Step |
Hyp |
Ref |
Expression |
1 |
|
lo1resb.1 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ ℝ ) |
2 |
|
lo1resb.2 |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ ) |
3 |
|
lo1resb.3 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
4 |
|
lo1res |
⊢ ( 𝐹 ∈ ≤𝑂(1) → ( 𝐹 ↾ ( 𝐵 [,) +∞ ) ) ∈ ≤𝑂(1) ) |
5 |
1
|
feqmptd |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
6 |
5
|
reseq1d |
⊢ ( 𝜑 → ( 𝐹 ↾ ( 𝐵 [,) +∞ ) ) = ( ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) ↾ ( 𝐵 [,) +∞ ) ) ) |
7 |
|
resmpt3 |
⊢ ( ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) ↾ ( 𝐵 [,) +∞ ) ) = ( 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) |
8 |
6 7
|
eqtrdi |
⊢ ( 𝜑 → ( 𝐹 ↾ ( 𝐵 [,) +∞ ) ) = ( 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
9 |
8
|
eleq1d |
⊢ ( 𝜑 → ( ( 𝐹 ↾ ( 𝐵 [,) +∞ ) ) ∈ ≤𝑂(1) ↔ ( 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ ≤𝑂(1) ) ) |
10 |
|
inss1 |
⊢ ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) ⊆ 𝐴 |
11 |
10 2
|
sstrid |
⊢ ( 𝜑 → ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) ⊆ ℝ ) |
12 |
|
elinel1 |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) → 𝑥 ∈ 𝐴 ) |
13 |
|
ffvelrn |
⊢ ( ( 𝐹 : 𝐴 ⟶ ℝ ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
14 |
1 12 13
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
15 |
11 14
|
ello1mpt |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ ≤𝑂(1) ↔ ∃ 𝑦 ∈ ℝ ∃ 𝑧 ∈ ℝ ∀ 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) ( 𝑦 ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) ) |
16 |
|
elin |
⊢ ( 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ( 𝐵 [,) +∞ ) ) ) |
17 |
16
|
imbi1i |
⊢ ( ( 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) → ( 𝑦 ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ( 𝐵 [,) +∞ ) ) → ( 𝑦 ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) ) |
18 |
|
impexp |
⊢ ( ( ( 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ( 𝐵 [,) +∞ ) ) → ( 𝑦 ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ ( 𝐵 [,) +∞ ) → ( 𝑦 ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) ) ) |
19 |
17 18
|
bitri |
⊢ ( ( 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) → ( 𝑦 ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ ( 𝐵 [,) +∞ ) → ( 𝑦 ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) ) ) |
20 |
|
impexp |
⊢ ( ( ( 𝑥 ∈ ( 𝐵 [,) +∞ ) ∧ 𝑦 ≤ 𝑥 ) → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ↔ ( 𝑥 ∈ ( 𝐵 [,) +∞ ) → ( 𝑦 ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) ) |
21 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
22 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) → 𝐴 ⊆ ℝ ) |
23 |
22
|
sselda |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ℝ ) |
24 |
|
elicopnf |
⊢ ( 𝐵 ∈ ℝ → ( 𝑥 ∈ ( 𝐵 [,) +∞ ) ↔ ( 𝑥 ∈ ℝ ∧ 𝐵 ≤ 𝑥 ) ) ) |
25 |
24
|
baibd |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( 𝑥 ∈ ( 𝐵 [,) +∞ ) ↔ 𝐵 ≤ 𝑥 ) ) |
26 |
21 23 25
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑥 ∈ ( 𝐵 [,) +∞ ) ↔ 𝐵 ≤ 𝑥 ) ) |
27 |
26
|
anbi1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑥 ∈ ( 𝐵 [,) +∞ ) ∧ 𝑦 ≤ 𝑥 ) ↔ ( 𝐵 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥 ) ) ) |
28 |
|
simplrl |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ ℝ ) |
29 |
|
maxle |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( if ( 𝐵 ≤ 𝑦 , 𝑦 , 𝐵 ) ≤ 𝑥 ↔ ( 𝐵 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥 ) ) ) |
30 |
21 28 23 29
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( if ( 𝐵 ≤ 𝑦 , 𝑦 , 𝐵 ) ≤ 𝑥 ↔ ( 𝐵 ≤ 𝑥 ∧ 𝑦 ≤ 𝑥 ) ) ) |
31 |
27 30
|
bitr4d |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑥 ∈ ( 𝐵 [,) +∞ ) ∧ 𝑦 ≤ 𝑥 ) ↔ if ( 𝐵 ≤ 𝑦 , 𝑦 , 𝐵 ) ≤ 𝑥 ) ) |
32 |
31
|
imbi1d |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ( ( 𝑥 ∈ ( 𝐵 [,) +∞ ) ∧ 𝑦 ≤ 𝑥 ) → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ↔ ( if ( 𝐵 ≤ 𝑦 , 𝑦 , 𝐵 ) ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) ) |
33 |
20 32
|
bitr3id |
⊢ ( ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑥 ∈ ( 𝐵 [,) +∞ ) → ( 𝑦 ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) ↔ ( if ( 𝐵 ≤ 𝑦 , 𝑦 , 𝐵 ) ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) ) |
34 |
33
|
pm5.74da |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) → ( ( 𝑥 ∈ 𝐴 → ( 𝑥 ∈ ( 𝐵 [,) +∞ ) → ( 𝑦 ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) ) ↔ ( 𝑥 ∈ 𝐴 → ( if ( 𝐵 ≤ 𝑦 , 𝑦 , 𝐵 ) ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) ) ) |
35 |
19 34
|
syl5bb |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) → ( ( 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) → ( 𝑦 ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) ↔ ( 𝑥 ∈ 𝐴 → ( if ( 𝐵 ≤ 𝑦 , 𝑦 , 𝐵 ) ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) ) ) |
36 |
35
|
ralbidv2 |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) → ( ∀ 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) ( 𝑦 ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ↔ ∀ 𝑥 ∈ 𝐴 ( if ( 𝐵 ≤ 𝑦 , 𝑦 , 𝐵 ) ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) ) |
37 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) → 𝐹 : 𝐴 ⟶ ℝ ) |
38 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) → 𝑦 ∈ ℝ ) |
39 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) → 𝐵 ∈ ℝ ) |
40 |
38 39
|
ifcld |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) → if ( 𝐵 ≤ 𝑦 , 𝑦 , 𝐵 ) ∈ ℝ ) |
41 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) → 𝑧 ∈ ℝ ) |
42 |
|
ello12r |
⊢ ( ( ( 𝐹 : 𝐴 ⟶ ℝ ∧ 𝐴 ⊆ ℝ ) ∧ ( if ( 𝐵 ≤ 𝑦 , 𝑦 , 𝐵 ) ∈ ℝ ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 ( if ( 𝐵 ≤ 𝑦 , 𝑦 , 𝐵 ) ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) ) → 𝐹 ∈ ≤𝑂(1) ) |
43 |
42
|
3expia |
⊢ ( ( ( 𝐹 : 𝐴 ⟶ ℝ ∧ 𝐴 ⊆ ℝ ) ∧ ( if ( 𝐵 ≤ 𝑦 , 𝑦 , 𝐵 ) ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) → ( ∀ 𝑥 ∈ 𝐴 ( if ( 𝐵 ≤ 𝑦 , 𝑦 , 𝐵 ) ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) → 𝐹 ∈ ≤𝑂(1) ) ) |
44 |
37 22 40 41 43
|
syl22anc |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) → ( ∀ 𝑥 ∈ 𝐴 ( if ( 𝐵 ≤ 𝑦 , 𝑦 , 𝐵 ) ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) → 𝐹 ∈ ≤𝑂(1) ) ) |
45 |
36 44
|
sylbid |
⊢ ( ( 𝜑 ∧ ( 𝑦 ∈ ℝ ∧ 𝑧 ∈ ℝ ) ) → ( ∀ 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) ( 𝑦 ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) → 𝐹 ∈ ≤𝑂(1) ) ) |
46 |
45
|
rexlimdvva |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ℝ ∃ 𝑧 ∈ ℝ ∀ 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) ( 𝑦 ≤ 𝑥 → ( 𝐹 ‘ 𝑥 ) ≤ 𝑧 ) → 𝐹 ∈ ≤𝑂(1) ) ) |
47 |
15 46
|
sylbid |
⊢ ( 𝜑 → ( ( 𝑥 ∈ ( 𝐴 ∩ ( 𝐵 [,) +∞ ) ) ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ ≤𝑂(1) → 𝐹 ∈ ≤𝑂(1) ) ) |
48 |
9 47
|
sylbid |
⊢ ( 𝜑 → ( ( 𝐹 ↾ ( 𝐵 [,) +∞ ) ) ∈ ≤𝑂(1) → 𝐹 ∈ ≤𝑂(1) ) ) |
49 |
4 48
|
impbid2 |
⊢ ( 𝜑 → ( 𝐹 ∈ ≤𝑂(1) ↔ ( 𝐹 ↾ ( 𝐵 [,) +∞ ) ) ∈ ≤𝑂(1) ) ) |