Step |
Hyp |
Ref |
Expression |
1 |
|
ovolshft.1 |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ ) |
2 |
|
ovolshft.2 |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
3 |
|
ovolshft.3 |
⊢ ( 𝜑 → 𝐵 = { 𝑥 ∈ ℝ ∣ ( 𝑥 − 𝐶 ) ∈ 𝐴 } ) |
4 |
|
ovolshft.4 |
⊢ 𝑀 = { 𝑦 ∈ ℝ* ∣ ∃ 𝑓 ∈ ( ( ≤ ∩ ( ℝ × ℝ ) ) ↑m ℕ ) ( 𝐵 ⊆ ∪ ran ( (,) ∘ 𝑓 ) ∧ 𝑦 = sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝑓 ) ) , ℝ* , < ) ) } |
5 |
|
ovolshft.5 |
⊢ 𝑆 = seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐹 ) ) |
6 |
|
ovolshft.6 |
⊢ 𝐺 = ( 𝑛 ∈ ℕ ↦ 〈 ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) 〉 ) |
7 |
|
ovolshft.7 |
⊢ ( 𝜑 → 𝐹 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ) |
8 |
|
ovolshft.8 |
⊢ ( 𝜑 → 𝐴 ⊆ ∪ ran ( (,) ∘ 𝐹 ) ) |
9 |
|
ovolfcl |
⊢ ( ( 𝐹 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ∧ 𝑛 ∈ ℕ ) → ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
10 |
7 9
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ∧ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ∧ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
11 |
10
|
simp1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ) |
12 |
10
|
simp2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ) |
13 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐶 ∈ ℝ ) |
14 |
10
|
simp3d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ≤ ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) |
15 |
11 12 13 14
|
leadd1dd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ≤ ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ) |
16 |
|
df-br |
⊢ ( ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ≤ ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ↔ 〈 ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) 〉 ∈ ≤ ) |
17 |
15 16
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 〈 ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) 〉 ∈ ≤ ) |
18 |
11 13
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ∈ ℝ ) |
19 |
12 13
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ∈ ℝ ) |
20 |
18 19
|
opelxpd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 〈 ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) 〉 ∈ ( ℝ × ℝ ) ) |
21 |
17 20
|
elind |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 〈 ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) 〉 ∈ ( ≤ ∩ ( ℝ × ℝ ) ) ) |
22 |
21 6
|
fmptd |
⊢ ( 𝜑 → 𝐺 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ) |
23 |
|
eqid |
⊢ ( ( abs ∘ − ) ∘ 𝐺 ) = ( ( abs ∘ − ) ∘ 𝐺 ) |
24 |
23
|
ovolfsf |
⊢ ( 𝐺 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) → ( ( abs ∘ − ) ∘ 𝐺 ) : ℕ ⟶ ( 0 [,) +∞ ) ) |
25 |
|
ffn |
⊢ ( ( ( abs ∘ − ) ∘ 𝐺 ) : ℕ ⟶ ( 0 [,) +∞ ) → ( ( abs ∘ − ) ∘ 𝐺 ) Fn ℕ ) |
26 |
22 24 25
|
3syl |
⊢ ( 𝜑 → ( ( abs ∘ − ) ∘ 𝐺 ) Fn ℕ ) |
27 |
|
eqid |
⊢ ( ( abs ∘ − ) ∘ 𝐹 ) = ( ( abs ∘ − ) ∘ 𝐹 ) |
28 |
27
|
ovolfsf |
⊢ ( 𝐹 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) → ( ( abs ∘ − ) ∘ 𝐹 ) : ℕ ⟶ ( 0 [,) +∞ ) ) |
29 |
|
ffn |
⊢ ( ( ( abs ∘ − ) ∘ 𝐹 ) : ℕ ⟶ ( 0 [,) +∞ ) → ( ( abs ∘ − ) ∘ 𝐹 ) Fn ℕ ) |
30 |
7 28 29
|
3syl |
⊢ ( 𝜑 → ( ( abs ∘ − ) ∘ 𝐹 ) Fn ℕ ) |
31 |
|
opex |
⊢ 〈 ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) 〉 ∈ V |
32 |
6
|
fvmpt2 |
⊢ ( ( 𝑛 ∈ ℕ ∧ 〈 ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) 〉 ∈ V ) → ( 𝐺 ‘ 𝑛 ) = 〈 ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) 〉 ) |
33 |
31 32
|
mpan2 |
⊢ ( 𝑛 ∈ ℕ → ( 𝐺 ‘ 𝑛 ) = 〈 ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) 〉 ) |
34 |
33
|
fveq2d |
⊢ ( 𝑛 ∈ ℕ → ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) = ( 2nd ‘ 〈 ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) 〉 ) ) |
35 |
|
ovex |
⊢ ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ∈ V |
36 |
|
ovex |
⊢ ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ∈ V |
37 |
35 36
|
op2nd |
⊢ ( 2nd ‘ 〈 ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) 〉 ) = ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) |
38 |
34 37
|
eqtrdi |
⊢ ( 𝑛 ∈ ℕ → ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) = ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ) |
39 |
33
|
fveq2d |
⊢ ( 𝑛 ∈ ℕ → ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) = ( 1st ‘ 〈 ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) 〉 ) ) |
40 |
35 36
|
op1st |
⊢ ( 1st ‘ 〈 ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) , ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) 〉 ) = ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) |
41 |
39 40
|
eqtrdi |
⊢ ( 𝑛 ∈ ℕ → ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) = ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ) |
42 |
38 41
|
oveq12d |
⊢ ( 𝑛 ∈ ℕ → ( ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) ) = ( ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) − ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ) ) |
43 |
42
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) ) = ( ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) − ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ) ) |
44 |
12
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℂ ) |
45 |
11
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℂ ) |
46 |
13
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐶 ∈ ℂ ) |
47 |
44 45 46
|
pnpcan2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) − ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ) = ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
48 |
43 47
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) ) = ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
49 |
23
|
ovolfsval |
⊢ ( ( 𝐺 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ∧ 𝑛 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝐺 ) ‘ 𝑛 ) = ( ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) |
50 |
22 49
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝐺 ) ‘ 𝑛 ) = ( ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) |
51 |
27
|
ovolfsval |
⊢ ( ( 𝐹 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ∧ 𝑛 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝐹 ) ‘ 𝑛 ) = ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
52 |
7 51
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝐹 ) ‘ 𝑛 ) = ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) − ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
53 |
48 50 52
|
3eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( ( abs ∘ − ) ∘ 𝐺 ) ‘ 𝑛 ) = ( ( ( abs ∘ − ) ∘ 𝐹 ) ‘ 𝑛 ) ) |
54 |
26 30 53
|
eqfnfvd |
⊢ ( 𝜑 → ( ( abs ∘ − ) ∘ 𝐺 ) = ( ( abs ∘ − ) ∘ 𝐹 ) ) |
55 |
54
|
seqeq3d |
⊢ ( 𝜑 → seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐺 ) ) = seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐹 ) ) ) |
56 |
55 5
|
eqtr4di |
⊢ ( 𝜑 → seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐺 ) ) = 𝑆 ) |
57 |
56
|
rneqd |
⊢ ( 𝜑 → ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐺 ) ) = ran 𝑆 ) |
58 |
57
|
supeq1d |
⊢ ( 𝜑 → sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐺 ) ) , ℝ* , < ) = sup ( ran 𝑆 , ℝ* , < ) ) |
59 |
3
|
eleq2d |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐵 ↔ 𝑦 ∈ { 𝑥 ∈ ℝ ∣ ( 𝑥 − 𝐶 ) ∈ 𝐴 } ) ) |
60 |
|
oveq1 |
⊢ ( 𝑥 = 𝑦 → ( 𝑥 − 𝐶 ) = ( 𝑦 − 𝐶 ) ) |
61 |
60
|
eleq1d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 − 𝐶 ) ∈ 𝐴 ↔ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) |
62 |
61
|
elrab |
⊢ ( 𝑦 ∈ { 𝑥 ∈ ℝ ∣ ( 𝑥 − 𝐶 ) ∈ 𝐴 } ↔ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) |
63 |
59 62
|
bitrdi |
⊢ ( 𝜑 → ( 𝑦 ∈ 𝐵 ↔ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) ) |
64 |
63
|
biimpa |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) |
65 |
|
breq2 |
⊢ ( 𝑥 = ( 𝑦 − 𝐶 ) → ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ↔ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < ( 𝑦 − 𝐶 ) ) ) |
66 |
|
breq1 |
⊢ ( 𝑥 = ( 𝑦 − 𝐶 ) → ( 𝑥 < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ↔ ( 𝑦 − 𝐶 ) < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
67 |
65 66
|
anbi12d |
⊢ ( 𝑥 = ( 𝑦 − 𝐶 ) → ( ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ∧ 𝑥 < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ↔ ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < ( 𝑦 − 𝐶 ) ∧ ( 𝑦 − 𝐶 ) < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) ) |
68 |
67
|
rexbidv |
⊢ ( 𝑥 = ( 𝑦 − 𝐶 ) → ( ∃ 𝑛 ∈ ℕ ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ∧ 𝑥 < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ↔ ∃ 𝑛 ∈ ℕ ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < ( 𝑦 − 𝐶 ) ∧ ( 𝑦 − 𝐶 ) < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) ) |
69 |
|
ovolfioo |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐹 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ) → ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝐹 ) ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑛 ∈ ℕ ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ∧ 𝑥 < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) ) |
70 |
1 7 69
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 ⊆ ∪ ran ( (,) ∘ 𝐹 ) ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑛 ∈ ℕ ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ∧ 𝑥 < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) ) |
71 |
8 70
|
mpbid |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑛 ∈ ℕ ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ∧ 𝑥 < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
72 |
71
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) → ∀ 𝑥 ∈ 𝐴 ∃ 𝑛 ∈ ℕ ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ∧ 𝑥 < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
73 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) → ( 𝑦 − 𝐶 ) ∈ 𝐴 ) |
74 |
68 72 73
|
rspcdva |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) → ∃ 𝑛 ∈ ℕ ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < ( 𝑦 − 𝐶 ) ∧ ( 𝑦 − 𝐶 ) < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
75 |
41
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) = ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ) |
76 |
75
|
breq1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) < 𝑦 ↔ ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) < 𝑦 ) ) |
77 |
11
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ) |
78 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → 𝐶 ∈ ℝ ) |
79 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → 𝑦 ∈ ℝ ) |
80 |
77 78 79
|
ltaddsubd |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) < 𝑦 ↔ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < ( 𝑦 − 𝐶 ) ) ) |
81 |
76 80
|
bitrd |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) < 𝑦 ↔ ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < ( 𝑦 − 𝐶 ) ) ) |
82 |
38
|
adantl |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) = ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ) |
83 |
82
|
breq2d |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝑦 < ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) ↔ 𝑦 < ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ) ) |
84 |
12
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ) |
85 |
79 78 84
|
ltsubaddd |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝑦 − 𝐶 ) < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ↔ 𝑦 < ( ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) + 𝐶 ) ) ) |
86 |
83 85
|
bitr4d |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( 𝑦 < ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) ↔ ( 𝑦 − 𝐶 ) < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) |
87 |
81 86
|
anbi12d |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) ∧ 𝑛 ∈ ℕ ) → ( ( ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) < 𝑦 ∧ 𝑦 < ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) ) ↔ ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < ( 𝑦 − 𝐶 ) ∧ ( 𝑦 − 𝐶 ) < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) ) |
88 |
87
|
rexbidva |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) → ( ∃ 𝑛 ∈ ℕ ( ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) < 𝑦 ∧ 𝑦 < ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) ) ↔ ∃ 𝑛 ∈ ℕ ( ( 1st ‘ ( 𝐹 ‘ 𝑛 ) ) < ( 𝑦 − 𝐶 ) ∧ ( 𝑦 − 𝐶 ) < ( 2nd ‘ ( 𝐹 ‘ 𝑛 ) ) ) ) ) |
89 |
74 88
|
mpbird |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ ( 𝑦 − 𝐶 ) ∈ 𝐴 ) ) → ∃ 𝑛 ∈ ℕ ( ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) < 𝑦 ∧ 𝑦 < ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) |
90 |
64 89
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐵 ) → ∃ 𝑛 ∈ ℕ ( ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) < 𝑦 ∧ 𝑦 < ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) |
91 |
90
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝐵 ∃ 𝑛 ∈ ℕ ( ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) < 𝑦 ∧ 𝑦 < ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) |
92 |
|
ssrab2 |
⊢ { 𝑥 ∈ ℝ ∣ ( 𝑥 − 𝐶 ) ∈ 𝐴 } ⊆ ℝ |
93 |
3 92
|
eqsstrdi |
⊢ ( 𝜑 → 𝐵 ⊆ ℝ ) |
94 |
|
ovolfioo |
⊢ ( ( 𝐵 ⊆ ℝ ∧ 𝐺 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ) → ( 𝐵 ⊆ ∪ ran ( (,) ∘ 𝐺 ) ↔ ∀ 𝑦 ∈ 𝐵 ∃ 𝑛 ∈ ℕ ( ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) < 𝑦 ∧ 𝑦 < ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) ) |
95 |
93 22 94
|
syl2anc |
⊢ ( 𝜑 → ( 𝐵 ⊆ ∪ ran ( (,) ∘ 𝐺 ) ↔ ∀ 𝑦 ∈ 𝐵 ∃ 𝑛 ∈ ℕ ( ( 1st ‘ ( 𝐺 ‘ 𝑛 ) ) < 𝑦 ∧ 𝑦 < ( 2nd ‘ ( 𝐺 ‘ 𝑛 ) ) ) ) ) |
96 |
91 95
|
mpbird |
⊢ ( 𝜑 → 𝐵 ⊆ ∪ ran ( (,) ∘ 𝐺 ) ) |
97 |
|
eqid |
⊢ seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐺 ) ) = seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐺 ) ) |
98 |
4 97
|
elovolmr |
⊢ ( ( 𝐺 : ℕ ⟶ ( ≤ ∩ ( ℝ × ℝ ) ) ∧ 𝐵 ⊆ ∪ ran ( (,) ∘ 𝐺 ) ) → sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐺 ) ) , ℝ* , < ) ∈ 𝑀 ) |
99 |
22 96 98
|
syl2anc |
⊢ ( 𝜑 → sup ( ran seq 1 ( + , ( ( abs ∘ − ) ∘ 𝐺 ) ) , ℝ* , < ) ∈ 𝑀 ) |
100 |
58 99
|
eqeltrrd |
⊢ ( 𝜑 → sup ( ran 𝑆 , ℝ* , < ) ∈ 𝑀 ) |