Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem41.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
fourierdlem41.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
fourierdlem41.altb |
⊢ ( 𝜑 → 𝐴 < 𝐵 ) |
4 |
|
fourierdlem41.t |
⊢ 𝑇 = ( 𝐵 − 𝐴 ) |
5 |
|
fourierdlem41.dper |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
6 |
|
fourierdlem41.x |
⊢ ( 𝜑 → 𝑋 ∈ ℝ ) |
7 |
|
fourierdlem41.z |
⊢ 𝑍 = ( 𝑥 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) |
8 |
|
fourierdlem41.e |
⊢ 𝐸 = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) ) |
9 |
|
fourierdlem41.p |
⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) |
10 |
|
fourierdlem41.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
11 |
|
fourierdlem41.q |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ) |
12 |
|
fourierdlem41.qssd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ 𝐷 ) |
13 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) |
14 |
9
|
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 ) ) ) ) ) ) |
15 |
10 14
|
syl |
⊢ ( 𝜑 → ( 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
16 |
11 15
|
mpbid |
⊢ ( 𝜑 → ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
17 |
16
|
simpld |
⊢ ( 𝜑 → 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ) |
18 |
|
elmapi |
⊢ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
19 |
|
ffn |
⊢ ( 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ → 𝑄 Fn ( 0 ... 𝑀 ) ) |
20 |
17 18 19
|
3syl |
⊢ ( 𝜑 → 𝑄 Fn ( 0 ... 𝑀 ) ) |
21 |
20
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → 𝑄 Fn ( 0 ... 𝑀 ) ) |
22 |
|
fvelrnb |
⊢ ( 𝑄 Fn ( 0 ... 𝑀 ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ) |
23 |
21 22
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ) |
24 |
13 23
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) |
25 |
|
0zd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 0 ∈ ℤ ) |
26 |
|
elfzelz |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ∈ ℤ ) |
27 |
26
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑗 ∈ ℤ ) |
28 |
|
1zzd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 1 ∈ ℤ ) |
29 |
27 28
|
zsubcld |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ∈ ℤ ) |
30 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 0 < 𝑗 ) → 𝜑 ) |
31 |
|
elfzle1 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 0 ≤ 𝑗 ) |
32 |
31
|
anim1i |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ¬ 0 < 𝑗 ) → ( 0 ≤ 𝑗 ∧ ¬ 0 < 𝑗 ) ) |
33 |
|
0red |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ¬ 0 < 𝑗 ) → 0 ∈ ℝ ) |
34 |
26
|
zred |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ∈ ℝ ) |
35 |
34
|
adantr |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ¬ 0 < 𝑗 ) → 𝑗 ∈ ℝ ) |
36 |
33 35
|
eqleltd |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ¬ 0 < 𝑗 ) → ( 0 = 𝑗 ↔ ( 0 ≤ 𝑗 ∧ ¬ 0 < 𝑗 ) ) ) |
37 |
32 36
|
mpbird |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ¬ 0 < 𝑗 ) → 0 = 𝑗 ) |
38 |
37
|
eqcomd |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ¬ 0 < 𝑗 ) → 𝑗 = 0 ) |
39 |
38
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 0 < 𝑗 ) → 𝑗 = 0 ) |
40 |
|
fveq2 |
⊢ ( 𝑗 = 0 → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ 0 ) ) |
41 |
16
|
simprld |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ) |
42 |
41
|
simpld |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) = 𝐴 ) |
43 |
40 42
|
sylan9eqr |
⊢ ( ( 𝜑 ∧ 𝑗 = 0 ) → ( 𝑄 ‘ 𝑗 ) = 𝐴 ) |
44 |
30 39 43
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 0 < 𝑗 ) → ( 𝑄 ‘ 𝑗 ) = 𝐴 ) |
45 |
44
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ ¬ 0 < 𝑗 ) → ( 𝑄 ‘ 𝑗 ) = 𝐴 ) |
46 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) |
47 |
1
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
48 |
2
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
49 |
|
eqid |
⊢ ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
50 |
1 2 3 4 49
|
fourierdlem4 |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) : ℝ ⟶ ( 𝐴 (,] 𝐵 ) ) |
51 |
8
|
a1i |
⊢ ( 𝜑 → 𝐸 = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) ) ) |
52 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → 𝑥 ∈ ℝ ) |
53 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → 𝐵 ∈ ℝ ) |
54 |
53 52
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝐵 − 𝑥 ) ∈ ℝ ) |
55 |
2 1
|
resubcld |
⊢ ( 𝜑 → ( 𝐵 − 𝐴 ) ∈ ℝ ) |
56 |
4 55
|
eqeltrid |
⊢ ( 𝜑 → 𝑇 ∈ ℝ ) |
57 |
56
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → 𝑇 ∈ ℝ ) |
58 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
59 |
1 2
|
posdifd |
⊢ ( 𝜑 → ( 𝐴 < 𝐵 ↔ 0 < ( 𝐵 − 𝐴 ) ) ) |
60 |
3 59
|
mpbid |
⊢ ( 𝜑 → 0 < ( 𝐵 − 𝐴 ) ) |
61 |
4
|
eqcomi |
⊢ ( 𝐵 − 𝐴 ) = 𝑇 |
62 |
61
|
a1i |
⊢ ( 𝜑 → ( 𝐵 − 𝐴 ) = 𝑇 ) |
63 |
60 62
|
breqtrd |
⊢ ( 𝜑 → 0 < 𝑇 ) |
64 |
58 63
|
gtned |
⊢ ( 𝜑 → 𝑇 ≠ 0 ) |
65 |
64
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → 𝑇 ≠ 0 ) |
66 |
54 57 65
|
redivcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( ( 𝐵 − 𝑥 ) / 𝑇 ) ∈ ℝ ) |
67 |
66
|
flcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) ∈ ℤ ) |
68 |
67
|
zred |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) ∈ ℝ ) |
69 |
68 57
|
remulcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ∈ ℝ ) |
70 |
7
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ ℝ ∧ ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ∈ ℝ ) → ( 𝑍 ‘ 𝑥 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) |
71 |
52 69 70
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝑍 ‘ 𝑥 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) |
72 |
71
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) = ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
73 |
72
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) ) |
74 |
51 73
|
eqtrd |
⊢ ( 𝜑 → 𝐸 = ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) ) |
75 |
74
|
feq1d |
⊢ ( 𝜑 → ( 𝐸 : ℝ ⟶ ( 𝐴 (,] 𝐵 ) ↔ ( 𝑥 ∈ ℝ ↦ ( 𝑥 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) : ℝ ⟶ ( 𝐴 (,] 𝐵 ) ) ) |
76 |
50 75
|
mpbird |
⊢ ( 𝜑 → 𝐸 : ℝ ⟶ ( 𝐴 (,] 𝐵 ) ) |
77 |
76 6
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) |
78 |
|
iocgtlb |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐴 < ( 𝐸 ‘ 𝑋 ) ) |
79 |
47 48 77 78
|
syl3anc |
⊢ ( 𝜑 → 𝐴 < ( 𝐸 ‘ 𝑋 ) ) |
80 |
1 79
|
gtned |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ≠ 𝐴 ) |
81 |
80
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ≠ 𝐴 ) |
82 |
46 81
|
eqnetrd |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ 𝑗 ) ≠ 𝐴 ) |
83 |
82
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ ¬ 0 < 𝑗 ) → ( 𝑄 ‘ 𝑗 ) ≠ 𝐴 ) |
84 |
83
|
3adantl2 |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ ¬ 0 < 𝑗 ) → ( 𝑄 ‘ 𝑗 ) ≠ 𝐴 ) |
85 |
84
|
neneqd |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ ¬ 0 < 𝑗 ) → ¬ ( 𝑄 ‘ 𝑗 ) = 𝐴 ) |
86 |
45 85
|
condan |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 0 < 𝑗 ) |
87 |
|
zltlem1 |
⊢ ( ( 0 ∈ ℤ ∧ 𝑗 ∈ ℤ ) → ( 0 < 𝑗 ↔ 0 ≤ ( 𝑗 − 1 ) ) ) |
88 |
25 27 87
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 0 < 𝑗 ↔ 0 ≤ ( 𝑗 − 1 ) ) ) |
89 |
86 88
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 0 ≤ ( 𝑗 − 1 ) ) |
90 |
|
eluz2 |
⊢ ( ( 𝑗 − 1 ) ∈ ( ℤ≥ ‘ 0 ) ↔ ( 0 ∈ ℤ ∧ ( 𝑗 − 1 ) ∈ ℤ ∧ 0 ≤ ( 𝑗 − 1 ) ) ) |
91 |
25 29 89 90
|
syl3anbrc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ∈ ( ℤ≥ ‘ 0 ) ) |
92 |
|
elfzel2 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑀 ∈ ℤ ) |
93 |
92
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑀 ∈ ℤ ) |
94 |
|
1red |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 1 ∈ ℝ ) |
95 |
34 94
|
resubcld |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 − 1 ) ∈ ℝ ) |
96 |
92
|
zred |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑀 ∈ ℝ ) |
97 |
34
|
ltm1d |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 − 1 ) < 𝑗 ) |
98 |
|
elfzle2 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ≤ 𝑀 ) |
99 |
95 34 96 97 98
|
ltletrd |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 − 1 ) < 𝑀 ) |
100 |
99
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) < 𝑀 ) |
101 |
|
elfzo2 |
⊢ ( ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ↔ ( ( 𝑗 − 1 ) ∈ ( ℤ≥ ‘ 0 ) ∧ 𝑀 ∈ ℤ ∧ ( 𝑗 − 1 ) < 𝑀 ) ) |
102 |
91 93 100 101
|
syl3anbrc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ) |
103 |
17 18
|
syl |
⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
104 |
103
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
105 |
95 96 99
|
ltled |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 − 1 ) ≤ 𝑀 ) |
106 |
105
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ≤ 𝑀 ) |
107 |
25 93 29 89 106
|
elfzd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑗 − 1 ) ∈ ( 0 ... 𝑀 ) ) |
108 |
104 107
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) ∈ ℝ ) |
109 |
108
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) ∈ ℝ* ) |
110 |
34
|
recnd |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ∈ ℂ ) |
111 |
|
1cnd |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 1 ∈ ℂ ) |
112 |
110 111
|
npcand |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 𝑗 − 1 ) + 1 ) = 𝑗 ) |
113 |
112
|
fveq2d |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) = ( 𝑄 ‘ 𝑗 ) ) |
114 |
113
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) = ( 𝑄 ‘ 𝑗 ) ) |
115 |
103
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ ) |
116 |
115
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ* ) |
117 |
114 116
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ∈ ℝ* ) |
118 |
117
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ∈ ℝ* ) |
119 |
|
id |
⊢ ( 𝑥 = 𝑋 → 𝑥 = 𝑋 ) |
120 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝑍 ‘ 𝑥 ) = ( 𝑍 ‘ 𝑋 ) ) |
121 |
119 120
|
oveq12d |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) = ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) |
122 |
121
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( 𝑥 + ( 𝑍 ‘ 𝑥 ) ) = ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) |
123 |
7
|
a1i |
⊢ ( 𝜑 → 𝑍 = ( 𝑥 ∈ ℝ ↦ ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) ) ) |
124 |
|
oveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝐵 − 𝑥 ) = ( 𝐵 − 𝑋 ) ) |
125 |
124
|
oveq1d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝐵 − 𝑥 ) / 𝑇 ) = ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) |
126 |
125
|
fveq2d |
⊢ ( 𝑥 = 𝑋 → ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) = ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ) |
127 |
126
|
oveq1d |
⊢ ( 𝑥 = 𝑋 → ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
128 |
127
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( ( ⌊ ‘ ( ( 𝐵 − 𝑥 ) / 𝑇 ) ) · 𝑇 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
129 |
2 6
|
resubcld |
⊢ ( 𝜑 → ( 𝐵 − 𝑋 ) ∈ ℝ ) |
130 |
129 56 64
|
redivcld |
⊢ ( 𝜑 → ( ( 𝐵 − 𝑋 ) / 𝑇 ) ∈ ℝ ) |
131 |
130
|
flcld |
⊢ ( 𝜑 → ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) |
132 |
131
|
zred |
⊢ ( 𝜑 → ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℝ ) |
133 |
132 56
|
remulcld |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ∈ ℝ ) |
134 |
123 128 6 133
|
fvmptd |
⊢ ( 𝜑 → ( 𝑍 ‘ 𝑋 ) = ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
135 |
134 133
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑍 ‘ 𝑋 ) ∈ ℝ ) |
136 |
6 135
|
readdcld |
⊢ ( 𝜑 → ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ) |
137 |
51 122 6 136
|
fvmptd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) |
138 |
137 136
|
eqeltrd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
139 |
138
|
rexrd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
140 |
139
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
141 |
|
simp1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝜑 ) |
142 |
|
ovex |
⊢ ( 𝑗 − 1 ) ∈ V |
143 |
|
eleq1 |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↔ ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ) ) |
144 |
143
|
anbi2d |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ↔ ( 𝜑 ∧ ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ) ) ) |
145 |
|
fveq2 |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ ( 𝑗 − 1 ) ) ) |
146 |
|
oveq1 |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( 𝑖 + 1 ) = ( ( 𝑗 − 1 ) + 1 ) ) |
147 |
146
|
fveq2d |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) |
148 |
145 147
|
breq12d |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ↔ ( 𝑄 ‘ ( 𝑗 − 1 ) ) < ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) |
149 |
144 148
|
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 ) ) ) ) ) |
150 |
16
|
simprrd |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
151 |
150
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
152 |
142 149 151
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) < ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) |
153 |
141 102 152
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) < ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) |
154 |
113
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) = ( 𝑄 ‘ 𝑗 ) ) |
155 |
153 154
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) < ( 𝑄 ‘ 𝑗 ) ) |
156 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) |
157 |
155 156
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑗 − 1 ) ) < ( 𝐸 ‘ 𝑋 ) ) |
158 |
138
|
leidd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
159 |
158
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
160 |
46
|
eqcomd |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ 𝑗 ) ) |
161 |
159 160
|
breqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ 𝑗 ) ) |
162 |
161
|
3adant2 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ 𝑗 ) ) |
163 |
112
|
eqcomd |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 = ( ( 𝑗 − 1 ) + 1 ) ) |
164 |
163
|
fveq2d |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) |
165 |
164
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) |
166 |
162 165
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) |
167 |
109 118 140 157 166
|
eliocd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) |
168 |
145 147
|
oveq12d |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) |
169 |
168
|
eleq2d |
⊢ ( 𝑖 = ( 𝑗 − 1 ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) ) |
170 |
169
|
rspcev |
⊢ ( ( ( 𝑗 − 1 ) ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ ( 𝑗 − 1 ) ) (,] ( 𝑄 ‘ ( ( 𝑗 − 1 ) + 1 ) ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
171 |
102 167 170
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
172 |
171
|
3exp |
⊢ ( 𝜑 → ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
173 |
172
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
174 |
173
|
rexlimdv |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
175 |
24 174
|
mpd |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
176 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → 𝑀 ∈ ℕ ) |
177 |
103
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
178 |
|
iocssicc |
⊢ ( ( 𝑄 ‘ 0 ) (,] ( 𝑄 ‘ 𝑀 ) ) ⊆ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) |
179 |
41
|
simprd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) = 𝐵 ) |
180 |
42 179
|
oveq12d |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) (,] ( 𝑄 ‘ 𝑀 ) ) = ( 𝐴 (,] 𝐵 ) ) |
181 |
77 180
|
eleqtrrd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 0 ) (,] ( 𝑄 ‘ 𝑀 ) ) ) |
182 |
178 181
|
sselid |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) |
183 |
182
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) |
184 |
|
simpr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) |
185 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝑄 ‘ 𝑘 ) = ( 𝑄 ‘ 𝑗 ) ) |
186 |
185
|
breq1d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝑄 ‘ 𝑘 ) < ( 𝐸 ‘ 𝑋 ) ↔ ( 𝑄 ‘ 𝑗 ) < ( 𝐸 ‘ 𝑋 ) ) ) |
187 |
186
|
cbvrabv |
⊢ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < ( 𝐸 ‘ 𝑋 ) } = { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) < ( 𝐸 ‘ 𝑋 ) } |
188 |
187
|
supeq1i |
⊢ sup ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < ( 𝐸 ‘ 𝑋 ) } , ℝ , < ) = sup ( { 𝑗 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑗 ) < ( 𝐸 ‘ 𝑋 ) } , ℝ , < ) |
189 |
176 177 183 184 188
|
fourierdlem25 |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
190 |
|
ioossioc |
⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
191 |
190
|
a1i |
⊢ ( ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
192 |
191
|
sseld |
⊢ ( ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
193 |
192
|
reximdva |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
194 |
189 193
|
mpd |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
195 |
175 194
|
pm2.61dan |
⊢ ( 𝜑 → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
196 |
103
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
197 |
|
elfzofz |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
198 |
197
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
199 |
196 198
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
200 |
199
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
201 |
135
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℝ ) |
202 |
200 201
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ) |
203 |
138
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
204 |
200
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
205 |
|
fzofzp1 |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
206 |
205
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
207 |
196 206
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
208 |
207
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
209 |
208
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
210 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
211 |
|
iocgtlb |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝐸 ‘ 𝑋 ) ) |
212 |
204 209 210 211
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝐸 ‘ 𝑋 ) ) |
213 |
200 203 201 212
|
ltsub1dd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < ( ( 𝐸 ‘ 𝑋 ) − ( 𝑍 ‘ 𝑋 ) ) ) |
214 |
137
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝐸 ‘ 𝑋 ) − ( 𝑍 ‘ 𝑋 ) ) = ( ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) |
215 |
6
|
recnd |
⊢ ( 𝜑 → 𝑋 ∈ ℂ ) |
216 |
135
|
recnd |
⊢ ( 𝜑 → ( 𝑍 ‘ 𝑋 ) ∈ ℂ ) |
217 |
215 216
|
pncand |
⊢ ( 𝜑 → ( ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) − ( 𝑍 ‘ 𝑋 ) ) = 𝑋 ) |
218 |
214 217
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝐸 ‘ 𝑋 ) − ( 𝑍 ‘ 𝑋 ) ) = 𝑋 ) |
219 |
218
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐸 ‘ 𝑋 ) − ( 𝑍 ‘ 𝑋 ) ) = 𝑋 ) |
220 |
213 219
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑋 ) |
221 |
|
elioore |
⊢ ( 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) → 𝑦 ∈ ℝ ) |
222 |
134
|
oveq2d |
⊢ ( 𝜑 → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) = ( 𝑦 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
223 |
132
|
recnd |
⊢ ( 𝜑 → ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℂ ) |
224 |
56
|
recnd |
⊢ ( 𝜑 → 𝑇 ∈ ℂ ) |
225 |
223 224
|
mulneg1d |
⊢ ( 𝜑 → ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) = - ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
226 |
222 225
|
oveq12d |
⊢ ( 𝜑 → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) = ( ( 𝑦 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) + - ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
227 |
226
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) = ( ( 𝑦 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) + - ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
228 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ℝ ) |
229 |
133
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ∈ ℝ ) |
230 |
228 229
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑦 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ∈ ℝ ) |
231 |
230
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑦 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ∈ ℂ ) |
232 |
229
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ∈ ℂ ) |
233 |
231 232
|
negsubd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ( 𝑦 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) + - ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) = ( ( 𝑦 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) − ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
234 |
228
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ℂ ) |
235 |
234 232
|
pncand |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ( 𝑦 + ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) − ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) = 𝑦 ) |
236 |
227 233 235
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑦 = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
237 |
221 236
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
238 |
237
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
239 |
|
simpl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝜑 ) |
240 |
12
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ 𝐷 ) |
241 |
240
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ 𝐷 ) |
242 |
204
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
243 |
209
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
244 |
221
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 ∈ ℝ ) |
245 |
135
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℝ ) |
246 |
244 245
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ) |
247 |
246
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ) |
248 |
135
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℝ ) |
249 |
199 248
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ) |
250 |
249
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ) |
251 |
250
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ) |
252 |
6
|
rexrd |
⊢ ( 𝜑 → 𝑋 ∈ ℝ* ) |
253 |
252
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑋 ∈ ℝ* ) |
254 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) |
255 |
|
ioogtlb |
⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ∧ 𝑋 ∈ ℝ* ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑦 ) |
256 |
251 253 254 255
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑦 ) |
257 |
199
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
258 |
135
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℝ ) |
259 |
221
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 ∈ ℝ ) |
260 |
257 258 259
|
ltsubaddd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑦 ↔ ( 𝑄 ‘ 𝑖 ) < ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ) ) |
261 |
256 260
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ) |
262 |
261
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ) |
263 |
239 138
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
264 |
207
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
265 |
264
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
266 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑋 ∈ ℝ ) |
267 |
|
iooltub |
⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ∧ 𝑋 ∈ ℝ* ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 < 𝑋 ) |
268 |
251 253 254 267
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 < 𝑋 ) |
269 |
259 266 258 268
|
ltadd1dd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) < ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) |
270 |
137
|
eqcomd |
⊢ ( 𝜑 → ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) = ( 𝐸 ‘ 𝑋 ) ) |
271 |
270
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) = ( 𝐸 ‘ 𝑋 ) ) |
272 |
269 271
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) < ( 𝐸 ‘ 𝑋 ) ) |
273 |
272
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) < ( 𝐸 ‘ 𝑋 ) ) |
274 |
|
iocleub |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
275 |
204 209 210 274
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
276 |
275
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ≤ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
277 |
247 263 265 273 276
|
ltletrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
278 |
242 243 247 262 277
|
eliood |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
279 |
241 278
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ) |
280 |
239 130
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ( 𝐵 − 𝑋 ) / 𝑇 ) ∈ ℝ ) |
281 |
280
|
flcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) |
282 |
281
|
znegcld |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) |
283 |
|
negex |
⊢ - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ V |
284 |
|
eleq1 |
⊢ ( 𝑘 = - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( 𝑘 ∈ ℤ ↔ - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) ) |
285 |
284
|
3anbi3d |
⊢ ( 𝑘 = - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( ( 𝜑 ∧ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) ↔ ( 𝜑 ∧ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ∧ - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) ) ) |
286 |
|
oveq1 |
⊢ ( 𝑘 = - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( 𝑘 · 𝑇 ) = ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) |
287 |
286
|
oveq2d |
⊢ ( 𝑘 = - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( 𝑘 · 𝑇 ) ) = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
288 |
287
|
eleq1d |
⊢ ( 𝑘 = - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) → ( ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ↔ ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ∈ 𝐷 ) ) |
289 |
285 288
|
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 ) ) ) |
290 |
|
ovex |
⊢ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ V |
291 |
|
eleq1 |
⊢ ( 𝑥 = ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) → ( 𝑥 ∈ 𝐷 ↔ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ) ) |
292 |
291
|
3anbi2d |
⊢ ( 𝑥 = ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) → ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) ↔ ( 𝜑 ∧ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) ) ) |
293 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) → ( 𝑥 + ( 𝑘 · 𝑇 ) ) = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( 𝑘 · 𝑇 ) ) ) |
294 |
293
|
eleq1d |
⊢ ( 𝑥 = ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) → ( ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ↔ ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ) |
295 |
292 294
|
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 ) ) ) |
296 |
290 295 5
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
297 |
283 289 296
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ∧ - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ∈ 𝐷 ) |
298 |
239 279 282 297
|
syl3anc |
⊢ ( ( ( 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 ) |
299 |
238 298
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) → 𝑦 ∈ 𝐷 ) |
300 |
299
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ∀ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) 𝑦 ∈ 𝐷 ) |
301 |
|
dfss3 |
⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ⊆ 𝐷 ↔ ∀ 𝑦 ∈ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) 𝑦 ∈ 𝐷 ) |
302 |
300 301
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ⊆ 𝐷 ) |
303 |
|
breq1 |
⊢ ( 𝑦 = ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) → ( 𝑦 < 𝑋 ↔ ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑋 ) ) |
304 |
|
oveq1 |
⊢ ( 𝑦 = ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) → ( 𝑦 (,) 𝑋 ) = ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ) |
305 |
304
|
sseq1d |
⊢ ( 𝑦 = ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) → ( ( 𝑦 (,) 𝑋 ) ⊆ 𝐷 ↔ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ⊆ 𝐷 ) ) |
306 |
303 305
|
anbi12d |
⊢ ( 𝑦 = ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) → ( ( 𝑦 < 𝑋 ∧ ( 𝑦 (,) 𝑋 ) ⊆ 𝐷 ) ↔ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑋 ∧ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ⊆ 𝐷 ) ) ) |
307 |
306
|
rspcev |
⊢ ( ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ∧ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) < 𝑋 ∧ ( ( ( 𝑄 ‘ 𝑖 ) − ( 𝑍 ‘ 𝑋 ) ) (,) 𝑋 ) ⊆ 𝐷 ) ) → ∃ 𝑦 ∈ ℝ ( 𝑦 < 𝑋 ∧ ( 𝑦 (,) 𝑋 ) ⊆ 𝐷 ) ) |
308 |
202 220 302 307
|
syl12anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ∃ 𝑦 ∈ ℝ ( 𝑦 < 𝑋 ∧ ( 𝑦 (,) 𝑋 ) ⊆ 𝐷 ) ) |
309 |
308
|
3exp |
⊢ ( 𝜑 → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ∃ 𝑦 ∈ ℝ ( 𝑦 < 𝑋 ∧ ( 𝑦 (,) 𝑋 ) ⊆ 𝐷 ) ) ) ) |
310 |
309
|
rexlimdv |
⊢ ( 𝜑 → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,] ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ∃ 𝑦 ∈ ℝ ( 𝑦 < 𝑋 ∧ ( 𝑦 (,) 𝑋 ) ⊆ 𝐷 ) ) ) |
311 |
195 310
|
mpd |
⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ( 𝑦 < 𝑋 ∧ ( 𝑦 (,) 𝑋 ) ⊆ 𝐷 ) ) |
312 |
|
0zd |
⊢ ( 𝜑 → 0 ∈ ℤ ) |
313 |
10
|
nnzd |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
314 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
315 |
|
0le1 |
⊢ 0 ≤ 1 |
316 |
315
|
a1i |
⊢ ( 𝜑 → 0 ≤ 1 ) |
317 |
10
|
nnge1d |
⊢ ( 𝜑 → 1 ≤ 𝑀 ) |
318 |
312 313 314 316 317
|
elfzd |
⊢ ( 𝜑 → 1 ∈ ( 0 ... 𝑀 ) ) |
319 |
103 318
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑄 ‘ 1 ) ∈ ℝ ) |
320 |
135 56
|
resubcld |
⊢ ( 𝜑 → ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ∈ ℝ ) |
321 |
319 320
|
resubcld |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ ℝ ) |
322 |
321
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ ℝ ) |
323 |
1
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
324 |
323 224
|
pncand |
⊢ ( 𝜑 → ( ( 𝐴 + 𝑇 ) − 𝑇 ) = 𝐴 ) |
325 |
324
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( ( 𝐴 + 𝑇 ) − 𝑇 ) = 𝐴 ) |
326 |
4
|
oveq2i |
⊢ ( 𝐴 + 𝑇 ) = ( 𝐴 + ( 𝐵 − 𝐴 ) ) |
327 |
326
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( 𝐴 + 𝑇 ) = ( 𝐴 + ( 𝐵 − 𝐴 ) ) ) |
328 |
2
|
recnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
329 |
323 328
|
pncan3d |
⊢ ( 𝜑 → ( 𝐴 + ( 𝐵 − 𝐴 ) ) = 𝐵 ) |
330 |
329
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( 𝐴 + ( 𝐵 − 𝐴 ) ) = 𝐵 ) |
331 |
|
id |
⊢ ( ( 𝐸 ‘ 𝑋 ) = 𝐵 → ( 𝐸 ‘ 𝑋 ) = 𝐵 ) |
332 |
331
|
eqcomd |
⊢ ( ( 𝐸 ‘ 𝑋 ) = 𝐵 → 𝐵 = ( 𝐸 ‘ 𝑋 ) ) |
333 |
332
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → 𝐵 = ( 𝐸 ‘ 𝑋 ) ) |
334 |
327 330 333
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( 𝐸 ‘ 𝑋 ) = ( 𝐴 + 𝑇 ) ) |
335 |
334
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) = ( ( 𝐴 + 𝑇 ) − 𝑇 ) ) |
336 |
42
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( 𝑄 ‘ 0 ) = 𝐴 ) |
337 |
325 335 336
|
3eqtr4rd |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( 𝑄 ‘ 0 ) = ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) ) |
338 |
337
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( ( 𝑄 ‘ 0 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) = ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) |
339 |
138
|
recnd |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑋 ) ∈ ℂ ) |
340 |
339 216 224
|
nnncan2d |
⊢ ( 𝜑 → ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) = ( ( 𝐸 ‘ 𝑋 ) − ( 𝑍 ‘ 𝑋 ) ) ) |
341 |
340
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( ( ( 𝐸 ‘ 𝑋 ) − 𝑇 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) = ( ( 𝐸 ‘ 𝑋 ) − ( 𝑍 ‘ 𝑋 ) ) ) |
342 |
218
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( ( 𝐸 ‘ 𝑋 ) − ( 𝑍 ‘ 𝑋 ) ) = 𝑋 ) |
343 |
338 341 342
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → 𝑋 = ( ( 𝑄 ‘ 0 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) |
344 |
42 1
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) ∈ ℝ ) |
345 |
10
|
nngt0d |
⊢ ( 𝜑 → 0 < 𝑀 ) |
346 |
|
fzolb |
⊢ ( 0 ∈ ( 0 ..^ 𝑀 ) ↔ ( 0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀 ) ) |
347 |
312 313 345 346
|
syl3anbrc |
⊢ ( 𝜑 → 0 ∈ ( 0 ..^ 𝑀 ) ) |
348 |
|
0re |
⊢ 0 ∈ ℝ |
349 |
|
eleq1 |
⊢ ( 𝑖 = 0 → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↔ 0 ∈ ( 0 ..^ 𝑀 ) ) ) |
350 |
349
|
anbi2d |
⊢ ( 𝑖 = 0 → ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ↔ ( 𝜑 ∧ 0 ∈ ( 0 ..^ 𝑀 ) ) ) ) |
351 |
|
fveq2 |
⊢ ( 𝑖 = 0 → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 0 ) ) |
352 |
|
oveq1 |
⊢ ( 𝑖 = 0 → ( 𝑖 + 1 ) = ( 0 + 1 ) ) |
353 |
352
|
fveq2d |
⊢ ( 𝑖 = 0 → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ ( 0 + 1 ) ) ) |
354 |
351 353
|
breq12d |
⊢ ( 𝑖 = 0 → ( ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ↔ ( 𝑄 ‘ 0 ) < ( 𝑄 ‘ ( 0 + 1 ) ) ) ) |
355 |
350 354
|
imbi12d |
⊢ ( 𝑖 = 0 → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( ( 𝜑 ∧ 0 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 0 ) < ( 𝑄 ‘ ( 0 + 1 ) ) ) ) ) |
356 |
355 151
|
vtoclg |
⊢ ( 0 ∈ ℝ → ( ( 𝜑 ∧ 0 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 0 ) < ( 𝑄 ‘ ( 0 + 1 ) ) ) ) |
357 |
348 356
|
ax-mp |
⊢ ( ( 𝜑 ∧ 0 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 0 ) < ( 𝑄 ‘ ( 0 + 1 ) ) ) |
358 |
347 357
|
mpdan |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) < ( 𝑄 ‘ ( 0 + 1 ) ) ) |
359 |
|
0p1e1 |
⊢ ( 0 + 1 ) = 1 |
360 |
359
|
fveq2i |
⊢ ( 𝑄 ‘ ( 0 + 1 ) ) = ( 𝑄 ‘ 1 ) |
361 |
360
|
a1i |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 0 + 1 ) ) = ( 𝑄 ‘ 1 ) ) |
362 |
358 361
|
breqtrd |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) < ( 𝑄 ‘ 1 ) ) |
363 |
344 319 320 362
|
ltsub1dd |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) < ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) |
364 |
363
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( ( 𝑄 ‘ 0 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) < ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) |
365 |
343 364
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → 𝑋 < ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) |
366 |
|
elioore |
⊢ ( 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) → 𝑦 ∈ ℝ ) |
367 |
134
|
eqcomd |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) = ( 𝑍 ‘ 𝑋 ) ) |
368 |
367
|
negeqd |
⊢ ( 𝜑 → - ( ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) = - ( 𝑍 ‘ 𝑋 ) ) |
369 |
225 368
|
eqtrd |
⊢ ( 𝜑 → ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) = - ( 𝑍 ‘ 𝑋 ) ) |
370 |
224
|
mulid2d |
⊢ ( 𝜑 → ( 1 · 𝑇 ) = 𝑇 ) |
371 |
369 370
|
oveq12d |
⊢ ( 𝜑 → ( ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) + ( 1 · 𝑇 ) ) = ( - ( 𝑍 ‘ 𝑋 ) + 𝑇 ) ) |
372 |
223
|
negcld |
⊢ ( 𝜑 → - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℂ ) |
373 |
|
1cnd |
⊢ ( 𝜑 → 1 ∈ ℂ ) |
374 |
372 373 224
|
adddird |
⊢ ( 𝜑 → ( ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) · 𝑇 ) = ( ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) + ( 1 · 𝑇 ) ) ) |
375 |
216 224
|
negsubdid |
⊢ ( 𝜑 → - ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) = ( - ( 𝑍 ‘ 𝑋 ) + 𝑇 ) ) |
376 |
371 374 375
|
3eqtr4d |
⊢ ( 𝜑 → ( ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) · 𝑇 ) = - ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) |
377 |
376
|
oveq2d |
⊢ ( 𝜑 → ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) · 𝑇 ) ) = ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + - ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) |
378 |
377
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) · 𝑇 ) ) = ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + - ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) |
379 |
320
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ∈ ℝ ) |
380 |
228 379
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ ℝ ) |
381 |
380
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ ℂ ) |
382 |
379
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ∈ ℂ ) |
383 |
381 382
|
negsubd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + - ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) = ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) |
384 |
234 382
|
pncand |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) = 𝑦 ) |
385 |
378 383 384
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑦 = ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) · 𝑇 ) ) ) |
386 |
366 385
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → 𝑦 = ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) · 𝑇 ) ) ) |
387 |
386
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → 𝑦 = ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) · 𝑇 ) ) ) |
388 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → 𝜑 ) |
389 |
361
|
eqcomd |
⊢ ( 𝜑 → ( 𝑄 ‘ 1 ) = ( 𝑄 ‘ ( 0 + 1 ) ) ) |
390 |
389
|
oveq2d |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) (,) ( 𝑄 ‘ 1 ) ) = ( ( 𝑄 ‘ 0 ) (,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ) |
391 |
351 353
|
oveq12d |
⊢ ( 𝑖 = 0 → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑄 ‘ 0 ) (,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ) |
392 |
391
|
sseq1d |
⊢ ( 𝑖 = 0 → ( ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ 𝐷 ↔ ( ( 𝑄 ‘ 0 ) (,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ⊆ 𝐷 ) ) |
393 |
350 392
|
imbi12d |
⊢ ( 𝑖 = 0 → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ 𝐷 ) ↔ ( ( 𝜑 ∧ 0 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 0 ) (,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ⊆ 𝐷 ) ) ) |
394 |
393 12
|
vtoclg |
⊢ ( 0 ∈ ℝ → ( ( 𝜑 ∧ 0 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 0 ) (,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ⊆ 𝐷 ) ) |
395 |
348 394
|
ax-mp |
⊢ ( ( 𝜑 ∧ 0 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ 0 ) (,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ⊆ 𝐷 ) |
396 |
347 395
|
mpdan |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) (,) ( 𝑄 ‘ ( 0 + 1 ) ) ) ⊆ 𝐷 ) |
397 |
390 396
|
eqsstrd |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) (,) ( 𝑄 ‘ 1 ) ) ⊆ 𝐷 ) |
398 |
397
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( ( 𝑄 ‘ 0 ) (,) ( 𝑄 ‘ 1 ) ) ⊆ 𝐷 ) |
399 |
42 47
|
eqeltrd |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) ∈ ℝ* ) |
400 |
399
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( 𝑄 ‘ 0 ) ∈ ℝ* ) |
401 |
319
|
rexrd |
⊢ ( 𝜑 → ( 𝑄 ‘ 1 ) ∈ ℝ* ) |
402 |
401
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( 𝑄 ‘ 1 ) ∈ ℝ* ) |
403 |
366 380
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ ℝ ) |
404 |
403
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ ℝ ) |
405 |
339 215 216
|
subaddd |
⊢ ( 𝜑 → ( ( ( 𝐸 ‘ 𝑋 ) − 𝑋 ) = ( 𝑍 ‘ 𝑋 ) ↔ ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) = ( 𝐸 ‘ 𝑋 ) ) ) |
406 |
270 405
|
mpbird |
⊢ ( 𝜑 → ( ( 𝐸 ‘ 𝑋 ) − 𝑋 ) = ( 𝑍 ‘ 𝑋 ) ) |
407 |
|
oveq1 |
⊢ ( ( 𝐸 ‘ 𝑋 ) = 𝐵 → ( ( 𝐸 ‘ 𝑋 ) − 𝑋 ) = ( 𝐵 − 𝑋 ) ) |
408 |
406 407
|
sylan9req |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( 𝑍 ‘ 𝑋 ) = ( 𝐵 − 𝑋 ) ) |
409 |
408
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) = ( ( 𝐵 − 𝑋 ) − 𝑇 ) ) |
410 |
409
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( 𝑋 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) = ( 𝑋 + ( ( 𝐵 − 𝑋 ) − 𝑇 ) ) ) |
411 |
129
|
recnd |
⊢ ( 𝜑 → ( 𝐵 − 𝑋 ) ∈ ℂ ) |
412 |
215 411 224
|
addsubassd |
⊢ ( 𝜑 → ( ( 𝑋 + ( 𝐵 − 𝑋 ) ) − 𝑇 ) = ( 𝑋 + ( ( 𝐵 − 𝑋 ) − 𝑇 ) ) ) |
413 |
412
|
eqcomd |
⊢ ( 𝜑 → ( 𝑋 + ( ( 𝐵 − 𝑋 ) − 𝑇 ) ) = ( ( 𝑋 + ( 𝐵 − 𝑋 ) ) − 𝑇 ) ) |
414 |
413
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( 𝑋 + ( ( 𝐵 − 𝑋 ) − 𝑇 ) ) = ( ( 𝑋 + ( 𝐵 − 𝑋 ) ) − 𝑇 ) ) |
415 |
328 224 323
|
subsub23d |
⊢ ( 𝜑 → ( ( 𝐵 − 𝑇 ) = 𝐴 ↔ ( 𝐵 − 𝐴 ) = 𝑇 ) ) |
416 |
62 415
|
mpbird |
⊢ ( 𝜑 → ( 𝐵 − 𝑇 ) = 𝐴 ) |
417 |
215 328
|
pncan3d |
⊢ ( 𝜑 → ( 𝑋 + ( 𝐵 − 𝑋 ) ) = 𝐵 ) |
418 |
417
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝑋 + ( 𝐵 − 𝑋 ) ) − 𝑇 ) = ( 𝐵 − 𝑇 ) ) |
419 |
416 418 42
|
3eqtr4d |
⊢ ( 𝜑 → ( ( 𝑋 + ( 𝐵 − 𝑋 ) ) − 𝑇 ) = ( 𝑄 ‘ 0 ) ) |
420 |
419
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( ( 𝑋 + ( 𝐵 − 𝑋 ) ) − 𝑇 ) = ( 𝑄 ‘ 0 ) ) |
421 |
410 414 420
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( 𝑄 ‘ 0 ) = ( 𝑋 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) |
422 |
421
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( 𝑄 ‘ 0 ) = ( 𝑋 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) |
423 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → 𝑋 ∈ ℝ ) |
424 |
366
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → 𝑦 ∈ ℝ ) |
425 |
320
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ∈ ℝ ) |
426 |
252
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → 𝑋 ∈ ℝ* ) |
427 |
321
|
rexrd |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ ℝ* ) |
428 |
427
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ ℝ* ) |
429 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) |
430 |
|
ioogtlb |
⊢ ( ( 𝑋 ∈ ℝ* ∧ ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ ℝ* ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → 𝑋 < 𝑦 ) |
431 |
426 428 429 430
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → 𝑋 < 𝑦 ) |
432 |
423 424 425 431
|
ltadd1dd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( 𝑋 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) < ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) |
433 |
432
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( 𝑋 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) < ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) |
434 |
422 433
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( 𝑄 ‘ 0 ) < ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) |
435 |
|
iooltub |
⊢ ( ( 𝑋 ∈ ℝ* ∧ ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ ℝ* ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → 𝑦 < ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) |
436 |
426 428 429 435
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → 𝑦 < ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) |
437 |
319
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( 𝑄 ‘ 1 ) ∈ ℝ ) |
438 |
424 425 437
|
ltaddsubd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) < ( 𝑄 ‘ 1 ) ↔ 𝑦 < ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) |
439 |
436 438
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) < ( 𝑄 ‘ 1 ) ) |
440 |
439
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) < ( 𝑄 ‘ 1 ) ) |
441 |
400 402 404 434 440
|
eliood |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ ( ( 𝑄 ‘ 0 ) (,) ( 𝑄 ‘ 1 ) ) ) |
442 |
398 441
|
sseldd |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ 𝐷 ) |
443 |
131
|
znegcld |
⊢ ( 𝜑 → - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) |
444 |
443
|
peano2zd |
⊢ ( 𝜑 → ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) ∈ ℤ ) |
445 |
444
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) ∈ ℤ ) |
446 |
|
ovex |
⊢ ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) ∈ V |
447 |
|
eleq1 |
⊢ ( 𝑘 = ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) → ( 𝑘 ∈ ℤ ↔ ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) ∈ ℤ ) ) |
448 |
447
|
3anbi3d |
⊢ ( 𝑘 = ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) → ( ( 𝜑 ∧ ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) ↔ ( 𝜑 ∧ ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ 𝐷 ∧ ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) ∈ ℤ ) ) ) |
449 |
|
oveq1 |
⊢ ( 𝑘 = ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) → ( 𝑘 · 𝑇 ) = ( ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) · 𝑇 ) ) |
450 |
449
|
oveq2d |
⊢ ( 𝑘 = ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) → ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( 𝑘 · 𝑇 ) ) = ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) · 𝑇 ) ) ) |
451 |
450
|
eleq1d |
⊢ ( 𝑘 = ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) → ( ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ↔ ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) · 𝑇 ) ) ∈ 𝐷 ) ) |
452 |
448 451
|
imbi12d |
⊢ ( 𝑘 = ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) → ( ( ( 𝜑 ∧ ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ↔ ( ( 𝜑 ∧ ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ 𝐷 ∧ ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) ∈ ℤ ) → ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) · 𝑇 ) ) ∈ 𝐷 ) ) ) |
453 |
|
ovex |
⊢ ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ V |
454 |
|
eleq1 |
⊢ ( 𝑥 = ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) → ( 𝑥 ∈ 𝐷 ↔ ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ 𝐷 ) ) |
455 |
454
|
3anbi2d |
⊢ ( 𝑥 = ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) → ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) ↔ ( 𝜑 ∧ ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) ) ) |
456 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) → ( 𝑥 + ( 𝑘 · 𝑇 ) ) = ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( 𝑘 · 𝑇 ) ) ) |
457 |
456
|
eleq1d |
⊢ ( 𝑥 = ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) → ( ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ↔ ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ) |
458 |
455 457
|
imbi12d |
⊢ ( 𝑥 = ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( 𝑥 + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ↔ ( ( 𝜑 ∧ ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) ) ) |
459 |
453 458 5
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ 𝐷 ∧ 𝑘 ∈ ℤ ) → ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( 𝑘 · 𝑇 ) ) ∈ 𝐷 ) |
460 |
446 452 459
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ 𝐷 ∧ ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) ∈ ℤ ) → ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) · 𝑇 ) ) ∈ 𝐷 ) |
461 |
388 442 445 460
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → ( ( 𝑦 + ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) + ( ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) + 1 ) · 𝑇 ) ) ∈ 𝐷 ) |
462 |
387 461
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) → 𝑦 ∈ 𝐷 ) |
463 |
462
|
ralrimiva |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ∀ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) 𝑦 ∈ 𝐷 ) |
464 |
|
dfss3 |
⊢ ( ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ⊆ 𝐷 ↔ ∀ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) 𝑦 ∈ 𝐷 ) |
465 |
463 464
|
sylibr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ⊆ 𝐷 ) |
466 |
|
breq2 |
⊢ ( 𝑦 = ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) → ( 𝑋 < 𝑦 ↔ 𝑋 < ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) |
467 |
|
oveq2 |
⊢ ( 𝑦 = ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) → ( 𝑋 (,) 𝑦 ) = ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ) |
468 |
467
|
sseq1d |
⊢ ( 𝑦 = ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) → ( ( 𝑋 (,) 𝑦 ) ⊆ 𝐷 ↔ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ⊆ 𝐷 ) ) |
469 |
466 468
|
anbi12d |
⊢ ( 𝑦 = ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) → ( ( 𝑋 < 𝑦 ∧ ( 𝑋 (,) 𝑦 ) ⊆ 𝐷 ) ↔ ( 𝑋 < ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∧ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ⊆ 𝐷 ) ) ) |
470 |
469
|
rspcev |
⊢ ( ( ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∈ ℝ ∧ ( 𝑋 < ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ∧ ( 𝑋 (,) ( ( 𝑄 ‘ 1 ) − ( ( 𝑍 ‘ 𝑋 ) − 𝑇 ) ) ) ⊆ 𝐷 ) ) → ∃ 𝑦 ∈ ℝ ( 𝑋 < 𝑦 ∧ ( 𝑋 (,) 𝑦 ) ⊆ 𝐷 ) ) |
471 |
322 365 465 470
|
syl12anc |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) → ∃ 𝑦 ∈ ℝ ( 𝑋 < 𝑦 ∧ ( 𝑋 (,) 𝑦 ) ⊆ 𝐷 ) ) |
472 |
24
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) |
473 |
|
simp2 |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑗 ∈ ( 0 ... 𝑀 ) ) |
474 |
34
|
3ad2ant2 |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑗 ∈ ℝ ) |
475 |
96
|
3ad2ant2 |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑀 ∈ ℝ ) |
476 |
98
|
3ad2ant2 |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑗 ≤ 𝑀 ) |
477 |
|
id |
⊢ ( ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) → ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) |
478 |
477
|
eqcomd |
⊢ ( ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ 𝑗 ) ) |
479 |
478
|
adantr |
⊢ ( ( ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ∧ 𝑀 = 𝑗 ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ 𝑗 ) ) |
480 |
479
|
3ad2antl3 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑀 = 𝑗 ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ 𝑗 ) ) |
481 |
|
fveq2 |
⊢ ( 𝑀 = 𝑗 → ( 𝑄 ‘ 𝑀 ) = ( 𝑄 ‘ 𝑗 ) ) |
482 |
481
|
eqcomd |
⊢ ( 𝑀 = 𝑗 → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ 𝑀 ) ) |
483 |
482
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑀 = 𝑗 ) → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ 𝑀 ) ) |
484 |
179
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑀 = 𝑗 ) → ( 𝑄 ‘ 𝑀 ) = 𝐵 ) |
485 |
484
|
3ad2antl1 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑀 = 𝑗 ) → ( 𝑄 ‘ 𝑀 ) = 𝐵 ) |
486 |
480 483 485
|
3eqtrd |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑀 = 𝑗 ) → ( 𝐸 ‘ 𝑋 ) = 𝐵 ) |
487 |
|
neneq |
⊢ ( ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 → ¬ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) |
488 |
487
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑀 = 𝑗 ) → ¬ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) |
489 |
488
|
3ad2antl1 |
⊢ ( ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) ∧ 𝑀 = 𝑗 ) → ¬ ( 𝐸 ‘ 𝑋 ) = 𝐵 ) |
490 |
486 489
|
pm2.65da |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ¬ 𝑀 = 𝑗 ) |
491 |
490
|
neqned |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑀 ≠ 𝑗 ) |
492 |
474 475 476 491
|
leneltd |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑗 < 𝑀 ) |
493 |
|
elfzfzo |
⊢ ( 𝑗 ∈ ( 0 ..^ 𝑀 ) ↔ ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑗 < 𝑀 ) ) |
494 |
473 492 493
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝑗 ∈ ( 0 ..^ 𝑀 ) ) |
495 |
116
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ* ) |
496 |
495
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ* ) |
497 |
|
simp1l |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → 𝜑 ) |
498 |
103
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
499 |
|
fzofzp1 |
⊢ ( 𝑗 ∈ ( 0 ..^ 𝑀 ) → ( 𝑗 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
500 |
499
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑗 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
501 |
498 500
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ∈ ℝ ) |
502 |
501
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ∈ ℝ* ) |
503 |
497 494 502
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ ( 𝑗 + 1 ) ) ∈ ℝ* ) |
504 |
140
|
3adant1r |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ* ) |
505 |
46 159
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ 𝑗 ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
506 |
505
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ 𝑗 ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
507 |
506
|
3adant2 |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ 𝑗 ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
508 |
478
|
3ad2ant3 |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑄 ‘ 𝑗 ) ) |
509 |
|
eleq1 |
⊢ ( 𝑖 = 𝑗 → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↔ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) ) |
510 |
509
|
anbi2d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ↔ ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) ) ) |
511 |
|
fveq2 |
⊢ ( 𝑖 = 𝑗 → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 𝑗 ) ) |
512 |
|
oveq1 |
⊢ ( 𝑖 = 𝑗 → ( 𝑖 + 1 ) = ( 𝑗 + 1 ) ) |
513 |
512
|
fveq2d |
⊢ ( 𝑖 = 𝑗 → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
514 |
511 513
|
breq12d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ↔ ( 𝑄 ‘ 𝑗 ) < ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) |
515 |
510 514
|
imbi12d |
⊢ ( 𝑖 = 𝑗 → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑗 ) < ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ) |
516 |
515 151
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑗 ) < ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
517 |
497 494 516
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝑄 ‘ 𝑗 ) < ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
518 |
508 517
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) < ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) |
519 |
496 503 504 507 518
|
elicod |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) [,) ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) |
520 |
511 513
|
oveq12d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑄 ‘ 𝑗 ) [,) ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) |
521 |
520
|
eleq2d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) [,) ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) ) |
522 |
521
|
rspcev |
⊢ ( ( 𝑗 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑗 ) [,) ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
523 |
494 519 522
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
524 |
523
|
3exp |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) → ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
525 |
524
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
526 |
525
|
rexlimdv |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = ( 𝐸 ‘ 𝑋 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
527 |
472 526
|
mpd |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
528 |
|
ioossico |
⊢ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
529 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
530 |
528 529
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
531 |
530
|
ex |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
532 |
531
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
533 |
532
|
reximdva |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
534 |
189 533
|
mpd |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
535 |
534
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) ∧ ¬ ( 𝐸 ‘ 𝑋 ) ∈ ran 𝑄 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
536 |
527 535
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
537 |
207 248
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ) |
538 |
537
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ) |
539 |
218
|
eqcomd |
⊢ ( 𝜑 → 𝑋 = ( ( 𝐸 ‘ 𝑋 ) − ( 𝑍 ‘ 𝑋 ) ) ) |
540 |
539
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑋 = ( ( 𝐸 ‘ 𝑋 ) − ( 𝑍 ‘ 𝑋 ) ) ) |
541 |
138
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
542 |
207
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
543 |
135
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℝ ) |
544 |
199
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
545 |
544
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
546 |
208
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
547 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
548 |
|
icoltub |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
549 |
545 546 547 548
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
550 |
541 542 543 549
|
ltsub1dd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝐸 ‘ 𝑋 ) − ( 𝑍 ‘ 𝑋 ) ) < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) |
551 |
540 550
|
eqbrtrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → 𝑋 < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) |
552 |
|
elioore |
⊢ ( 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) → 𝑦 ∈ ℝ ) |
553 |
552 236
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → 𝑦 = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
554 |
553
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → 𝑦 = ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ) |
555 |
|
simpl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → 𝜑 ) |
556 |
12
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ 𝐷 ) |
557 |
556
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ⊆ 𝐷 ) |
558 |
545
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ) |
559 |
546
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ) |
560 |
552
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → 𝑦 ∈ ℝ ) |
561 |
135
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℝ ) |
562 |
560 561
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ) |
563 |
562
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ) |
564 |
199
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
565 |
564
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
566 |
555 138
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) ∈ ℝ ) |
567 |
|
icogelb |
⊢ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
568 |
545 546 547 567
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
569 |
568
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) ≤ ( 𝐸 ‘ 𝑋 ) ) |
570 |
137
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) = ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) ) |
571 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → 𝑋 ∈ ℝ ) |
572 |
552
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → 𝑦 ∈ ℝ ) |
573 |
135
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℝ ) |
574 |
252
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → 𝑋 ∈ ℝ* ) |
575 |
537
|
rexrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ) |
576 |
575
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ) |
577 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) |
578 |
|
ioogtlb |
⊢ ( ( 𝑋 ∈ ℝ* ∧ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → 𝑋 < 𝑦 ) |
579 |
574 576 577 578
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → 𝑋 < 𝑦 ) |
580 |
571 572 573 579
|
ltadd1dd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝑋 + ( 𝑍 ‘ 𝑋 ) ) < ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ) |
581 |
570 580
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) < ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ) |
582 |
581
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝐸 ‘ 𝑋 ) < ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ) |
583 |
565 566 563 569 582
|
lelttrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ) |
584 |
537
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ) |
585 |
|
iooltub |
⊢ ( ( 𝑋 ∈ ℝ* ∧ ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ* ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → 𝑦 < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) |
586 |
574 576 577 585
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → 𝑦 < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) |
587 |
572 584 573 586
|
ltadd1dd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) < ( ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) + ( 𝑍 ‘ 𝑋 ) ) ) |
588 |
207
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℂ ) |
589 |
216
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑍 ‘ 𝑋 ) ∈ ℂ ) |
590 |
588 589
|
npcand |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) + ( 𝑍 ‘ 𝑋 ) ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
591 |
590
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) + ( 𝑍 ‘ 𝑋 ) ) = ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
592 |
587 591
|
breqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
593 |
592
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
594 |
558 559 563 583 593
|
eliood |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
595 |
557 594
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) ∈ 𝐷 ) |
596 |
555 443
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) ∈ ℤ ) |
597 |
555 595 596 297
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → ( ( 𝑦 + ( 𝑍 ‘ 𝑋 ) ) + ( - ( ⌊ ‘ ( ( 𝐵 − 𝑋 ) / 𝑇 ) ) · 𝑇 ) ) ∈ 𝐷 ) |
598 |
554 597
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ∧ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) → 𝑦 ∈ 𝐷 ) |
599 |
598
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ∀ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) 𝑦 ∈ 𝐷 ) |
600 |
|
dfss3 |
⊢ ( ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ⊆ 𝐷 ↔ ∀ 𝑦 ∈ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) 𝑦 ∈ 𝐷 ) |
601 |
599 600
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ⊆ 𝐷 ) |
602 |
|
breq2 |
⊢ ( 𝑦 = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) → ( 𝑋 < 𝑦 ↔ 𝑋 < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) |
603 |
|
oveq2 |
⊢ ( 𝑦 = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) → ( 𝑋 (,) 𝑦 ) = ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ) |
604 |
603
|
sseq1d |
⊢ ( 𝑦 = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) → ( ( 𝑋 (,) 𝑦 ) ⊆ 𝐷 ↔ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ⊆ 𝐷 ) ) |
605 |
602 604
|
anbi12d |
⊢ ( 𝑦 = ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) → ( ( 𝑋 < 𝑦 ∧ ( 𝑋 (,) 𝑦 ) ⊆ 𝐷 ) ↔ ( 𝑋 < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ∧ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ⊆ 𝐷 ) ) ) |
606 |
605
|
rspcev |
⊢ ( ( ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ∈ ℝ ∧ ( 𝑋 < ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ∧ ( 𝑋 (,) ( ( 𝑄 ‘ ( 𝑖 + 1 ) ) − ( 𝑍 ‘ 𝑋 ) ) ) ⊆ 𝐷 ) ) → ∃ 𝑦 ∈ ℝ ( 𝑋 < 𝑦 ∧ ( 𝑋 (,) 𝑦 ) ⊆ 𝐷 ) ) |
607 |
538 551 601 606
|
syl12anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) → ∃ 𝑦 ∈ ℝ ( 𝑋 < 𝑦 ∧ ( 𝑋 (,) 𝑦 ) ⊆ 𝐷 ) ) |
608 |
607
|
3exp |
⊢ ( 𝜑 → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ∃ 𝑦 ∈ ℝ ( 𝑋 < 𝑦 ∧ ( 𝑋 (,) 𝑦 ) ⊆ 𝐷 ) ) ) ) |
609 |
608
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ∃ 𝑦 ∈ ℝ ( 𝑋 < 𝑦 ∧ ( 𝑋 (,) 𝑦 ) ⊆ 𝐷 ) ) ) ) |
610 |
609
|
rexlimdv |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) → ( ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝐸 ‘ 𝑋 ) ∈ ( ( 𝑄 ‘ 𝑖 ) [,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) → ∃ 𝑦 ∈ ℝ ( 𝑋 < 𝑦 ∧ ( 𝑋 (,) 𝑦 ) ⊆ 𝐷 ) ) ) |
611 |
536 610
|
mpd |
⊢ ( ( 𝜑 ∧ ( 𝐸 ‘ 𝑋 ) ≠ 𝐵 ) → ∃ 𝑦 ∈ ℝ ( 𝑋 < 𝑦 ∧ ( 𝑋 (,) 𝑦 ) ⊆ 𝐷 ) ) |
612 |
471 611
|
pm2.61dane |
⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ( 𝑋 < 𝑦 ∧ ( 𝑋 (,) 𝑦 ) ⊆ 𝐷 ) ) |
613 |
311 612
|
jca |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ℝ ( 𝑦 < 𝑋 ∧ ( 𝑦 (,) 𝑋 ) ⊆ 𝐷 ) ∧ ∃ 𝑦 ∈ ℝ ( 𝑋 < 𝑦 ∧ ( 𝑋 (,) 𝑦 ) ⊆ 𝐷 ) ) ) |