| Step |
Hyp |
Ref |
Expression |
| 1 |
|
liminflelimsuplem.1 |
⊢ ( 𝜑 → 𝐹 ∈ 𝑉 ) |
| 2 |
|
liminflelimsuplem.2 |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ ( 𝑘 [,) +∞ ) ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) |
| 3 |
|
simpr |
⊢ ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) → 𝑙 ∈ ℝ ) |
| 4 |
|
simpl |
⊢ ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) → 𝑖 ∈ ℝ ) |
| 5 |
3 4
|
ifcld |
⊢ ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) → if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) ∈ ℝ ) |
| 6 |
5
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ℝ ) ∧ 𝑙 ∈ ℝ ) → if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) ∈ ℝ ) |
| 7 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ℝ ) ∧ 𝑙 ∈ ℝ ) → ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ ( 𝑘 [,) +∞ ) ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) |
| 8 |
|
oveq1 |
⊢ ( 𝑘 = if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) → ( 𝑘 [,) +∞ ) = ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) |
| 9 |
8
|
rexeqdv |
⊢ ( 𝑘 = if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) → ( ∃ 𝑗 ∈ ( 𝑘 [,) +∞ ) ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ↔ ∃ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) ) |
| 10 |
9
|
rspcva |
⊢ ( ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) ∈ ℝ ∧ ∀ 𝑘 ∈ ℝ ∃ 𝑗 ∈ ( 𝑘 [,) +∞ ) ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) → ∃ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) |
| 11 |
6 7 10
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ℝ ) ∧ 𝑙 ∈ ℝ ) → ∃ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) |
| 12 |
|
inss2 |
⊢ ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* |
| 13 |
|
infxrcl |
⊢ ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* → inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* ) |
| 14 |
12 13
|
ax-mp |
⊢ inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* |
| 15 |
14
|
a1i |
⊢ ( ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) ∧ ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) → inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* ) |
| 16 |
|
inss2 |
⊢ ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* |
| 17 |
|
infxrcl |
⊢ ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* → inf ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* ) |
| 18 |
16 17
|
ax-mp |
⊢ inf ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* |
| 19 |
18
|
a1i |
⊢ ( ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) ∧ ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) → inf ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* ) |
| 20 |
|
inss2 |
⊢ ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* |
| 21 |
|
supxrcl |
⊢ ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* → sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* ) |
| 22 |
20 21
|
ax-mp |
⊢ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* |
| 23 |
22
|
a1i |
⊢ ( ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) ∧ ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) → sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* ) |
| 24 |
|
rexr |
⊢ ( 𝑖 ∈ ℝ → 𝑖 ∈ ℝ* ) |
| 25 |
24
|
ad2antrr |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → 𝑖 ∈ ℝ* ) |
| 26 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
| 27 |
26
|
a1i |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → +∞ ∈ ℝ* ) |
| 28 |
5
|
rexrd |
⊢ ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) → if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) ∈ ℝ* ) |
| 29 |
28
|
adantr |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) ∈ ℝ* ) |
| 30 |
|
icossxr |
⊢ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ⊆ ℝ* |
| 31 |
|
id |
⊢ ( 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) → 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) |
| 32 |
30 31
|
sselid |
⊢ ( 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) → 𝑗 ∈ ℝ* ) |
| 33 |
32
|
adantl |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → 𝑗 ∈ ℝ* ) |
| 34 |
|
max1 |
⊢ ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) → 𝑖 ≤ if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) ) |
| 35 |
34
|
adantr |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → 𝑖 ≤ if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) ) |
| 36 |
|
simpr |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) |
| 37 |
29 27 36
|
icogelbd |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) ≤ 𝑗 ) |
| 38 |
25 29 33 35 37
|
xrletrd |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → 𝑖 ≤ 𝑗 ) |
| 39 |
25 27 38
|
icossico2 |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → ( 𝑗 [,) +∞ ) ⊆ ( 𝑖 [,) +∞ ) ) |
| 40 |
39
|
imass2d |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ⊆ ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ) |
| 41 |
40
|
ssrind |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ⊆ ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) ) |
| 42 |
12
|
a1i |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* ) |
| 43 |
|
infxrss |
⊢ ( ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ⊆ ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) ∧ ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* ) → inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ inf ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 44 |
41 42 43
|
syl2anc |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ inf ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 45 |
44
|
adantr |
⊢ ( ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) ∧ ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) → inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ inf ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 46 |
|
supxrcl |
⊢ ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* → sup ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* ) |
| 47 |
16 46
|
ax-mp |
⊢ sup ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* |
| 48 |
47
|
a1i |
⊢ ( ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) ∧ ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) → sup ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* ) |
| 49 |
16
|
a1i |
⊢ ( ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) ∧ ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) → ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* ) |
| 50 |
|
simpr |
⊢ ( ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) ∧ ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) → ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) |
| 51 |
49 50
|
infxrlesupxr |
⊢ ( ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) ∧ ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) → inf ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 52 |
|
rexr |
⊢ ( 𝑙 ∈ ℝ → 𝑙 ∈ ℝ* ) |
| 53 |
52
|
ad2antlr |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → 𝑙 ∈ ℝ* ) |
| 54 |
|
max2 |
⊢ ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) → 𝑙 ≤ if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) ) |
| 55 |
54
|
adantr |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → 𝑙 ≤ if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) ) |
| 56 |
53 29 33 55 37
|
xrletrd |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → 𝑙 ≤ 𝑗 ) |
| 57 |
53 27 56
|
icossico2 |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → ( 𝑗 [,) +∞ ) ⊆ ( 𝑙 [,) +∞ ) ) |
| 58 |
57
|
imass2d |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ⊆ ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ) |
| 59 |
58
|
ssrind |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ⊆ ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) ) |
| 60 |
20
|
a1i |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* ) |
| 61 |
|
supxrss |
⊢ ( ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ⊆ ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) ∧ ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* ) → sup ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 62 |
59 60 61
|
syl2anc |
⊢ ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) → sup ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 63 |
62
|
adantr |
⊢ ( ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) ∧ ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) → sup ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 64 |
19 48 23 51 63
|
xrletrd |
⊢ ( ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) ∧ ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) → inf ( ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 65 |
15 19 23 45 64
|
xrletrd |
⊢ ( ( ( ( 𝑖 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) ∧ ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) → inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 66 |
65
|
ad5ant2345 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ℝ ) ∧ 𝑙 ∈ ℝ ) ∧ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ) ∧ ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ ) → inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 67 |
66
|
rexlimdva2 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ℝ ) ∧ 𝑙 ∈ ℝ ) → ( ∃ 𝑗 ∈ ( if ( 𝑖 ≤ 𝑙 , 𝑙 , 𝑖 ) [,) +∞ ) ( ( 𝐹 “ ( 𝑗 [,) +∞ ) ) ∩ ℝ* ) ≠ ∅ → inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) ) |
| 68 |
11 67
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ℝ ) ∧ 𝑙 ∈ ℝ ) → inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 69 |
68
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℝ ) → ∀ 𝑙 ∈ ℝ inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 70 |
|
nfv |
⊢ Ⅎ 𝑙 𝜑 |
| 71 |
|
xrltso |
⊢ < Or ℝ* |
| 72 |
71
|
supex |
⊢ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ V |
| 73 |
72
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑙 ∈ ℝ ) → sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ V ) |
| 74 |
|
breq2 |
⊢ ( 𝑦 = sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) → ( inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ 𝑦 ↔ inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) ) |
| 75 |
70 73 74
|
ralrnmpt3 |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ ran ( 𝑙 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ 𝑦 ↔ ∀ 𝑙 ∈ ℝ inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) ) |
| 76 |
75
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℝ ) → ( ∀ 𝑦 ∈ ran ( 𝑙 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ 𝑦 ↔ ∀ 𝑙 ∈ ℝ inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) ) |
| 77 |
69 76
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℝ ) → ∀ 𝑦 ∈ ran ( 𝑙 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ 𝑦 ) |
| 78 |
|
oveq1 |
⊢ ( 𝑙 = 𝑖 → ( 𝑙 [,) +∞ ) = ( 𝑖 [,) +∞ ) ) |
| 79 |
78
|
imaeq2d |
⊢ ( 𝑙 = 𝑖 → ( 𝐹 “ ( 𝑙 [,) +∞ ) ) = ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ) |
| 80 |
79
|
ineq1d |
⊢ ( 𝑙 = 𝑖 → ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) = ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) ) |
| 81 |
80
|
supeq1d |
⊢ ( 𝑙 = 𝑖 → sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) = sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 82 |
81
|
cbvmptv |
⊢ ( 𝑙 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) = ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 83 |
82
|
rneqi |
⊢ ran ( 𝑙 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) = ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 84 |
83
|
raleqi |
⊢ ( ∀ 𝑦 ∈ ran ( 𝑙 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ 𝑦 ↔ ∀ 𝑦 ∈ ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ 𝑦 ) |
| 85 |
84
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℝ ) → ( ∀ 𝑦 ∈ ran ( 𝑙 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑙 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ 𝑦 ↔ ∀ 𝑦 ∈ ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ 𝑦 ) ) |
| 86 |
77 85
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℝ ) → ∀ 𝑦 ∈ ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ 𝑦 ) |
| 87 |
|
supxrcl |
⊢ ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) ⊆ ℝ* → sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* ) |
| 88 |
12 87
|
ax-mp |
⊢ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* |
| 89 |
88
|
rgenw |
⊢ ∀ 𝑖 ∈ ℝ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* |
| 90 |
|
eqid |
⊢ ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) = ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 91 |
90
|
rnmptss |
⊢ ( ∀ 𝑖 ∈ ℝ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* → ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) ⊆ ℝ* ) |
| 92 |
89 91
|
ax-mp |
⊢ ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) ⊆ ℝ* |
| 93 |
92
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℝ ) → ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) ⊆ ℝ* ) |
| 94 |
14
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℝ ) → inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* ) |
| 95 |
|
infxrgelb |
⊢ ( ( ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) ⊆ ℝ* ∧ inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ∈ ℝ* ) → ( inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ inf ( ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) ↔ ∀ 𝑦 ∈ ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ 𝑦 ) ) |
| 96 |
93 94 95
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℝ ) → ( inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ inf ( ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) ↔ ∀ 𝑦 ∈ ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ 𝑦 ) ) |
| 97 |
86 96
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ℝ ) → inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ inf ( ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) ) |
| 98 |
97
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ℝ inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ inf ( ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) ) |
| 99 |
|
nfv |
⊢ Ⅎ 𝑖 𝜑 |
| 100 |
|
nfcv |
⊢ Ⅎ 𝑖 ℝ |
| 101 |
|
nfmpt1 |
⊢ Ⅎ 𝑖 ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 102 |
101
|
nfrn |
⊢ Ⅎ 𝑖 ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 103 |
|
nfcv |
⊢ Ⅎ 𝑖 ℝ* |
| 104 |
|
nfcv |
⊢ Ⅎ 𝑖 < |
| 105 |
102 103 104
|
nfinf |
⊢ Ⅎ 𝑖 inf ( ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) |
| 106 |
|
infxrcl |
⊢ ( ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) ⊆ ℝ* → inf ( ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) ∈ ℝ* ) |
| 107 |
92 106
|
ax-mp |
⊢ inf ( ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) ∈ ℝ* |
| 108 |
107
|
a1i |
⊢ ( 𝜑 → inf ( ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) ∈ ℝ* ) |
| 109 |
99 100 105 94 108
|
supxrleubrnmptf |
⊢ ( 𝜑 → ( sup ( ran ( 𝑖 ∈ ℝ ↦ inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) ≤ inf ( ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) ↔ ∀ 𝑖 ∈ ℝ inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ≤ inf ( ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) ) ) |
| 110 |
98 109
|
mpbird |
⊢ ( 𝜑 → sup ( ran ( 𝑖 ∈ ℝ ↦ inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) ≤ inf ( ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) ) |
| 111 |
|
eqid |
⊢ ( 𝑖 ∈ ℝ ↦ inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) = ( 𝑖 ∈ ℝ ↦ inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) |
| 112 |
1 111
|
liminfvald |
⊢ ( 𝜑 → ( lim inf ‘ 𝐹 ) = sup ( ran ( 𝑖 ∈ ℝ ↦ inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) ) |
| 113 |
1 90
|
limsupvald |
⊢ ( 𝜑 → ( lim sup ‘ 𝐹 ) = inf ( ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) ) |
| 114 |
112 113
|
breq12d |
⊢ ( 𝜑 → ( ( lim inf ‘ 𝐹 ) ≤ ( lim sup ‘ 𝐹 ) ↔ sup ( ran ( 𝑖 ∈ ℝ ↦ inf ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) ≤ inf ( ran ( 𝑖 ∈ ℝ ↦ sup ( ( ( 𝐹 “ ( 𝑖 [,) +∞ ) ) ∩ ℝ* ) , ℝ* , < ) ) , ℝ* , < ) ) ) |
| 115 |
110 114
|
mpbird |
⊢ ( 𝜑 → ( lim inf ‘ 𝐹 ) ≤ ( lim sup ‘ 𝐹 ) ) |