Step |
Hyp |
Ref |
Expression |
1 |
|
o1lo1.1 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
2 |
|
o1dm |
⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ ) |
3 |
2
|
a1i |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ ) ) |
4 |
|
lo1dm |
⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ≤𝑂(1) → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ ) |
5 |
4
|
adantr |
⊢ ( ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ≤𝑂(1) ∧ ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ∈ ≤𝑂(1) ) → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ ) |
6 |
5
|
a1i |
⊢ ( 𝜑 → ( ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ≤𝑂(1) ∧ ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ∈ ≤𝑂(1) ) → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ ) ) |
7 |
1
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ℝ ) |
8 |
|
dmmptg |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ℝ → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 ) |
9 |
7 8
|
syl |
⊢ ( 𝜑 → dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 ) |
10 |
9
|
sseq1d |
⊢ ( 𝜑 → ( dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ ↔ 𝐴 ⊆ ℝ ) ) |
11 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ 𝑚 ∈ ℝ ) → 𝑚 ∈ ℝ ) |
12 |
1
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
13 |
12
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ 𝑚 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
14 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ 𝑚 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → 𝑚 ∈ ℝ ) |
15 |
13 14
|
absled |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ 𝑚 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → ( ( abs ‘ 𝐵 ) ≤ 𝑚 ↔ ( - 𝑚 ≤ 𝐵 ∧ 𝐵 ≤ 𝑚 ) ) ) |
16 |
|
ancom |
⊢ ( ( - 𝑚 ≤ 𝐵 ∧ 𝐵 ≤ 𝑚 ) ↔ ( 𝐵 ≤ 𝑚 ∧ - 𝑚 ≤ 𝐵 ) ) |
17 |
|
lenegcon1 |
⊢ ( ( 𝑚 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( - 𝑚 ≤ 𝐵 ↔ - 𝐵 ≤ 𝑚 ) ) |
18 |
14 13 17
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ 𝑚 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → ( - 𝑚 ≤ 𝐵 ↔ - 𝐵 ≤ 𝑚 ) ) |
19 |
18
|
anbi2d |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ 𝑚 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐵 ≤ 𝑚 ∧ - 𝑚 ≤ 𝐵 ) ↔ ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑚 ) ) ) |
20 |
16 19
|
syl5bb |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ 𝑚 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → ( ( - 𝑚 ≤ 𝐵 ∧ 𝐵 ≤ 𝑚 ) ↔ ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑚 ) ) ) |
21 |
15 20
|
bitrd |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ 𝑚 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → ( ( abs ‘ 𝐵 ) ≤ 𝑚 ↔ ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑚 ) ) ) |
22 |
21
|
imbi2d |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ 𝑚 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ↔ ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑚 ) ) ) ) |
23 |
22
|
ralbidva |
⊢ ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ 𝑚 ∈ ℝ ) → ( ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ↔ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑚 ) ) ) ) |
24 |
23
|
rexbidv |
⊢ ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ 𝑚 ∈ ℝ ) → ( ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ↔ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑚 ) ) ) ) |
25 |
24
|
biimpd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ 𝑚 ∈ ℝ ) → ( ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) → ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑚 ) ) ) ) |
26 |
|
breq2 |
⊢ ( 𝑛 = 𝑚 → ( 𝐵 ≤ 𝑛 ↔ 𝐵 ≤ 𝑚 ) ) |
27 |
26
|
anbi1d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) ↔ ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑝 ) ) ) |
28 |
27
|
imbi2d |
⊢ ( 𝑛 = 𝑚 → ( ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) ) ↔ ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑝 ) ) ) ) |
29 |
28
|
rexralbidv |
⊢ ( 𝑛 = 𝑚 → ( ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) ) ↔ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑝 ) ) ) ) |
30 |
|
breq2 |
⊢ ( 𝑝 = 𝑚 → ( - 𝐵 ≤ 𝑝 ↔ - 𝐵 ≤ 𝑚 ) ) |
31 |
30
|
anbi2d |
⊢ ( 𝑝 = 𝑚 → ( ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑝 ) ↔ ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑚 ) ) ) |
32 |
31
|
imbi2d |
⊢ ( 𝑝 = 𝑚 → ( ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑝 ) ) ↔ ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑚 ) ) ) ) |
33 |
32
|
rexralbidv |
⊢ ( 𝑝 = 𝑚 → ( ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑝 ) ) ↔ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑚 ) ) ) ) |
34 |
29 33
|
rspc2ev |
⊢ ( ( 𝑚 ∈ ℝ ∧ 𝑚 ∈ ℝ ∧ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑚 ) ) ) → ∃ 𝑛 ∈ ℝ ∃ 𝑝 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) ) ) |
35 |
34
|
3anidm12 |
⊢ ( ( 𝑚 ∈ ℝ ∧ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑚 ∧ - 𝐵 ≤ 𝑚 ) ) ) → ∃ 𝑛 ∈ ℝ ∃ 𝑝 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) ) ) |
36 |
11 25 35
|
syl6an |
⊢ ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ 𝑚 ∈ ℝ ) → ( ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) → ∃ 𝑛 ∈ ℝ ∃ 𝑝 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) ) ) ) |
37 |
36
|
rexlimdva |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) → ( ∃ 𝑚 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) → ∃ 𝑛 ∈ ℝ ∃ 𝑝 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) ) ) ) |
38 |
|
simplrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑛 ≤ 𝑝 ) → 𝑝 ∈ ℝ ) |
39 |
|
simplrl |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ ¬ 𝑛 ≤ 𝑝 ) → 𝑛 ∈ ℝ ) |
40 |
38 39
|
ifclda |
⊢ ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) → if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ∈ ℝ ) |
41 |
|
max2 |
⊢ ( ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) → 𝑝 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) |
42 |
41
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → 𝑝 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) |
43 |
12
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
44 |
43
|
renegcld |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → - 𝐵 ∈ ℝ ) |
45 |
|
simplrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → 𝑝 ∈ ℝ ) |
46 |
|
simplrl |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → 𝑛 ∈ ℝ ) |
47 |
45 46
|
ifcld |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ∈ ℝ ) |
48 |
|
letr |
⊢ ( ( - 𝐵 ∈ ℝ ∧ 𝑝 ∈ ℝ ∧ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ∈ ℝ ) → ( ( - 𝐵 ≤ 𝑝 ∧ 𝑝 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) → - 𝐵 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) |
49 |
44 45 47 48
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ( - 𝐵 ≤ 𝑝 ∧ 𝑝 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) → - 𝐵 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) |
50 |
42 49
|
mpan2d |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( - 𝐵 ≤ 𝑝 → - 𝐵 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) |
51 |
|
lenegcon1 |
⊢ ( ( 𝐵 ∈ ℝ ∧ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ∈ ℝ ) → ( - 𝐵 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ↔ - if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ≤ 𝐵 ) ) |
52 |
43 47 51
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( - 𝐵 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ↔ - if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ≤ 𝐵 ) ) |
53 |
50 52
|
sylibd |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( - 𝐵 ≤ 𝑝 → - if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ≤ 𝐵 ) ) |
54 |
|
max1 |
⊢ ( ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) → 𝑛 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) |
55 |
54
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → 𝑛 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) |
56 |
|
letr |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ∈ ℝ ) → ( ( 𝐵 ≤ 𝑛 ∧ 𝑛 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) → 𝐵 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) |
57 |
43 46 47 56
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐵 ≤ 𝑛 ∧ 𝑛 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) → 𝐵 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) |
58 |
55 57
|
mpan2d |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐵 ≤ 𝑛 → 𝐵 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) |
59 |
53 58
|
anim12d |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ( - 𝐵 ≤ 𝑝 ∧ 𝐵 ≤ 𝑛 ) → ( - if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ≤ 𝐵 ∧ 𝐵 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) ) |
60 |
59
|
ancomsd |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) → ( - if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ≤ 𝐵 ∧ 𝐵 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) ) |
61 |
43 47
|
absled |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ( abs ‘ 𝐵 ) ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ↔ ( - if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ≤ 𝐵 ∧ 𝐵 ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) ) |
62 |
60 61
|
sylibrd |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) → ( abs ‘ 𝐵 ) ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) |
63 |
62
|
imim2d |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) ) → ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) ) |
64 |
63
|
ralimdva |
⊢ ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) → ( ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) ) → ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) ) |
65 |
64
|
reximdv |
⊢ ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) → ( ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) ) → ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) ) |
66 |
|
breq2 |
⊢ ( 𝑚 = if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) → ( ( abs ‘ 𝐵 ) ≤ 𝑚 ↔ ( abs ‘ 𝐵 ) ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) |
67 |
66
|
imbi2d |
⊢ ( 𝑚 = if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) → ( ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ↔ ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) ) |
68 |
67
|
rexralbidv |
⊢ ( 𝑚 = if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) → ( ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ↔ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) ) |
69 |
68
|
rspcev |
⊢ ( ( if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ∈ ℝ ∧ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ if ( 𝑛 ≤ 𝑝 , 𝑝 , 𝑛 ) ) ) → ∃ 𝑚 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ) |
70 |
40 65 69
|
syl6an |
⊢ ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ ( 𝑛 ∈ ℝ ∧ 𝑝 ∈ ℝ ) ) → ( ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) ) → ∃ 𝑚 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ) ) |
71 |
70
|
rexlimdvva |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) → ( ∃ 𝑛 ∈ ℝ ∃ 𝑝 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) ) → ∃ 𝑚 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ) ) |
72 |
37 71
|
impbid |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) → ( ∃ 𝑚 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ↔ ∃ 𝑛 ∈ ℝ ∃ 𝑝 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) ) ) ) |
73 |
|
rexanre |
⊢ ( 𝐴 ⊆ ℝ → ( ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) ) ↔ ( ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → 𝐵 ≤ 𝑛 ) ∧ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → - 𝐵 ≤ 𝑝 ) ) ) ) |
74 |
73
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) → ( ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) ) ↔ ( ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → 𝐵 ≤ 𝑛 ) ∧ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → - 𝐵 ≤ 𝑝 ) ) ) ) |
75 |
74
|
2rexbidv |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) → ( ∃ 𝑛 ∈ ℝ ∃ 𝑝 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( 𝐵 ≤ 𝑛 ∧ - 𝐵 ≤ 𝑝 ) ) ↔ ∃ 𝑛 ∈ ℝ ∃ 𝑝 ∈ ℝ ( ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → 𝐵 ≤ 𝑛 ) ∧ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → - 𝐵 ≤ 𝑝 ) ) ) ) |
76 |
72 75
|
bitrd |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) → ( ∃ 𝑚 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ↔ ∃ 𝑛 ∈ ℝ ∃ 𝑝 ∈ ℝ ( ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → 𝐵 ≤ 𝑛 ) ∧ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → - 𝐵 ≤ 𝑝 ) ) ) ) |
77 |
|
reeanv |
⊢ ( ∃ 𝑛 ∈ ℝ ∃ 𝑝 ∈ ℝ ( ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → 𝐵 ≤ 𝑛 ) ∧ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → - 𝐵 ≤ 𝑝 ) ) ↔ ( ∃ 𝑛 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → 𝐵 ≤ 𝑛 ) ∧ ∃ 𝑝 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → - 𝐵 ≤ 𝑝 ) ) ) |
78 |
76 77
|
bitrdi |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) → ( ∃ 𝑚 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ↔ ( ∃ 𝑛 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → 𝐵 ≤ 𝑛 ) ∧ ∃ 𝑝 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → - 𝐵 ≤ 𝑝 ) ) ) ) |
79 |
|
rexcom |
⊢ ( ∃ 𝑐 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ↔ ∃ 𝑚 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ) |
80 |
|
rexcom |
⊢ ( ∃ 𝑐 ∈ ℝ ∃ 𝑛 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → 𝐵 ≤ 𝑛 ) ↔ ∃ 𝑛 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → 𝐵 ≤ 𝑛 ) ) |
81 |
|
rexcom |
⊢ ( ∃ 𝑐 ∈ ℝ ∃ 𝑝 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → - 𝐵 ≤ 𝑝 ) ↔ ∃ 𝑝 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → - 𝐵 ≤ 𝑝 ) ) |
82 |
80 81
|
anbi12i |
⊢ ( ( ∃ 𝑐 ∈ ℝ ∃ 𝑛 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → 𝐵 ≤ 𝑛 ) ∧ ∃ 𝑐 ∈ ℝ ∃ 𝑝 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → - 𝐵 ≤ 𝑝 ) ) ↔ ( ∃ 𝑛 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → 𝐵 ≤ 𝑛 ) ∧ ∃ 𝑝 ∈ ℝ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → - 𝐵 ≤ 𝑝 ) ) ) |
83 |
78 79 82
|
3bitr4g |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) → ( ∃ 𝑐 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ↔ ( ∃ 𝑐 ∈ ℝ ∃ 𝑛 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → 𝐵 ≤ 𝑛 ) ∧ ∃ 𝑐 ∈ ℝ ∃ 𝑝 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → - 𝐵 ≤ 𝑝 ) ) ) ) |
84 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) → 𝐴 ⊆ ℝ ) |
85 |
12
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
86 |
84 85
|
elo1mpt |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) ↔ ∃ 𝑐 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ) ) |
87 |
84 12
|
ello1mpt |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ≤𝑂(1) ↔ ∃ 𝑐 ∈ ℝ ∃ 𝑛 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → 𝐵 ≤ 𝑛 ) ) ) |
88 |
12
|
renegcld |
⊢ ( ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → - 𝐵 ∈ ℝ ) |
89 |
84 88
|
ello1mpt |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) → ( ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ∈ ≤𝑂(1) ↔ ∃ 𝑐 ∈ ℝ ∃ 𝑝 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → - 𝐵 ≤ 𝑝 ) ) ) |
90 |
87 89
|
anbi12d |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) → ( ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ≤𝑂(1) ∧ ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ∈ ≤𝑂(1) ) ↔ ( ∃ 𝑐 ∈ ℝ ∃ 𝑛 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → 𝐵 ≤ 𝑛 ) ∧ ∃ 𝑐 ∈ ℝ ∃ 𝑝 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑐 ≤ 𝑥 → - 𝐵 ≤ 𝑝 ) ) ) ) |
91 |
83 86 90
|
3bitr4d |
⊢ ( ( 𝜑 ∧ 𝐴 ⊆ ℝ ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) ↔ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ≤𝑂(1) ∧ ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ∈ ≤𝑂(1) ) ) ) |
92 |
91
|
ex |
⊢ ( 𝜑 → ( 𝐴 ⊆ ℝ → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) ↔ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ≤𝑂(1) ∧ ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ∈ ≤𝑂(1) ) ) ) ) |
93 |
10 92
|
sylbid |
⊢ ( 𝜑 → ( dom ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) ↔ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ≤𝑂(1) ∧ ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ∈ ≤𝑂(1) ) ) ) ) |
94 |
3 6 93
|
pm5.21ndd |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) ↔ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ≤𝑂(1) ∧ ( 𝑥 ∈ 𝐴 ↦ - 𝐵 ) ∈ ≤𝑂(1) ) ) ) |