Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem49.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
fourierdlem49.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
fourierdlem49.altb |
⊢ ( 𝜑 → 𝐴 < 𝐵 ) |
4 |
|
fourierdlem49.p |
⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) |
5 |
|
fourierdlem49.t |
⊢ 𝑇 = ( 𝐵 − 𝐴 ) |
6 |
|
fourierdlem49.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
7 |
|
fourierdlem49.q |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ) |
8 |
|
fourierdlem49.d |
⊢ ( 𝜑 → 𝐷 ⊆ ℝ ) |
9 |
|
fourierdlem49.f |
⊢ ( 𝜑 → 𝐹 : 𝐷 ⟶ ℝ ) |
10 |
|
fourierdlem49.dper |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
11 |
|
fourierdlem49.per |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
12 |
|
fourierdlem49.cn |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
13 |
|
fourierdlem49.l |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
14 |
|
fourierdlem49.x |
⊢ ( 𝜑 → 𝑋 ∈ ℝ ) |
15 |
|
fourierdlem49.z |
⊢ 𝑍 = ( 𝑥 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) |
16 |
|
fourierdlem49.e |
⊢ 𝐸 = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) ) |
17 |
|
ovex |
⊢ ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ∈ V |
18 |
15
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ ℝ ∧ ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ∈ V ) → ( 𝑍 ‘ 𝑥 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) |
19 |
17 18
|
mpan2 |
⊢ ( 𝑥 ∈ ℝ → ( 𝑍 ‘ 𝑥 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) |
20 |
19
|
oveq2d |
⊢ ( 𝑥 ∈ ℝ → ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) = ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
21 |
20
|
mpteq2ia |
⊢ ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
22 |
16 21
|
eqtri |
⊢ 𝐸 = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
23 |
1 2 3 5 22
|
fourierdlem4 |
⊢ ( 𝜑 → 𝐸 : ℝ ⟶ ( 𝐴 (,] 𝐵 ) ) |
24 |
23 14
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) |
25 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) |
26 |
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 ) ) ) ) ) ) |
27 |
6 26
|
syl |
⊢ ( 𝜑 → ( 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
28 |
7 27
|
mpbid |
⊢ ( 𝜑 → ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
29 |
28
|
simpld |
⊢ ( 𝜑 → 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ) |
30 |
|
elmapi |
⊢ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
31 |
29 30
|
syl |
⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
32 |
|
ffn |
⊢ ( 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ → 𝑄 Fn ( 0 ... 𝑀 ) ) |
33 |
31 32
|
syl |
⊢ ( 𝜑 → 𝑄 Fn ( 0 ... 𝑀 ) ) |
34 |
33
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → 𝑄 Fn ( 0 ... 𝑀 ) ) |
35 |
|
fvelrnb |
⊢ ( 𝑄 Fn ( 0 ... 𝑀 ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ) |
36 |
34 35
|
syl |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ) |
37 |
25 36
|
mpbid |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) |
38 |
|
1zzd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 1 ∈ ℤ ) |
39 |
|
elfzelz |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ∈ ℤ ) |
40 |
39
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑗 ∈ ℤ ) |
41 |
|
1e0p1 |
⊢ 1 = ( 0 + 1 ) |
42 |
41
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 1 = ( 0 + 1 ) ) |
43 |
40
|
zred |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑗 ∈ ℝ ) |
44 |
|
elfzle1 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 0 ≤ 𝑗 ) |
45 |
44
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 0 ≤ 𝑗 ) |
46 |
|
id |
⊢ ( ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) → ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) |
47 |
46
|
eqcomd |
⊢ ( ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ 𝑗 ) ) |
48 |
47
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑗 = 0 ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ 𝑗 ) ) |
49 |
|
fveq2 |
⊢ ( 𝑗 = 0 → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ 0 ) ) |
50 |
49
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑗 = 0 ) → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ 0 ) ) |
51 |
28
|
simprld |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ) |
52 |
51
|
simpld |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) = 𝐴 ) |
53 |
52
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑗 = 0 ) → ( 𝑄 ‘ 0 ) = 𝐴 ) |
54 |
48 50 53
|
3eqtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑗 = 0 ) → ( 𝐸 ‘ 𝑋 ) = 𝐴 ) |
55 |
54
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑗 = 0 ) → ( 𝐸 ‘ 𝑋 ) = 𝐴 ) |
56 |
55
|
adantllr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑗 = 0 ) → ( 𝐸 ‘ 𝑋 ) = 𝐴 ) |
57 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐴 ∈ ℝ ) |
58 |
1
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
59 |
58
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐴 ∈ ℝ* ) |
60 |
2
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
61 |
60
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐵 ∈ ℝ* ) |
62 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) |
63 |
|
iocgtlb |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐴 < ( 𝐸 ‘ 𝑋 ) ) |
64 |
59 61 62 63
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐴 < ( 𝐸 ‘ 𝑋 ) ) |
65 |
57 64
|
gtned |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → ( 𝐸 ‘ 𝑋 ) ≠ 𝐴 ) |
66 |
65
|
neneqd |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → ¬ ( 𝐸 ‘ 𝑋 ) = 𝐴 ) |
67 |
66
|
ad3antrrr |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑗 = 0 ) → ¬ ( 𝐸 ‘ 𝑋 ) = 𝐴 ) |
68 |
56 67
|
pm2.65da |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ¬ 𝑗 = 0 ) |
69 |
68
|
neqned |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑗 ≠ 0 ) |
70 |
43 45 69
|
ne0gt0d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 0 < 𝑗 ) |
71 |
|
0zd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 0 ∈ ℤ ) |
72 |
|
zltp1le |
⊢ ( ( 0 ∈ ℤ ∧ 𝑗 ∈ ℤ ) → ( 0 < 𝑗 ↔ ( 0 + 1 ) ≤ 𝑗 ) ) |
73 |
71 40 72
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 0 < 𝑗 ↔ ( 0 + 1 ) ≤ 𝑗 ) ) |
74 |
70 73
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 0 + 1 ) ≤ 𝑗 ) |
75 |
42 74
|
eqbrtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 1 ≤ 𝑗 ) |
76 |
|
eluz2 |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 1 ) ↔ ( 1 ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ 1 ≤ 𝑗 ) ) |
77 |
38 40 75 76
|
syl3anbrc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑗 ∈ ( ℤ≥ ‘ 1 ) ) |
78 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
79 |
77 78
|
eleqtrrdi |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑗 ∈ ℕ ) |
80 |
|
nnm1nn0 |
⊢ ( 𝑗 ∈ ℕ → ( 𝑗 − 1 ) ∈ ℕ0 ) |
81 |
79 80
|
syl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ∈ ℕ0 ) |
82 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
83 |
82
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ℕ0 = ( ℤ≥ ‘ 0 ) ) |
84 |
81 83
|
eleqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ∈ ( ℤ≥ ‘ 0 ) ) |
85 |
6
|
nnzd |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
86 |
85
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑀 ∈ ℤ ) |
87 |
|
peano2zm |
⊢ ( 𝑗 ∈ ℤ → ( 𝑗 − 1 ) ∈ ℤ ) |
88 |
39 87
|
syl |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 − 1 ) ∈ ℤ ) |
89 |
88
|
zred |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 − 1 ) ∈ ℝ ) |
90 |
39
|
zred |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ∈ ℝ ) |
91 |
|
elfzel2 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑀 ∈ ℤ ) |
92 |
91
|
zred |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑀 ∈ ℝ ) |
93 |
90
|
ltm1d |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 − 1 ) < 𝑗 ) |
94 |
|
elfzle2 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ≤ 𝑀 ) |
95 |
89 90 92 93 94
|
ltletrd |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 − 1 ) < 𝑀 ) |
96 |
95
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) < 𝑀 ) |
97 |
|
elfzo2 |
⊢ ( ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ↔ ( ( 𝑗 − 1 ) ∈ ( ℤ≥ ‘ 0 ) ∧ 𝑀 ∈ ℤ ∧ ( 𝑗 − 1 ) < 𝑀 ) ) |
98 |
84 86 96 97
|
syl3anbrc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ) |
99 |
31
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
100 |
40 87
|
syl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ∈ ℤ ) |
101 |
81
|
nn0ge0d |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 0 ≤ ( 𝑗 − 1 ) ) |
102 |
89 92 95
|
ltled |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 − 1 ) ≤ 𝑀 ) |
103 |
102
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ≤ 𝑀 ) |
104 |
71 86 100 101 103
|
elfzd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ∈ ( 0 ... 𝑀 ) ) |
105 |
99 104
|
ffvelrnd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) ∈ ℝ ) |
106 |
105
|
rexrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) ∈ ℝ* ) |
107 |
31
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ ) |
108 |
107
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ* ) |
109 |
108
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ* ) |
110 |
109
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ* ) |
111 |
|
iocssre |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ) → ( 𝐴 (,] 𝐵 ) ⊆ ℝ ) |
112 |
58 2 111
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 (,] 𝐵 ) ⊆ ℝ ) |
113 |
112
|
sselda |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
114 |
113
|
rexrd |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
115 |
114
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
116 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝜑 ) |
117 |
|
ovex |
⊢ ( 𝑗 − 1 ) ∈ V |
118 |
|
eleq1 |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↔ ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ) ) |
119 |
118
|
anbi2d |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ↔ ( 𝜑 ∧ ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ) ) ) |
120 |
|
fveq2 |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ ( 𝑗 − 1 ) ) ) |
121 |
|
oveq1 |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( 𝑖 + 1 ) = ( ( 𝑗 − 1 ) + 1 ) ) |
122 |
121
|
fveq2d |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) |
123 |
120 122
|
breq12d |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ↔ ( 𝑄 ‘ ( 𝑗 − 1 ) ) < ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) |
124 |
119 123
|
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 ) ) ) ) ) |
125 |
28
|
simprrd |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
126 |
125
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
127 |
117 124 126
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) < ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) |
128 |
116 98 127
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) < ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) |
129 |
39
|
zcnd |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ∈ ℂ ) |
130 |
|
1cnd |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 1 ∈ ℂ ) |
131 |
129 130
|
npcand |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 𝑗 − 1 ) + 1 ) = 𝑗 ) |
132 |
131
|
eqcomd |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 = ( ( 𝑗 − 1 ) + 1 ) ) |
133 |
132
|
fveq2d |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) |
134 |
133
|
eqcomd |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) = ( 𝑄 ‘ 𝑗 ) ) |
135 |
134
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) = ( 𝑄 ‘ 𝑗 ) ) |
136 |
128 135
|
breqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) < ( 𝑄 ‘ 𝑗 ) ) |
137 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) |
138 |
136 137
|
breqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) < ( 𝐸 ‘ 𝑋 ) ) |
139 |
112 24
|
sseldd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
140 |
139
|
leidd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
141 |
140
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
142 |
47
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ 𝑗 ) ) |
143 |
141 142
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ 𝑗 ) ) |
144 |
143
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ 𝑗 ) ) |
145 |
106 110 115 138 144
|
eliocd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ 𝑗 ) ) ) |
146 |
133
|
oveq2d |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ 𝑗 ) ) = ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) |
147 |
146
|
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 ) ) ) ) |
148 |
145 147
|
eleqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) |
149 |
120 122
|
oveq12d |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) |
150 |
149
|
eleq2d |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) ) |
151 |
150
|
rspcev |
⊢ ( ( ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
152 |
98 148 151
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
153 |
152
|
ex |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
154 |
153
|
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 ) ) ) ) ) |
155 |
154
|
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 ) ) ) ) ) |
156 |
37 155
|
mpd |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
157 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → 𝑀 ∈ ℕ ) |
158 |
31
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
159 |
|
iocssicc |
⊢ ( 𝐴 (,] 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) |
160 |
52
|
eqcomd |
⊢ ( 𝜑 → 𝐴 = ( 𝑄 ‘ 0 ) ) |
161 |
51
|
simprd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) = 𝐵 ) |
162 |
161
|
eqcomd |
⊢ ( 𝜑 → 𝐵 = ( 𝑄 ‘ 𝑀 ) ) |
163 |
160 162
|
oveq12d |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) = ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) |
164 |
159 163
|
sseqtrid |
⊢ ( 𝜑 → ( 𝐴 (,] 𝐵 ) ⊆ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) |
165 |
164
|
sselda |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) |
166 |
165
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) |
167 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) |
168 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝑄 ‘ 𝑘 ) = ( 𝑄 ‘ 𝑗 ) ) |
169 |
168
|
breq1d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝑄 ‘ 𝑘 ) < ( 𝐸 ‘ 𝑋 ) ↔ ( 𝑄 ‘ 𝑗 ) < ( 𝐸 ‘ 𝑋 ) ) ) |
170 |
169
|
cbvrabv |
⊢ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < ( 𝐸 ‘ 𝑋 ) } = { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) < ( 𝐸 ‘ 𝑋 ) } |
171 |
170
|
supeq1i |
⊢ sup ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < ( 𝐸 ‘ 𝑋 ) } , ℝ , < ) = sup ( { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) < ( 𝐸 ‘ 𝑋 ) } , ℝ , < ) |
172 |
157 158 166 167 171
|
fourierdlem25 |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
173 |
|
ioossioc |
⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
174 |
173
|
sseli |
⊢ ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
175 |
174
|
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 ) ) ) ) ) |
176 |
175
|
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 ) ) ) ) ) |
177 |
172 176
|
mpd |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
178 |
156 177
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
179 |
24 178
|
mpdan |
⊢ ( 𝜑 → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
180 |
|
frel |
⊢ ( 𝐹 : 𝐷 ⟶ ℝ → Rel 𝐹 ) |
181 |
9 180
|
syl |
⊢ ( 𝜑 → Rel 𝐹 ) |
182 |
|
resindm |
⊢ ( Rel 𝐹 → ( 𝐹 ↾ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ dom 𝐹 ) ) = ( 𝐹 ↾ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) ) |
183 |
182
|
eqcomd |
⊢ ( Rel 𝐹 → ( 𝐹 ↾ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) = ( 𝐹 ↾ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ dom 𝐹 ) ) ) |
184 |
181 183
|
syl |
⊢ ( 𝜑 → ( 𝐹 ↾ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) = ( 𝐹 ↾ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ dom 𝐹 ) ) ) |
185 |
|
fdm |
⊢ ( 𝐹 : 𝐷 ⟶ ℝ → dom 𝐹 = 𝐷 ) |
186 |
9 185
|
syl |
⊢ ( 𝜑 → dom 𝐹 = 𝐷 ) |
187 |
186
|
ineq2d |
⊢ ( 𝜑 → ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ dom 𝐹 ) = ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ) |
188 |
187
|
reseq2d |
⊢ ( 𝜑 → ( 𝐹 ↾ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ dom 𝐹 ) ) = ( 𝐹 ↾ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ) ) |
189 |
184 188
|
eqtrd |
⊢ ( 𝜑 → ( 𝐹 ↾ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) = ( 𝐹 ↾ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ) ) |
190 |
189
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ↾ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) = ( 𝐹 ↾ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ) ) |
191 |
190
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) = ( ( 𝐹 ↾ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) |
192 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
193 |
192
|
a1i |
⊢ ( 𝜑 → ℝ ⊆ ℂ ) |
194 |
9 193
|
fssd |
⊢ ( 𝜑 → 𝐹 : 𝐷 ⟶ ℂ ) |
195 |
194
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐹 : 𝐷 ⟶ ℂ ) |
196 |
|
inss2 |
⊢ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ⊆ 𝐷 |
197 |
196
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ⊆ 𝐷 ) |
198 |
195 197
|
fssresd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ↾ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ) : ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ⟶ ℂ ) |
199 |
|
mnfxr |
⊢ -∞ ∈ ℝ* |
200 |
199
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → -∞ ∈ ℝ* ) |
201 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
202 |
|
elfzofz |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
203 |
202
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
204 |
201 203
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
205 |
204
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
206 |
204
|
mnfltd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → -∞ < ( 𝑄 ‘ 𝑖 ) ) |
207 |
200 205 206
|
xrltled |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → -∞ ≤ ( 𝑄 ‘ 𝑖 ) ) |
208 |
|
iooss1 |
⊢ ( ( -∞ ∈ ℝ* ∧ -∞ ≤ ( 𝑄 ‘ 𝑖 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ⊆ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) |
209 |
199 207 208
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ⊆ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) |
210 |
209
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ⊆ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) |
211 |
|
fzofzp1 |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
212 |
211
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
213 |
201 212
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
214 |
213
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
215 |
214
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
216 |
204
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
217 |
216
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
218 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
219 |
|
iocleub |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
220 |
217 215 218 219
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
221 |
|
iooss2 |
⊢ ( ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
222 |
215 220 221
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
223 |
|
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 ) |
224 |
|
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 ) ) ) ) |
225 |
12 223 224
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
226 |
|
ssdmres |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ dom 𝐹 ↔ dom ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
227 |
225 226
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ dom 𝐹 ) |
228 |
186
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → dom 𝐹 = 𝐷 ) |
229 |
227 228
|
sseqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ 𝐷 ) |
230 |
229
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ 𝐷 ) |
231 |
222 230
|
sstrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ⊆ 𝐷 ) |
232 |
210 231
|
ssind |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ⊆ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ) |
233 |
8 193
|
sstrd |
⊢ ( 𝜑 → 𝐷 ⊆ ℂ ) |
234 |
233
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐷 ⊆ ℂ ) |
235 |
196 234
|
sstrid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ⊆ ℂ ) |
236 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
237 |
|
eqid |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) = ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) |
238 |
139
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
239 |
238
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
240 |
|
iocgtlb |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝐸 ‘ 𝑋 ) ) |
241 |
217 215 218 240
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝐸 ‘ 𝑋 ) ) |
242 |
238
|
leidd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
243 |
217 239 239 241 242
|
eliocd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) |
244 |
|
ioounsn |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ∧ ( 𝑄 ‘ 𝑖 ) < ( 𝐸 ‘ 𝑋 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) = ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) |
245 |
217 239 241 244
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) = ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) |
246 |
245
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) ‘ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) = ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ) |
247 |
236
|
cnfldtop |
⊢ ( TopOpen ‘ ℂfld ) ∈ Top |
248 |
|
ovex |
⊢ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∈ V |
249 |
248
|
inex1 |
⊢ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∈ V |
250 |
|
snex |
⊢ { ( 𝐸 ‘ 𝑋 ) } ∈ V |
251 |
249 250
|
unex |
⊢ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ∈ V |
252 |
|
resttop |
⊢ ( ( ( TopOpen ‘ ℂfld ) ∈ Top ∧ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ∈ V ) → ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ∈ Top ) |
253 |
247 251 252
|
mp2an |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ∈ Top |
254 |
|
retop |
⊢ ( topGen ‘ ran (,) ) ∈ Top |
255 |
254
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( topGen ‘ ran (,) ) ∈ Top ) |
256 |
251
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ∈ V ) |
257 |
|
iooretop |
⊢ ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∈ ( topGen ‘ ran (,) ) |
258 |
257
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∈ ( topGen ‘ ran (,) ) ) |
259 |
|
elrestr |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ Top ∧ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ∈ V ∧ ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∈ ( topGen ‘ ran (,) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ∈ ( ( topGen ‘ ran (,) ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) |
260 |
255 256 258 259
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ∈ ( ( topGen ‘ ran (,) ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) |
261 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 = ( 𝐸 ‘ 𝑋 ) ) |
262 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
263 |
262
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → +∞ ∈ ℝ* ) |
264 |
238
|
ltpnfd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) < +∞ ) |
265 |
217 263 238 241 264
|
eliood |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ) |
266 |
|
snidg |
⊢ ( ( 𝐸 ‘ 𝑋 ) ∈ ℝ → ( 𝐸 ‘ 𝑋 ) ∈ { ( 𝐸 ‘ 𝑋 ) } ) |
267 |
|
elun2 |
⊢ ( ( 𝐸 ‘ 𝑋 ) ∈ { ( 𝐸 ‘ 𝑋 ) } → ( 𝐸 ‘ 𝑋 ) ∈ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) |
268 |
266 267
|
syl |
⊢ ( ( 𝐸 ‘ 𝑋 ) ∈ ℝ → ( 𝐸 ‘ 𝑋 ) ∈ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) |
269 |
139 268
|
syl |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ∈ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) |
270 |
269
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) |
271 |
265 270
|
elind |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) |
272 |
271
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) |
273 |
261 272
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) |
274 |
273
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) |
275 |
217
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
276 |
262
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → +∞ ∈ ℝ* ) |
277 |
205
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
278 |
139
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
279 |
278
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
280 |
|
iocssre |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) → ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ⊆ ℝ ) |
281 |
277 279 280
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ⊆ ℝ ) |
282 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) |
283 |
281 282
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → 𝑥 ∈ ℝ ) |
284 |
283
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → 𝑥 ∈ ℝ ) |
285 |
279
|
rexrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
286 |
|
iocgtlb |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑥 ) |
287 |
277 285 282 286
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑥 ) |
288 |
287
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑥 ) |
289 |
284
|
ltpnfd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → 𝑥 < +∞ ) |
290 |
275 276 284 288 289
|
eliood |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ) |
291 |
290
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ) |
292 |
199
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → -∞ ∈ ℝ* ) |
293 |
285
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
294 |
283
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 ∈ ℝ ) |
295 |
294
|
mnfltd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → -∞ < 𝑥 ) |
296 |
139
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
297 |
|
iocleub |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → 𝑥 ≤ ( 𝐸 ‘ 𝑋 ) ) |
298 |
277 285 282 297
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → 𝑥 ≤ ( 𝐸 ‘ 𝑋 ) ) |
299 |
298
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 ≤ ( 𝐸 ‘ 𝑋 ) ) |
300 |
|
neqne |
⊢ ( ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) → 𝑥 ≠ ( 𝐸 ‘ 𝑋 ) ) |
301 |
300
|
necomd |
⊢ ( ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) → ( 𝐸 ‘ 𝑋 ) ≠ 𝑥 ) |
302 |
301
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ≠ 𝑥 ) |
303 |
294 296 299 302
|
leneltd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 < ( 𝐸 ‘ 𝑋 ) ) |
304 |
292 293 294 295 303
|
eliood |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 ∈ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) |
305 |
304
|
3adantll3 |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 ∈ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) |
306 |
230
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ 𝐷 ) |
307 |
275
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
308 |
215
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
309 |
284
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 ∈ ℝ ) |
310 |
288
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑥 ) |
311 |
238
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
312 |
214
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
313 |
303
|
3adantll3 |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 < ( 𝐸 ‘ 𝑋 ) ) |
314 |
220
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
315 |
309 311 312 313 314
|
ltletrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
316 |
307 308 309 310 315
|
eliood |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
317 |
306 316
|
sseldd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 ∈ 𝐷 ) |
318 |
305 317
|
elind |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 ∈ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ) |
319 |
|
elun1 |
⊢ ( 𝑥 ∈ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) → 𝑥 ∈ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) |
320 |
318 319
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 ∈ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) |
321 |
291 320
|
elind |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) ∧ ¬ 𝑥 = ( 𝐸 ‘ 𝑋 ) ) → 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) |
322 |
274 321
|
pm2.61dan |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) → 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) |
323 |
217
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
324 |
239
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
325 |
|
elinel1 |
⊢ ( 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ) |
326 |
|
elioore |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) → 𝑥 ∈ ℝ ) |
327 |
326
|
rexrd |
⊢ ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) → 𝑥 ∈ ℝ* ) |
328 |
325 327
|
syl |
⊢ ( 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) → 𝑥 ∈ ℝ* ) |
329 |
328
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) → 𝑥 ∈ ℝ* ) |
330 |
205
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
331 |
262
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) → +∞ ∈ ℝ* ) |
332 |
325
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ) |
333 |
|
ioogtlb |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑥 ) |
334 |
330 331 332 333
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑥 ) |
335 |
334
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) → ( 𝑄 ‘ 𝑖 ) < 𝑥 ) |
336 |
|
elinel2 |
⊢ ( 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) → 𝑥 ∈ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) |
337 |
|
elsni |
⊢ ( 𝑥 ∈ { ( 𝐸 ‘ 𝑋 ) } → 𝑥 = ( 𝐸 ‘ 𝑋 ) ) |
338 |
337
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ( 𝐸 ‘ 𝑋 ) } ) → 𝑥 = ( 𝐸 ‘ 𝑋 ) ) |
339 |
140
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ( 𝐸 ‘ 𝑋 ) } ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
340 |
338 339
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ( 𝐸 ‘ 𝑋 ) } ) → 𝑥 ≤ ( 𝐸 ‘ 𝑋 ) ) |
341 |
340
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ∧ 𝑥 ∈ { ( 𝐸 ‘ 𝑋 ) } ) → 𝑥 ≤ ( 𝐸 ‘ 𝑋 ) ) |
342 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ∧ ¬ 𝑥 ∈ { ( 𝐸 ‘ 𝑋 ) } ) → 𝜑 ) |
343 |
|
elunnel2 |
⊢ ( ( 𝑥 ∈ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ∧ ¬ 𝑥 ∈ { ( 𝐸 ‘ 𝑋 ) } ) → 𝑥 ∈ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ) |
344 |
343
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ∧ ¬ 𝑥 ∈ { ( 𝐸 ‘ 𝑋 ) } ) → 𝑥 ∈ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ) |
345 |
|
elinel1 |
⊢ ( 𝑥 ∈ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) → 𝑥 ∈ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) |
346 |
|
elioore |
⊢ ( 𝑥 ∈ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) → 𝑥 ∈ ℝ ) |
347 |
346
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) → 𝑥 ∈ ℝ ) |
348 |
139
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
349 |
199
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) → -∞ ∈ ℝ* ) |
350 |
348
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
351 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) → 𝑥 ∈ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) |
352 |
|
iooltub |
⊢ ( ( -∞ ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ∧ 𝑥 ∈ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) → 𝑥 < ( 𝐸 ‘ 𝑋 ) ) |
353 |
349 350 351 352
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) → 𝑥 < ( 𝐸 ‘ 𝑋 ) ) |
354 |
347 348 353
|
ltled |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) → 𝑥 ≤ ( 𝐸 ‘ 𝑋 ) ) |
355 |
345 354
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ) → 𝑥 ≤ ( 𝐸 ‘ 𝑋 ) ) |
356 |
342 344 355
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ∧ ¬ 𝑥 ∈ { ( 𝐸 ‘ 𝑋 ) } ) → 𝑥 ≤ ( 𝐸 ‘ 𝑋 ) ) |
357 |
341 356
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) → 𝑥 ≤ ( 𝐸 ‘ 𝑋 ) ) |
358 |
357
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) → 𝑥 ≤ ( 𝐸 ‘ 𝑋 ) ) |
359 |
336 358
|
sylan2 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) → 𝑥 ≤ ( 𝐸 ‘ 𝑋 ) ) |
360 |
359
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) → 𝑥 ≤ ( 𝐸 ‘ 𝑋 ) ) |
361 |
323 324 329 335 360
|
eliocd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) → 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) |
362 |
322 361
|
impbida |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ↔ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) ) |
363 |
362
|
eqrdv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) = ( ( ( 𝑄 ‘ 𝑖 ) (,) +∞ ) ∩ ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) |
364 |
|
ioossre |
⊢ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ⊆ ℝ |
365 |
|
ssinss1 |
⊢ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ⊆ ℝ → ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ⊆ ℝ ) |
366 |
364 365
|
mp1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ⊆ ℝ ) |
367 |
238
|
snssd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → { ( 𝐸 ‘ 𝑋 ) } ⊆ ℝ ) |
368 |
366 367
|
unssd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ⊆ ℝ ) |
369 |
|
eqid |
⊢ ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) ) |
370 |
236 369
|
rerest |
⊢ ( ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ⊆ ℝ → ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) = ( ( topGen ‘ ran (,) ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) |
371 |
368 370
|
syl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) = ( ( topGen ‘ ran (,) ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) |
372 |
260 363 371
|
3eltr4d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ∈ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) |
373 |
|
isopn3i |
⊢ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ∈ Top ∧ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ∈ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) → ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) |
374 |
253 372 373
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) ‘ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) = ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) ) |
375 |
246 374
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝐸 ‘ 𝑋 ) ) = ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) ‘ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) |
376 |
243 375
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) ‘ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ∪ { ( 𝐸 ‘ 𝑋 ) } ) ) ) |
377 |
198 232 235 236 237 376
|
limcres |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝐹 ↾ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) = ( ( 𝐹 ↾ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) |
378 |
232
|
resabs1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) = ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) ) |
379 |
378
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝐹 ↾ ( ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ∩ 𝐷 ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) |
380 |
191 377 379
|
3eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) |
381 |
186
|
feq2d |
⊢ ( 𝜑 → ( 𝐹 : dom 𝐹 ⟶ ℂ ↔ 𝐹 : 𝐷 ⟶ ℂ ) ) |
382 |
194 381
|
mpbird |
⊢ ( 𝜑 → 𝐹 : dom 𝐹 ⟶ ℂ ) |
383 |
382
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) → 𝐹 : dom 𝐹 ⟶ ℂ ) |
384 |
383
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) → 𝐹 : dom 𝐹 ⟶ ℂ ) |
385 |
|
ioosscn |
⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ⊆ ℂ |
386 |
385
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ⊆ ℂ ) |
387 |
186
|
eqcomd |
⊢ ( 𝜑 → 𝐷 = dom 𝐹 ) |
388 |
387
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐷 = dom 𝐹 ) |
389 |
231 388
|
sseqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ⊆ dom 𝐹 ) |
390 |
389
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ⊆ dom 𝐹 ) |
391 |
15
|
a1i |
⊢ ( 𝜑 → 𝑍 = ( 𝑥 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
392 |
|
oveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝐵 − 𝑥 ) = ( 𝐵 − 𝑋 ) ) |
393 |
392
|
oveq1d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝐵 − 𝑥 ) / 𝑇 ) = ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) |
394 |
393
|
fveq2d |
⊢ ( 𝑥 = 𝑋 → ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) = ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ) |
395 |
394
|
oveq1d |
⊢ ( 𝑥 = 𝑋 → ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
396 |
395
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
397 |
2 14
|
resubcld |
⊢ ( 𝜑 → ( 𝐵 − 𝑋 ) ∈ ℝ ) |
398 |
2 1
|
resubcld |
⊢ ( 𝜑 → ( 𝐵 − 𝐴 ) ∈ ℝ ) |
399 |
5 398
|
eqeltrid |
⊢ ( 𝜑 → 𝑇 ∈ ℝ ) |
400 |
1 2
|
posdifd |
⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ 0 < ( 𝐵 − 𝐴 ) ) ) |
401 |
3 400
|
mpbid |
⊢ ( 𝜑 → 0 < ( 𝐵 − 𝐴 ) ) |
402 |
5
|
eqcomi |
⊢ ( 𝐵 − 𝐴 ) = 𝑇 |
403 |
402
|
a1i |
⊢ ( 𝜑 → ( 𝐵 − 𝐴 ) = 𝑇 ) |
404 |
401 403
|
breqtrd |
⊢ ( 𝜑 → 0 < 𝑇 ) |
405 |
404
|
gt0ne0d |
⊢ ( 𝜑 → 𝑇 ≠ 0 ) |
406 |
397 399 405
|
redivcld |
⊢ ( 𝜑 → ( ( 𝐵 − 𝑋 ) / 𝑇 ) ∈ ℝ ) |
407 |
406
|
flcld |
⊢ ( 𝜑 → ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) |
408 |
407
|
zred |
⊢ ( 𝜑 → ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℝ ) |
409 |
408 399
|
remulcld |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ∈ ℝ ) |
410 |
391 396 14 409
|
fvmptd |
⊢ ( 𝜑 → ( 𝑍 ‘ 𝑋 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
411 |
410 409
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑍 ‘ 𝑋 ) ∈ ℝ ) |
412 |
411
|
recnd |
⊢ ( 𝜑 → ( 𝑍 ‘ 𝑋 ) ∈ ℂ ) |
413 |
412
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℂ ) |
414 |
413
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℂ ) |
415 |
414
|
negcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) → - ( 𝑍 ‘ 𝑋 ) ∈ ℂ ) |
416 |
|
eqid |
⊢ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) 𝑧 = ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) } = { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) 𝑧 = ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) } |
417 |
|
ioosscn |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ⊆ ℂ |
418 |
417
|
sseli |
⊢ ( 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) → 𝑦 ∈ ℂ ) |
419 |
418
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 ∈ ℂ ) |
420 |
412
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℂ ) |
421 |
419 420
|
pncand |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) − ( 𝑍 ‘ 𝑋 ) ) = 𝑦 ) |
422 |
421
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) |
423 |
422
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) |
424 |
410
|
oveq2d |
⊢ ( 𝜑 → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) − ( 𝑍 ‘ 𝑋 ) ) = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) − ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
425 |
424
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) − ( 𝑍 ‘ 𝑋 ) ) = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) − ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
426 |
419 420
|
addcld |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ ℂ ) |
427 |
409
|
recnd |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ∈ ℂ ) |
428 |
427
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ∈ ℂ ) |
429 |
426 428
|
negsubd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + - ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) − ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
430 |
407
|
zcnd |
⊢ ( 𝜑 → ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℂ ) |
431 |
399
|
recnd |
⊢ ( 𝜑 → 𝑇 ∈ ℂ ) |
432 |
430 431
|
mulneg1d |
⊢ ( 𝜑 → ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) = - ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
433 |
432
|
eqcomd |
⊢ ( 𝜑 → - ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) = ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
434 |
433
|
oveq2d |
⊢ ( 𝜑 → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + - ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
435 |
434
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + - ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
436 |
425 429 435
|
3eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) − ( 𝑍 ‘ 𝑋 ) ) = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
437 |
436
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) − ( 𝑍 ‘ 𝑋 ) ) = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
438 |
407
|
znegcld |
⊢ ( 𝜑 → - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) |
439 |
438
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) |
440 |
439
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) |
441 |
|
simpl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝜑 ) |
442 |
231
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ⊆ 𝐷 ) |
443 |
205
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
444 |
139
|
rexrd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
445 |
444
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
446 |
|
elioore |
⊢ ( 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) → 𝑦 ∈ ℝ ) |
447 |
446
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 ∈ ℝ ) |
448 |
411
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℝ ) |
449 |
447 448
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ) |
450 |
449
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ) |
451 |
411
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℝ ) |
452 |
204 451
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ) |
453 |
452
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ) |
454 |
453
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ) |
455 |
14
|
rexrd |
⊢ ( 𝜑 → 𝑋 ∈ ℝ* ) |
456 |
455
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑋 ∈ ℝ* ) |
457 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) |
458 |
|
ioogtlb |
⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ∧ 𝑋 ∈ ℝ* ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑦 ) |
459 |
454 456 457 458
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑦 ) |
460 |
204
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
461 |
451
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℝ ) |
462 |
446
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 ∈ ℝ ) |
463 |
460 461 462
|
ltsubaddd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑦 ↔ ( 𝑄 ‘ 𝑖 ) < ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ) ) |
464 |
459 463
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ) |
465 |
14
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑋 ∈ ℝ ) |
466 |
|
iooltub |
⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ∧ 𝑋 ∈ ℝ* ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 < 𝑋 ) |
467 |
454 456 457 466
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 < 𝑋 ) |
468 |
462 465 461 467
|
ltadd1dd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) < ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) |
469 |
16
|
a1i |
⊢ ( 𝜑 → 𝐸 = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) ) ) |
470 |
|
id |
⊢ ( 𝑥 = 𝑋 → 𝑥 = 𝑋 ) |
471 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝑍 ‘ 𝑥 ) = ( 𝑍 ‘ 𝑋 ) ) |
472 |
470 471
|
oveq12d |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) = ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) |
473 |
472
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) = ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) |
474 |
14 411
|
readdcld |
⊢ ( 𝜑 → ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ) |
475 |
469 473 14 474
|
fvmptd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) |
476 |
475
|
eqcomd |
⊢ ( 𝜑 → ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) = ( 𝐸 ‘ 𝑋 ) ) |
477 |
476
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) = ( 𝐸 ‘ 𝑋 ) ) |
478 |
468 477
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) < ( 𝐸 ‘ 𝑋 ) ) |
479 |
443 445 450 464 478
|
eliood |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) |
480 |
479
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) |
481 |
442 480
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ) |
482 |
441 481 440
|
3jca |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝜑 ∧ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ∧ - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) ) |
483 |
|
eleq1 |
⊢ ( 𝑘 = - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( 𝑘 ∈ ℤ ↔ - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) ) |
484 |
483
|
3anbi3d |
⊢ ( 𝑘 = - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( ( 𝜑 ∧ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) ↔ ( 𝜑 ∧ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ∧ - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) ) ) |
485 |
|
oveq1 |
⊢ ( 𝑘 = - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( 𝑘 · 𝑇 ) = ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
486 |
485
|
oveq2d |
⊢ ( 𝑘 = - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( 𝑘 · 𝑇 ) ) = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
487 |
486
|
eleq1d |
⊢ ( 𝑘 = - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ↔ ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ∈ 𝐷 ) ) |
488 |
484 487
|
imbi12d |
⊢ ( 𝑘 = - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( ( ( 𝜑 ∧ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ↔ ( ( 𝜑 ∧ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ∧ - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ∈ 𝐷 ) ) ) |
489 |
|
ovex |
⊢ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ V |
490 |
|
eleq1 |
⊢ ( 𝑥 = ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) → ( 𝑥 ∈ 𝐷 ↔ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ) ) |
491 |
490
|
3anbi2d |
⊢ ( 𝑥 = ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) → ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) ↔ ( 𝜑 ∧ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) ) ) |
492 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) → ( 𝑥 + ( 𝑘 · 𝑇 ) ) = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( 𝑘 · 𝑇 ) ) ) |
493 |
492
|
eleq1d |
⊢ ( 𝑥 = ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) → ( ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ↔ ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ) |
494 |
491 493
|
imbi12d |
⊢ ( 𝑥 = ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ↔ ( ( 𝜑 ∧ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ) ) |
495 |
489 494 10
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
496 |
488 495
|
vtoclg |
⊢ ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ → ( ( 𝜑 ∧ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ∧ - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ∈ 𝐷 ) ) |
497 |
440 482 496
|
sylc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ∈ 𝐷 ) |
498 |
437 497
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) − ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ) |
499 |
423 498
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 ∈ 𝐷 ) |
500 |
499
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ∀ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) 𝑦 ∈ 𝐷 ) |
501 |
|
dfss3 |
⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ⊆ 𝐷 ↔ ∀ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) 𝑦 ∈ 𝐷 ) |
502 |
500 501
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ⊆ 𝐷 ) |
503 |
204
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℂ ) |
504 |
412
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℂ ) |
505 |
503 504
|
negsubd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) + - ( 𝑍 ‘ 𝑋 ) ) = ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ) |
506 |
505
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) = ( ( 𝑄 ‘ 𝑖 ) + - ( 𝑍 ‘ 𝑋 ) ) ) |
507 |
475
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝐸 ‘ 𝑋 ) + - ( 𝑍 ‘ 𝑋 ) ) = ( ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) + - ( 𝑍 ‘ 𝑋 ) ) ) |
508 |
474
|
recnd |
⊢ ( 𝜑 → ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ∈ ℂ ) |
509 |
508 412
|
negsubd |
⊢ ( 𝜑 → ( ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) + - ( 𝑍 ‘ 𝑋 ) ) = ( ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) |
510 |
14
|
recnd |
⊢ ( 𝜑 → 𝑋 ∈ ℂ ) |
511 |
510 412
|
pncand |
⊢ ( 𝜑 → ( ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) − ( 𝑍 ‘ 𝑋 ) ) = 𝑋 ) |
512 |
507 509 511
|
3eqtrrd |
⊢ ( 𝜑 → 𝑋 = ( ( 𝐸 ‘ 𝑋 ) + - ( 𝑍 ‘ 𝑋 ) ) ) |
513 |
512
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑋 = ( ( 𝐸 ‘ 𝑋 ) + - ( 𝑍 ‘ 𝑋 ) ) ) |
514 |
506 513
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) = ( ( ( 𝑄 ‘ 𝑖 ) + - ( 𝑍 ‘ 𝑋 ) ) (,) ( ( 𝐸 ‘ 𝑋 ) + - ( 𝑍 ‘ 𝑋 ) ) ) ) |
515 |
451
|
renegcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → - ( 𝑍 ‘ 𝑋 ) ∈ ℝ ) |
516 |
204 278 515
|
iooshift |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) + - ( 𝑍 ‘ 𝑋 ) ) (,) ( ( 𝐸 ‘ 𝑋 ) + - ( 𝑍 ‘ 𝑋 ) ) ) = { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) 𝑧 = ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) } ) |
517 |
514 516
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) 𝑧 = ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) } = ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) |
518 |
517
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) 𝑧 = ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) } = ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) |
519 |
186
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → dom 𝐹 = 𝐷 ) |
520 |
502 518 519
|
3sstr4d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) 𝑧 = ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) } ⊆ dom 𝐹 ) |
521 |
520
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) 𝑧 = ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) } ⊆ dom 𝐹 ) |
522 |
410
|
negeqd |
⊢ ( 𝜑 → - ( 𝑍 ‘ 𝑋 ) = - ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
523 |
522 433
|
eqtrd |
⊢ ( 𝜑 → - ( 𝑍 ‘ 𝑋 ) = ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
524 |
523
|
oveq2d |
⊢ ( 𝜑 → ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) = ( 𝑥 + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
525 |
524
|
fveq2d |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) ) = ( 𝐹 ‘ ( 𝑥 + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) ) |
526 |
525
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝐹 ‘ ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) ) = ( 𝐹 ‘ ( 𝑥 + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) ) |
527 |
526
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝐹 ‘ ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) ) = ( 𝐹 ‘ ( 𝑥 + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) ) |
528 |
438
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) → - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) |
529 |
528
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) → - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) |
530 |
|
simpl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) → 𝜑 ) |
531 |
231
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) → 𝑥 ∈ 𝐷 ) |
532 |
530 531 529
|
3jca |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) ) |
533 |
483
|
3anbi3d |
⊢ ( 𝑘 = - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) ↔ ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) ) ) |
534 |
485
|
oveq2d |
⊢ ( 𝑘 = - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( 𝑥 + ( 𝑘 · 𝑇 ) ) = ( 𝑥 + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
535 |
534
|
fveq2d |
⊢ ( 𝑘 = - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ ( 𝑥 + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) ) |
536 |
535
|
eqeq1d |
⊢ ( 𝑘 = - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( ( 𝐹 ‘ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ ( 𝑥 + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) ) |
537 |
533 536
|
imbi12d |
⊢ ( 𝑘 = - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) ↔ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) → ( 𝐹 ‘ ( 𝑥 + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) |
538 |
537 11
|
vtoclg |
⊢ ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ → ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) → ( 𝐹 ‘ ( 𝑥 + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) ) |
539 |
529 532 538
|
sylc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝐹 ‘ ( 𝑥 + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
540 |
527 539
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝐹 ‘ ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
541 |
540
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) ∧ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) → ( 𝐹 ‘ ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
542 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) → 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) |
543 |
384 386 390 415 416 521 541 542
|
limcperiod |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) → 𝑦 ∈ ( ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) 𝑧 = ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) } ) limℂ ( ( 𝐸 ‘ 𝑋 ) + - ( 𝑍 ‘ 𝑋 ) ) ) ) |
544 |
517
|
reseq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) 𝑧 = ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) } ) = ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) ) |
545 |
513
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐸 ‘ 𝑋 ) + - ( 𝑍 ‘ 𝑋 ) ) = 𝑋 ) |
546 |
544 545
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) 𝑧 = ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) } ) limℂ ( ( 𝐸 ‘ 𝑋 ) + - ( 𝑍 ‘ 𝑋 ) ) ) = ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) |
547 |
546
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) 𝑧 = ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) } ) limℂ ( ( 𝐸 ‘ 𝑋 ) + - ( 𝑍 ‘ 𝑋 ) ) ) = ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) |
548 |
547
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) → ( ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) 𝑧 = ( 𝑥 + - ( 𝑍 ‘ 𝑋 ) ) } ) limℂ ( ( 𝐸 ‘ 𝑋 ) + - ( 𝑍 ‘ 𝑋 ) ) ) = ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) |
549 |
543 548
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) → 𝑦 ∈ ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) |
550 |
382
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) → 𝐹 : dom 𝐹 ⟶ ℂ ) |
551 |
550
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) → 𝐹 : dom 𝐹 ⟶ ℂ ) |
552 |
417
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ⊆ ℂ ) |
553 |
502 519
|
sseqtrrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ⊆ dom 𝐹 ) |
554 |
553
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ⊆ dom 𝐹 ) |
555 |
412
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℂ ) |
556 |
555
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℂ ) |
557 |
|
eqid |
⊢ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) 𝑧 = ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) } = { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) 𝑧 = ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) } |
558 |
503 504
|
npcand |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) + ( 𝑍 ‘ 𝑋 ) ) = ( 𝑄 ‘ 𝑖 ) ) |
559 |
558
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) = ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) + ( 𝑍 ‘ 𝑋 ) ) ) |
560 |
475
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) |
561 |
559 560
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) = ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) + ( 𝑍 ‘ 𝑋 ) ) (,) ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) ) |
562 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑋 ∈ ℝ ) |
563 |
452 562 451
|
iooshift |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) + ( 𝑍 ‘ 𝑋 ) ) (,) ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) = { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) 𝑧 = ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) } ) |
564 |
561 563
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) 𝑧 = ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) } = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) |
565 |
564
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) 𝑧 = ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) } = ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) |
566 |
231 565 519
|
3sstr4d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) 𝑧 = ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) } ⊆ dom 𝐹 ) |
567 |
566
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) → { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) 𝑧 = ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) } ⊆ dom 𝐹 ) |
568 |
410
|
oveq2d |
⊢ ( 𝜑 → ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) = ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
569 |
568
|
fveq2d |
⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) ) = ( 𝐹 ‘ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) ) |
570 |
569
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) ) = ( 𝐹 ‘ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) ) |
571 |
570
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) ) = ( 𝐹 ‘ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) ) |
572 |
407
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) |
573 |
572
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) |
574 |
|
simpl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝜑 ) |
575 |
502
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑥 ∈ 𝐷 ) |
576 |
574 575 573
|
3jca |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) ) |
577 |
|
eleq1 |
⊢ ( 𝑘 = ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( 𝑘 ∈ ℤ ↔ ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) ) |
578 |
577
|
3anbi3d |
⊢ ( 𝑘 = ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) ↔ ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) ) ) |
579 |
|
oveq1 |
⊢ ( 𝑘 = ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( 𝑘 · 𝑇 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
580 |
579
|
oveq2d |
⊢ ( 𝑘 = ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( 𝑥 + ( 𝑘 · 𝑇 ) ) = ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
581 |
580
|
fveq2d |
⊢ ( 𝑘 = ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) ) |
582 |
581
|
eqeq1d |
⊢ ( 𝑘 = ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( ( 𝐹 ‘ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) ) |
583 |
578 582
|
imbi12d |
⊢ ( 𝑘 = ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑘 · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) ↔ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) → ( 𝐹 ‘ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) |
584 |
583 11
|
vtoclg |
⊢ ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ → ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) → ( 𝐹 ‘ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) ) |
585 |
573 576 584
|
sylc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝐹 ‘ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
586 |
571 585
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
587 |
586
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝐹 ‘ ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) ) = ( 𝐹 ‘ 𝑥 ) ) |
588 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) → 𝑦 ∈ ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) |
589 |
551 552 554 556 557 567 587 588
|
limcperiod |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) → 𝑦 ∈ ( ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) 𝑧 = ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) } ) limℂ ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) ) |
590 |
564
|
reseq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) 𝑧 = ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) } ) = ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) ) |
591 |
476
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) = ( 𝐸 ‘ 𝑋 ) ) |
592 |
590 591
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) 𝑧 = ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) } ) limℂ ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) |
593 |
592
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) 𝑧 = ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) } ) limℂ ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) |
594 |
593
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) → ( ( 𝐹 ↾ { 𝑧 ∈ ℂ ∣ ∃ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) 𝑧 = ( 𝑥 + ( 𝑍 ‘ 𝑋 ) ) } ) limℂ ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) |
595 |
589 594
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) → 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) |
596 |
549 595
|
impbida |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑦 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ↔ 𝑦 ∈ ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) ) |
597 |
596
|
eqrdv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) = ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) |
598 |
|
resindm |
⊢ ( Rel 𝐹 → ( 𝐹 ↾ ( ( -∞ (,) 𝑋 ) ∩ dom 𝐹 ) ) = ( 𝐹 ↾ ( -∞ (,) 𝑋 ) ) ) |
599 |
598
|
eqcomd |
⊢ ( Rel 𝐹 → ( 𝐹 ↾ ( -∞ (,) 𝑋 ) ) = ( 𝐹 ↾ ( ( -∞ (,) 𝑋 ) ∩ dom 𝐹 ) ) ) |
600 |
181 599
|
syl |
⊢ ( 𝜑 → ( 𝐹 ↾ ( -∞ (,) 𝑋 ) ) = ( 𝐹 ↾ ( ( -∞ (,) 𝑋 ) ∩ dom 𝐹 ) ) ) |
601 |
186
|
ineq2d |
⊢ ( 𝜑 → ( ( -∞ (,) 𝑋 ) ∩ dom 𝐹 ) = ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ) |
602 |
601
|
reseq2d |
⊢ ( 𝜑 → ( 𝐹 ↾ ( ( -∞ (,) 𝑋 ) ∩ dom 𝐹 ) ) = ( 𝐹 ↾ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ) ) |
603 |
600 602
|
eqtrd |
⊢ ( 𝜑 → ( 𝐹 ↾ ( -∞ (,) 𝑋 ) ) = ( 𝐹 ↾ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ) ) |
604 |
603
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝐹 ↾ ( -∞ (,) 𝑋 ) ) limℂ 𝑋 ) = ( ( 𝐹 ↾ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ) limℂ 𝑋 ) ) |
605 |
604
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( -∞ (,) 𝑋 ) ) limℂ 𝑋 ) = ( ( 𝐹 ↾ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ) limℂ 𝑋 ) ) |
606 |
|
inss2 |
⊢ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ⊆ 𝐷 |
607 |
606
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ⊆ 𝐷 ) |
608 |
195 607
|
fssresd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ↾ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ) : ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ⟶ ℂ ) |
609 |
452
|
mnfltd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → -∞ < ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ) |
610 |
200 453 609
|
xrltled |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → -∞ ≤ ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ) |
611 |
|
iooss1 |
⊢ ( ( -∞ ∈ ℝ* ∧ -∞ ≤ ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ⊆ ( -∞ (,) 𝑋 ) ) |
612 |
199 610 611
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ⊆ ( -∞ (,) 𝑋 ) ) |
613 |
612
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ⊆ ( -∞ (,) 𝑋 ) ) |
614 |
613 502
|
ssind |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ⊆ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ) |
615 |
606 234
|
sstrid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ⊆ ℂ ) |
616 |
|
eqid |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) = ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
617 |
453
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ) |
618 |
455
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑋 ∈ ℝ* ) |
619 |
475
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) |
620 |
241 619
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) |
621 |
411
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℝ ) |
622 |
14
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑋 ∈ ℝ ) |
623 |
216 621 622
|
ltsubaddd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑋 ↔ ( 𝑄 ‘ 𝑖 ) < ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) ) |
624 |
620 623
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑋 ) |
625 |
14
|
leidd |
⊢ ( 𝜑 → 𝑋 ≤ 𝑋 ) |
626 |
625
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑋 ≤ 𝑋 ) |
627 |
617 618 618 624 626
|
eliocd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑋 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) |
628 |
|
ioounsn |
⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ∧ 𝑋 ∈ ℝ* ∧ ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑋 ) → ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ∪ { 𝑋 } ) = ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) |
629 |
617 618 624 628
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ∪ { 𝑋 } ) = ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) |
630 |
629
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) ‘ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ∪ { 𝑋 } ) ) = ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) ‘ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) ) |
631 |
|
ovex |
⊢ ( -∞ (,) 𝑋 ) ∈ V |
632 |
631
|
inex1 |
⊢ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∈ V |
633 |
|
snex |
⊢ { 𝑋 } ∈ V |
634 |
632 633
|
unex |
⊢ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ∈ V |
635 |
|
resttop |
⊢ ( ( ( TopOpen ‘ ℂfld ) ∈ Top ∧ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ∈ V ) → ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∈ Top ) |
636 |
247 634 635
|
mp2an |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∈ Top |
637 |
634
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ∈ V ) |
638 |
|
iooretop |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∈ ( topGen ‘ ran (,) ) |
639 |
638
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∈ ( topGen ‘ ran (,) ) ) |
640 |
|
elrestr |
⊢ ( ( ( topGen ‘ ran (,) ) ∈ Top ∧ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ∈ V ∧ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∈ ( topGen ‘ ran (,) ) ) → ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∈ ( ( topGen ‘ ran (,) ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) |
641 |
255 637 639 640
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∈ ( ( topGen ‘ ran (,) ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) |
642 |
453
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ) |
643 |
262
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) → +∞ ∈ ℝ* ) |
644 |
14
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) → 𝑋 ∈ ℝ ) |
645 |
|
iocssre |
⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ∧ 𝑋 ∈ ℝ ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ⊆ ℝ ) |
646 |
642 644 645
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ⊆ ℝ ) |
647 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) → 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) |
648 |
646 647
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) → 𝑥 ∈ ℝ ) |
649 |
455
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) → 𝑋 ∈ ℝ* ) |
650 |
|
iocgtlb |
⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ∧ 𝑋 ∈ ℝ* ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑥 ) |
651 |
642 649 647 650
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑥 ) |
652 |
648
|
ltpnfd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) → 𝑥 < +∞ ) |
653 |
642 643 648 651 652
|
eliood |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) → 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ) |
654 |
653
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) → 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ) |
655 |
|
eqvisset |
⊢ ( 𝑥 = 𝑋 → 𝑋 ∈ V ) |
656 |
|
snidg |
⊢ ( 𝑋 ∈ V → 𝑋 ∈ { 𝑋 } ) |
657 |
655 656
|
syl |
⊢ ( 𝑥 = 𝑋 → 𝑋 ∈ { 𝑋 } ) |
658 |
470 657
|
eqeltrd |
⊢ ( 𝑥 = 𝑋 → 𝑥 ∈ { 𝑋 } ) |
659 |
|
elun2 |
⊢ ( 𝑥 ∈ { 𝑋 } → 𝑥 ∈ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
660 |
658 659
|
syl |
⊢ ( 𝑥 = 𝑋 → 𝑥 ∈ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
661 |
660
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) ∧ 𝑥 = 𝑋 ) → 𝑥 ∈ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
662 |
|
simpll |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) ∧ ¬ 𝑥 = 𝑋 ) → ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
663 |
642
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) ∧ ¬ 𝑥 = 𝑋 ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ) |
664 |
455
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑋 ∈ ℝ* ) |
665 |
648
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 ∈ ℝ ) |
666 |
651
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) ∧ ¬ 𝑥 = 𝑋 ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑥 ) |
667 |
14
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑋 ∈ ℝ ) |
668 |
|
iocleub |
⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ∧ 𝑋 ∈ ℝ* ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) → 𝑥 ≤ 𝑋 ) |
669 |
642 649 647 668
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) → 𝑥 ≤ 𝑋 ) |
670 |
669
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 ≤ 𝑋 ) |
671 |
470
|
eqcoms |
⊢ ( 𝑋 = 𝑥 → 𝑥 = 𝑋 ) |
672 |
671
|
necon3bi |
⊢ ( ¬ 𝑥 = 𝑋 → 𝑋 ≠ 𝑥 ) |
673 |
672
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑋 ≠ 𝑥 ) |
674 |
665 667 670 673
|
leneltd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 < 𝑋 ) |
675 |
663 664 665 666 674
|
eliood |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) |
676 |
675
|
3adantll3 |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) |
677 |
614
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑥 ∈ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ) |
678 |
|
elun1 |
⊢ ( 𝑥 ∈ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) → 𝑥 ∈ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
679 |
677 678
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑥 ∈ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
680 |
662 676 679
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) ∧ ¬ 𝑥 = 𝑋 ) → 𝑥 ∈ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
681 |
661 680
|
pm2.61dan |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) → 𝑥 ∈ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
682 |
654 681
|
elind |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) → 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) |
683 |
617
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ) |
684 |
618
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → 𝑋 ∈ ℝ* ) |
685 |
|
elinel1 |
⊢ ( 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) → 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ) |
686 |
|
elioore |
⊢ ( 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) → 𝑥 ∈ ℝ ) |
687 |
685 686
|
syl |
⊢ ( 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) → 𝑥 ∈ ℝ ) |
688 |
687
|
rexrd |
⊢ ( 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) → 𝑥 ∈ ℝ* ) |
689 |
688
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → 𝑥 ∈ ℝ* ) |
690 |
453
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ) |
691 |
262
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → +∞ ∈ ℝ* ) |
692 |
685
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ) |
693 |
|
ioogtlb |
⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑥 ) |
694 |
690 691 692 693
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑥 ) |
695 |
694
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑥 ) |
696 |
|
elinel2 |
⊢ ( 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) → 𝑥 ∈ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) |
697 |
|
elsni |
⊢ ( 𝑥 ∈ { 𝑋 } → 𝑥 = 𝑋 ) |
698 |
697
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ { 𝑋 } ) → 𝑥 = 𝑋 ) |
699 |
625
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ { 𝑋 } ) → 𝑋 ≤ 𝑋 ) |
700 |
698 699
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ { 𝑋 } ) → 𝑥 ≤ 𝑋 ) |
701 |
700
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∧ 𝑥 ∈ { 𝑋 } ) → 𝑥 ≤ 𝑋 ) |
702 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∧ ¬ 𝑥 ∈ { 𝑋 } ) → 𝜑 ) |
703 |
|
elunnel2 |
⊢ ( ( 𝑥 ∈ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ∧ ¬ 𝑥 ∈ { 𝑋 } ) → 𝑥 ∈ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ) |
704 |
703
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∧ ¬ 𝑥 ∈ { 𝑋 } ) → 𝑥 ∈ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ) |
705 |
|
elinel1 |
⊢ ( 𝑥 ∈ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) → 𝑥 ∈ ( -∞ (,) 𝑋 ) ) |
706 |
704 705
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∧ ¬ 𝑥 ∈ { 𝑋 } ) → 𝑥 ∈ ( -∞ (,) 𝑋 ) ) |
707 |
|
elioore |
⊢ ( 𝑥 ∈ ( -∞ (,) 𝑋 ) → 𝑥 ∈ ℝ ) |
708 |
707
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( -∞ (,) 𝑋 ) ) → 𝑥 ∈ ℝ ) |
709 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( -∞ (,) 𝑋 ) ) → 𝑋 ∈ ℝ ) |
710 |
199
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( -∞ (,) 𝑋 ) ) → -∞ ∈ ℝ* ) |
711 |
455
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( -∞ (,) 𝑋 ) ) → 𝑋 ∈ ℝ* ) |
712 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( -∞ (,) 𝑋 ) ) → 𝑥 ∈ ( -∞ (,) 𝑋 ) ) |
713 |
|
iooltub |
⊢ ( ( -∞ ∈ ℝ* ∧ 𝑋 ∈ ℝ* ∧ 𝑥 ∈ ( -∞ (,) 𝑋 ) ) → 𝑥 < 𝑋 ) |
714 |
710 711 712 713
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( -∞ (,) 𝑋 ) ) → 𝑥 < 𝑋 ) |
715 |
708 709 714
|
ltled |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( -∞ (,) 𝑋 ) ) → 𝑥 ≤ 𝑋 ) |
716 |
702 706 715
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∧ ¬ 𝑥 ∈ { 𝑋 } ) → 𝑥 ≤ 𝑋 ) |
717 |
701 716
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) → 𝑥 ≤ 𝑋 ) |
718 |
696 717
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → 𝑥 ≤ 𝑋 ) |
719 |
718
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → 𝑥 ≤ 𝑋 ) |
720 |
683 684 689 695 719
|
eliocd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) |
721 |
682 720
|
impbida |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑥 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ↔ 𝑥 ∈ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) ) |
722 |
721
|
eqrdv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) = ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) +∞ ) ∩ ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) |
723 |
606 8
|
sstrid |
⊢ ( 𝜑 → ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ⊆ ℝ ) |
724 |
14
|
snssd |
⊢ ( 𝜑 → { 𝑋 } ⊆ ℝ ) |
725 |
723 724
|
unssd |
⊢ ( 𝜑 → ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ⊆ ℝ ) |
726 |
725
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ⊆ ℝ ) |
727 |
236 369
|
rerest |
⊢ ( ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ⊆ ℝ → ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) = ( ( topGen ‘ ran (,) ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) |
728 |
726 727
|
syl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) = ( ( topGen ‘ ran (,) ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) |
729 |
641 722 728
|
3eltr4d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ∈ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) |
730 |
|
isopn3i |
⊢ ( ( ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ∈ Top ∧ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ∈ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) → ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) ‘ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) = ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) |
731 |
636 729 730
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) ‘ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) = ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) ) |
732 |
630 731
|
eqtr2d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,] 𝑋 ) = ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) ‘ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ∪ { 𝑋 } ) ) ) |
733 |
627 732
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑋 ∈ ( ( int ‘ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ∪ { 𝑋 } ) ) ) ‘ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ∪ { 𝑋 } ) ) ) |
734 |
608 614 615 236 616 733
|
limcres |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝐹 ↾ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ) ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) = ( ( 𝐹 ↾ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ) limℂ 𝑋 ) ) |
735 |
734
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ) limℂ 𝑋 ) = ( ( ( 𝐹 ↾ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ) ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) |
736 |
614
|
resabs1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ) ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) = ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) ) |
737 |
736
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝐹 ↾ ( ( -∞ (,) 𝑋 ) ∩ 𝐷 ) ) ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) = ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) ) |
738 |
605 735 737
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) limℂ 𝑋 ) = ( ( 𝐹 ↾ ( -∞ (,) 𝑋 ) ) limℂ 𝑋 ) ) |
739 |
380 597 738
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( -∞ (,) 𝑋 ) ) limℂ 𝑋 ) = ( ( 𝐹 ↾ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) |
740 |
739
|
rexlimdv3a |
⊢ ( 𝜑 → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( -∞ (,) 𝑋 ) ) limℂ 𝑋 ) = ( ( 𝐹 ↾ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) ) |
741 |
179 740
|
mpd |
⊢ ( 𝜑 → ( ( 𝐹 ↾ ( -∞ (,) 𝑋 ) ) limℂ 𝑋 ) = ( ( 𝐹 ↾ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) |
742 |
126
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
743 |
12
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∈ ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) –cn→ ℂ ) ) |
744 |
13
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝐿 ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) limℂ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
745 |
|
eqid |
⊢ if ( ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝐸 ‘ 𝑋 ) ) ) = if ( ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝐸 ‘ 𝑋 ) ) ) |
746 |
|
eqid |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) = ( ( TopOpen ‘ ℂfld ) ↾t ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ∪ { ( 𝑄 ‘ ( 𝑖 + 1 ) ) } ) ) |
747 |
216 214 742 743 744 216 238 241 222 745 746
|
fourierdlem33 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → if ( ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝐸 ‘ 𝑋 ) ) ) ∈ ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) |
748 |
222
|
resabs1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) = ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) ) |
749 |
748
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) = ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) |
750 |
747 749
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → if ( ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝐸 ‘ 𝑋 ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ) |
751 |
|
ne0i |
⊢ ( if ( ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) , 𝐿 , ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ‘ ( 𝐸 ‘ 𝑋 ) ) ) ∈ ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ≠ ∅ ) |
752 |
750 751
|
syl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ≠ ∅ ) |
753 |
380 752
|
eqnetrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐹 ↾ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ≠ ∅ ) |
754 |
753
|
rexlimdv3a |
⊢ ( 𝜑 → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( ( 𝐹 ↾ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ≠ ∅ ) ) |
755 |
179 754
|
mpd |
⊢ ( 𝜑 → ( ( 𝐹 ↾ ( -∞ (,) ( 𝐸 ‘ 𝑋 ) ) ) limℂ ( 𝐸 ‘ 𝑋 ) ) ≠ ∅ ) |
756 |
741 755
|
eqnetrd |
⊢ ( 𝜑 → ( ( 𝐹 ↾ ( -∞ (,) 𝑋 ) ) limℂ 𝑋 ) ≠ ∅ ) |