Step |
Hyp |
Ref |
Expression |
1 |
|
ioodvbdlimc1.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
ioodvbdlimc1.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
ioodvbdlimc1.f |
⊢ ( 𝜑 → 𝐹 : ( 𝐴 (,) 𝐵 ) ⟶ ℝ ) |
4 |
|
ioodvbdlimc1.dmdv |
⊢ ( 𝜑 → dom ( ℝ D 𝐹 ) = ( 𝐴 (,) 𝐵 ) ) |
5 |
|
ioodvbdlimc1.dvbd |
⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ≤ 𝑦 ) |
6 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → 𝐴 ∈ ℝ ) |
7 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → 𝐵 ∈ ℝ ) |
8 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → 𝐴 < 𝐵 ) |
9 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → 𝐹 : ( 𝐴 (,) 𝐵 ) ⟶ ℝ ) |
10 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → dom ( ℝ D 𝐹 ) = ( 𝐴 (,) 𝐵 ) ) |
11 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ≤ 𝑦 ) |
12 |
|
2fveq3 |
⊢ ( 𝑦 = 𝑥 → ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) = ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ) |
13 |
12
|
cbvmptv |
⊢ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) ) = ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ) |
14 |
13
|
rneqi |
⊢ ran ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) ) = ran ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ) |
15 |
14
|
supeq1i |
⊢ sup ( ran ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) ) , ℝ , < ) = sup ( ran ( 𝑥 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑥 ) ) ) , ℝ , < ) |
16 |
|
eqid |
⊢ ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) = ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) |
17 |
|
oveq2 |
⊢ ( 𝑗 = 𝑘 → ( 1 / 𝑗 ) = ( 1 / 𝑘 ) ) |
18 |
17
|
oveq2d |
⊢ ( 𝑗 = 𝑘 → ( 𝐴 + ( 1 / 𝑗 ) ) = ( 𝐴 + ( 1 / 𝑘 ) ) ) |
19 |
18
|
fveq2d |
⊢ ( 𝑗 = 𝑘 → ( 𝐹 ‘ ( 𝐴 + ( 1 / 𝑗 ) ) ) = ( 𝐹 ‘ ( 𝐴 + ( 1 / 𝑘 ) ) ) ) |
20 |
19
|
cbvmptv |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ) ↦ ( 𝐹 ‘ ( 𝐴 + ( 1 / 𝑗 ) ) ) ) = ( 𝑘 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ) ↦ ( 𝐹 ‘ ( 𝐴 + ( 1 / 𝑘 ) ) ) ) |
21 |
18
|
cbvmptv |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ) ↦ ( 𝐴 + ( 1 / 𝑗 ) ) ) = ( 𝑘 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ) ↦ ( 𝐴 + ( 1 / 𝑘 ) ) ) |
22 |
|
eqid |
⊢ if ( ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ≤ ( ( ⌊ ‘ ( sup ( ran ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) ) , ℝ , < ) / ( 𝑥 / 2 ) ) ) + 1 ) , ( ( ⌊ ‘ ( sup ( ran ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) ) , ℝ , < ) / ( 𝑥 / 2 ) ) ) + 1 ) , ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ) = if ( ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ≤ ( ( ⌊ ‘ ( sup ( ran ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) ) , ℝ , < ) / ( 𝑥 / 2 ) ) ) + 1 ) , ( ( ⌊ ‘ ( sup ( ran ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) ) , ℝ , < ) / ( 𝑥 / 2 ) ) ) + 1 ) , ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ) |
23 |
|
biid |
⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝐴 < 𝐵 ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ if ( ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ≤ ( ( ⌊ ‘ ( sup ( ran ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) ) , ℝ , < ) / ( 𝑥 / 2 ) ) ) + 1 ) , ( ( ⌊ ‘ ( sup ( ran ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) ) , ℝ , < ) / ( 𝑥 / 2 ) ) ) + 1 ) , ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ) ) ) ∧ ( abs ‘ ( ( ( 𝑗 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ) ↦ ( 𝐹 ‘ ( 𝐴 + ( 1 / 𝑗 ) ) ) ) ‘ 𝑘 ) − ( lim sup ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ) ↦ ( 𝐹 ‘ ( 𝐴 + ( 1 / 𝑗 ) ) ) ) ) ) ) < ( 𝑥 / 2 ) ) ∧ 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ) ∧ ( abs ‘ ( 𝑧 − 𝐴 ) ) < ( 1 / 𝑘 ) ) ↔ ( ( ( ( ( ( 𝜑 ∧ 𝐴 < 𝐵 ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑘 ∈ ( ℤ≥ ‘ if ( ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ≤ ( ( ⌊ ‘ ( sup ( ran ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) ) , ℝ , < ) / ( 𝑥 / 2 ) ) ) + 1 ) , ( ( ⌊ ‘ ( sup ( ran ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ↦ ( abs ‘ ( ( ℝ D 𝐹 ) ‘ 𝑦 ) ) ) , ℝ , < ) / ( 𝑥 / 2 ) ) ) + 1 ) , ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ) ) ) ∧ ( abs ‘ ( ( ( 𝑗 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ) ↦ ( 𝐹 ‘ ( 𝐴 + ( 1 / 𝑗 ) ) ) ) ‘ 𝑘 ) − ( lim sup ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ) ↦ ( 𝐹 ‘ ( 𝐴 + ( 1 / 𝑗 ) ) ) ) ) ) ) < ( 𝑥 / 2 ) ) ∧ 𝑧 ∈ ( 𝐴 (,) 𝐵 ) ) ∧ ( abs ‘ ( 𝑧 − 𝐴 ) ) < ( 1 / 𝑘 ) ) ) |
24 |
6 7 8 9 10 11 15 16 20 21 22 23
|
ioodvbdlimc1lem2 |
⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → ( lim sup ‘ ( 𝑗 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 1 / ( 𝐵 − 𝐴 ) ) ) + 1 ) ) ↦ ( 𝐹 ‘ ( 𝐴 + ( 1 / 𝑗 ) ) ) ) ) ∈ ( 𝐹 limℂ 𝐴 ) ) |
25 |
24
|
ne0d |
⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → ( 𝐹 limℂ 𝐴 ) ≠ ∅ ) |
26 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
27 |
26
|
a1i |
⊢ ( 𝜑 → ℝ ⊆ ℂ ) |
28 |
3 27
|
fssd |
⊢ ( 𝜑 → 𝐹 : ( 𝐴 (,) 𝐵 ) ⟶ ℂ ) |
29 |
28
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → 𝐹 : ( 𝐴 (,) 𝐵 ) ⟶ ℂ ) |
30 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → 𝐵 ≤ 𝐴 ) |
31 |
1
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
32 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → 𝐴 ∈ ℝ* ) |
33 |
2
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
34 |
33
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → 𝐵 ∈ ℝ* ) |
35 |
|
ioo0 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 (,) 𝐵 ) = ∅ ↔ 𝐵 ≤ 𝐴 ) ) |
36 |
32 34 35
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → ( ( 𝐴 (,) 𝐵 ) = ∅ ↔ 𝐵 ≤ 𝐴 ) ) |
37 |
30 36
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → ( 𝐴 (,) 𝐵 ) = ∅ ) |
38 |
37
|
feq2d |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → ( 𝐹 : ( 𝐴 (,) 𝐵 ) ⟶ ℂ ↔ 𝐹 : ∅ ⟶ ℂ ) ) |
39 |
29 38
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → 𝐹 : ∅ ⟶ ℂ ) |
40 |
1
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
41 |
40
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → 𝐴 ∈ ℂ ) |
42 |
39 41
|
limcdm0 |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → ( 𝐹 limℂ 𝐴 ) = ℂ ) |
43 |
|
0cn |
⊢ 0 ∈ ℂ |
44 |
43
|
ne0ii |
⊢ ℂ ≠ ∅ |
45 |
44
|
a1i |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → ℂ ≠ ∅ ) |
46 |
42 45
|
eqnetrd |
⊢ ( ( 𝜑 ∧ 𝐵 ≤ 𝐴 ) → ( 𝐹 limℂ 𝐴 ) ≠ ∅ ) |
47 |
25 46 1 2
|
ltlecasei |
⊢ ( 𝜑 → ( 𝐹 limℂ 𝐴 ) ≠ ∅ ) |