( 𝑗 ∈ ( 0 ..^ 𝑀 ) → ( 𝑗 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
808 |
807
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑗 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
809 |
806 808
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ∈ ℝ ) |
810 |
809
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ∈ ℝ* ) |
811 |
810
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ∈ ℝ* ) |
812 |
811
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ∈ ℝ* ) |
813 |
812
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ∈ ℝ* ) |
814 |
|
simpl1l |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → 𝜑 ) |
815 |
814 48
|
syl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
816 |
|
fzofzp1 |
⊢ ( ( 𝑗 + 1 ) ∈ ( 0 ..^ 𝑀 ) → ( ( 𝑗 + 1 ) + 1 ) ∈ ( 0 ... 𝑀 ) ) |
817 |
805 816
|
syl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( ( 𝑗 + 1 ) + 1 ) ∈ ( 0 ... 𝑀 ) ) |
818 |
815 817
|
ffvelrnd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ ( ( 𝑗 + 1 ) + 1 ) ) ∈ ℝ ) |
819 |
818
|
rexrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ ( ( 𝑗 + 1 ) + 1 ) ) ∈ ℝ* ) |
820 |
624
|
rexrd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
821 |
820
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
822 |
821
|
3ad2antl1 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
823 |
809
|
leidd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ≤ ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
824 |
823
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ≤ ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
825 |
|
id |
⊢ ( ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
826 |
825
|
eqcomd |
⊢ ( ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) = ( 𝐸 ‘ 𝑋 ) ) |
827 |
826
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) = ( 𝐸 ‘ 𝑋 ) ) |
828 |
824 827
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
829 |
828
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
830 |
829
|
3adantl3 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
831 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
832 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
833 |
|
ovex |
⊢ ( 𝑗 + 1 ) ∈ V |
834 |
|
eleq1 |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↔ ( 𝑗 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) ) |
835 |
834
|
anbi2d |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ↔ ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) ) ) |
836 |
|
fveq2 |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
837 |
|
oveq1 |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( 𝑖 + 1 ) = ( ( 𝑗 + 1 ) + 1 ) ) |
838 |
837
|
fveq2d |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ ( ( 𝑗 + 1 ) + 1 ) ) ) |
839 |
836 838
|
breq12d |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ↔ ( 𝑄 ‘ ( 𝑗 + 1 ) ) < ( 𝑄 ‘ ( ( 𝑗 + 1 ) + 1 ) ) ) ) |
840 |
835 839
|
imbi12d |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) < ( 𝑄 ‘ ( ( 𝑗 + 1 ) + 1 ) ) ) ) ) |
841 |
833 840 78
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) < ( 𝑄 ‘ ( ( 𝑗 + 1 ) + 1 ) ) ) |
842 |
841
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) < ( 𝑄 ‘ ( ( 𝑗 + 1 ) + 1 ) ) ) |
843 |
832 842
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) < ( 𝑄 ‘ ( ( 𝑗 + 1 ) + 1 ) ) ) |
844 |
814 805 831 843
|
syl21anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) < ( 𝑄 ‘ ( ( 𝑗 + 1 ) + 1 ) ) ) |
845 |
813 819 822 830 844
|
elicod |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ ( 𝑗 + 1 ) ) [,) ( 𝑄 ‘ ( ( 𝑗 + 1 ) + 1 ) ) ) ) |
846 |
836 838
|
oveq12d |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑄 ‘ ( 𝑗 + 1 ) ) [,) ( 𝑄 ‘ ( ( 𝑗 + 1 ) + 1 ) ) ) ) |
847 |
846
|
eleq2d |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ ( 𝑗 + 1 ) ) [,) ( 𝑄 ‘ ( ( 𝑗 + 1 ) + 1 ) ) ) ) ) |
848 |
847
|
rspcev |
⊢ ( ( ( 𝑗 + 1 ) ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ ( 𝑗 + 1 ) ) [,) ( 𝑄 ‘ ( ( 𝑗 + 1 ) + 1 ) ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
849 |
805 845 848
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
850 |
|
simpl2 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → 𝑗 ∈ ( 0 ..^ 𝑀 ) ) |
851 |
|
id |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) → ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ) |
852 |
851
|
3adant1r |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) → ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ) |
853 |
|
elfzofz |
⊢ ( 𝑗 ∈ ( 0 ..^ 𝑀 ) → 𝑗 ∈ ( 0 ... 𝑀 ) ) |
854 |
853
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → 𝑗 ∈ ( 0 ... 𝑀 ) ) |
855 |
806 854
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ ) |
856 |
855
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ* ) |
857 |
856
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ* ) |
858 |
857
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ* ) |
859 |
810
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ∈ ℝ* ) |
860 |
859
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ∈ ℝ* ) |
861 |
820
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
862 |
861
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
863 |
855
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ ) |
864 |
624
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
865 |
856
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ* ) |
866 |
810
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ∈ ℝ* ) |
867 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) |
868 |
|
iocgtlb |
⊢ ( ( ( 𝑄 ‘ 𝑗 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑗 + 1 ) ) ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑗 ) < ( 𝐸 ‘ 𝑋 ) ) |
869 |
865 866 867 868
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑗 ) < ( 𝐸 ‘ 𝑋 ) ) |
870 |
863 864 869
|
ltled |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑗 ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
871 |
870
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ 𝑗 ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
872 |
864
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
873 |
809
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ∈ ℝ ) |
874 |
873
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ∈ ℝ ) |
875 |
|
iocleub |
⊢ ( ( ( 𝑄 ‘ 𝑗 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑗 + 1 ) ) ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
876 |
865 866 867 875
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
877 |
876
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
878 |
|
neqne |
⊢ ( ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) → ( 𝐸 ‘ 𝑋 ) ≠ ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
879 |
878
|
necomd |
⊢ ( ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ≠ ( 𝐸 ‘ 𝑋 ) ) |
880 |
879
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ≠ ( 𝐸 ‘ 𝑋 ) ) |
881 |
872 874 877 880
|
leneltd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) < ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
882 |
858 860 862 871 881
|
elicod |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) [,) ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) |
883 |
852 882
|
sylan |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) [,) ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) |
884 |
765 767
|
oveq12d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑄 ‘ 𝑗 ) [,) ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) |
885 |
884
|
eleq2d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) [,) ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ) |
886 |
885
|
rspcev |
⊢ ( ( 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) [,) ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
887 |
850 883 886
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
888 |
849 887
|
pm2.61dan |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
889 |
888
|
rexlimdv3a |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) → ( ∃ 𝑗 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) (,] ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
890 |
772 889
|
mpd |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
891 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
892 |
|
oveq1 |
⊢ ( 𝑘 = ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( 𝑘 · 𝑇 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
893 |
892
|
oveq2d |
⊢ ( 𝑘 = ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( 𝑋 + ( 𝑘 · 𝑇 ) ) = ( 𝑋 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
894 |
893
|
eqeq2d |
⊢ ( 𝑘 = ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ↔ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) ) |
895 |
894
|
rspcev |
⊢ ( ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) → ∃ 𝑘 ∈ ℤ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) |
896 |
102 110 895
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑘 ∈ ℤ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) |
897 |
896
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ∃ 𝑘 ∈ ℤ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) |
898 |
|
r19.42v |
⊢ ( ∃ 𝑘 ∈ ℤ ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ↔ ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ∃ 𝑘 ∈ ℤ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) |
899 |
891 897 898
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ∃ 𝑘 ∈ ℤ ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) |
900 |
899
|
ex |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ∃ 𝑘 ∈ ℤ ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) ) |
901 |
900
|
reximdv |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) ) |
902 |
890 901
|
mpd |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) |
903 |
626 902
|
jca |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) → ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) ) |
904 |
|
eleq1 |
⊢ ( 𝑦 = ( 𝐸 ‘ 𝑋 ) → ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
905 |
|
eqeq1 |
⊢ ( 𝑦 = ( 𝐸 ‘ 𝑋 ) → ( 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ↔ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) |
906 |
904 905
|
anbi12d |
⊢ ( 𝑦 = ( 𝐸 ‘ 𝑋 ) → ( ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ↔ ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) ) |
907 |
906
|
2rexbidv |
⊢ ( 𝑦 = ( 𝐸 ‘ 𝑋 ) → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ↔ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) ) |
908 |
907
|
anbi2d |
⊢ ( 𝑦 = ( 𝐸 ‘ 𝑋 ) → ( ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) ↔ ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) ) ) |
909 |
908
|
imbi1d |
⊢ ( 𝑦 = ( 𝐸 ‘ 𝑋 ) → ( ( ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( 𝑦 ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ 𝑦 = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) ↔ ( ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) ) ) |
910 |
909 611
|
vtoclg |
⊢ ( ( 𝐸 ‘ 𝑋 ) ∈ ℝ → ( ( 𝜑 ∧ ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∃ 𝑘 ∈ ℤ ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑘 · 𝑇 ) ) ) ) → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) ) |
911 |
625 903 910
|
sylc |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) |
912 |
613 911
|
pm2.61dane |
⊢ ( 𝜑 → ( ( 𝐹 ↾ ( 𝑋 (,) +∞ ) ) limℂ 𝑋 ) ≠ ∅ ) |