Step |
Hyp |
Ref |
Expression |
1 |
|
rexabslelem.1 |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
rexabslelem.2 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
3 |
|
simp2 |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) → 𝑦 ∈ ℝ ) |
4 |
|
nfv |
⊢ Ⅎ 𝑥 𝑦 ∈ ℝ |
5 |
|
nfra1 |
⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 |
6 |
1 4 5
|
nf3an |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) |
7 |
2
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
8 |
2
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
9 |
8
|
3ad2antl1 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
10 |
9
|
abscld |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ 𝐵 ) ∈ ℝ ) |
11 |
3
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ ℝ ) |
12 |
7
|
leabsd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≤ ( abs ‘ 𝐵 ) ) |
13 |
|
rspa |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ 𝐵 ) ≤ 𝑦 ) |
14 |
13
|
3ad2antl3 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ 𝐵 ) ≤ 𝑦 ) |
15 |
7 10 11 12 14
|
letrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≤ 𝑦 ) |
16 |
15
|
ex |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) → ( 𝑥 ∈ 𝐴 → 𝐵 ≤ 𝑦 ) ) |
17 |
6 16
|
ralrimi |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) → ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) |
18 |
|
brralrspcev |
⊢ ( ( 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) → ∃ 𝑤 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) |
19 |
3 17 18
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) → ∃ 𝑤 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) |
20 |
3
|
renegcld |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) → - 𝑦 ∈ ℝ ) |
21 |
2
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
22 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ ℝ ) |
23 |
|
absle |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( ( abs ‘ 𝐵 ) ≤ 𝑦 ↔ ( - 𝑦 ≤ 𝐵 ∧ 𝐵 ≤ 𝑦 ) ) ) |
24 |
21 22 23
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → ( ( abs ‘ 𝐵 ) ≤ 𝑦 ↔ ( - 𝑦 ≤ 𝐵 ∧ 𝐵 ≤ 𝑦 ) ) ) |
25 |
24
|
3adantl3 |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ) → ( ( abs ‘ 𝐵 ) ≤ 𝑦 ↔ ( - 𝑦 ≤ 𝐵 ∧ 𝐵 ≤ 𝑦 ) ) ) |
26 |
14 25
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ) → ( - 𝑦 ≤ 𝐵 ∧ 𝐵 ≤ 𝑦 ) ) |
27 |
26
|
simpld |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ) → - 𝑦 ≤ 𝐵 ) |
28 |
27
|
ex |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) → ( 𝑥 ∈ 𝐴 → - 𝑦 ≤ 𝐵 ) ) |
29 |
6 28
|
ralrimi |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) → ∀ 𝑥 ∈ 𝐴 - 𝑦 ≤ 𝐵 ) |
30 |
|
breq1 |
⊢ ( 𝑧 = - 𝑦 → ( 𝑧 ≤ 𝐵 ↔ - 𝑦 ≤ 𝐵 ) ) |
31 |
30
|
ralbidv |
⊢ ( 𝑧 = - 𝑦 → ( ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 - 𝑦 ≤ 𝐵 ) ) |
32 |
31
|
rspcev |
⊢ ( ( - 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 - 𝑦 ≤ 𝐵 ) → ∃ 𝑧 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) |
33 |
20 29 32
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) → ∃ 𝑧 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) |
34 |
19 33
|
jca |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) → ( ∃ 𝑤 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ∧ ∃ 𝑧 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ) |
35 |
34
|
3exp |
⊢ ( 𝜑 → ( 𝑦 ∈ ℝ → ( ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 → ( ∃ 𝑤 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ∧ ∃ 𝑧 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ) ) ) |
36 |
35
|
rexlimdv |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 → ( ∃ 𝑤 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ∧ ∃ 𝑧 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ) ) |
37 |
|
renegcl |
⊢ ( 𝑧 ∈ ℝ → - 𝑧 ∈ ℝ ) |
38 |
37
|
adantl |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → - 𝑧 ∈ ℝ ) |
39 |
|
simpl |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → 𝑤 ∈ ℝ ) |
40 |
38 39
|
ifcld |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ∈ ℝ ) |
41 |
40
|
ad5ant24 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) → if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ∈ ℝ ) |
42 |
|
nfv |
⊢ Ⅎ 𝑥 𝑤 ∈ ℝ |
43 |
1 42
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑤 ∈ ℝ ) |
44 |
|
nfra1 |
⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 |
45 |
43 44
|
nfan |
⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) |
46 |
|
nfv |
⊢ Ⅎ 𝑥 𝑧 ∈ ℝ |
47 |
45 46
|
nfan |
⊢ Ⅎ 𝑥 ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) |
48 |
|
nfra1 |
⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 |
49 |
47 48
|
nfan |
⊢ Ⅎ 𝑥 ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) |
50 |
40
|
ad5ant23 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ∈ ℝ ) |
51 |
50
|
renegcld |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → - if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ∈ ℝ ) |
52 |
|
simpllr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑧 ∈ ℝ ) |
53 |
2
|
ad5ant15 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
54 |
|
max2 |
⊢ ( ( 𝑤 ∈ ℝ ∧ - 𝑧 ∈ ℝ ) → - 𝑧 ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ) |
55 |
39 38 54
|
syl2anc |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → - 𝑧 ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ) |
56 |
38 40
|
lenegd |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → ( - 𝑧 ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ↔ - if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ≤ - - 𝑧 ) ) |
57 |
55 56
|
mpbid |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → - if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ≤ - - 𝑧 ) |
58 |
|
recn |
⊢ ( 𝑧 ∈ ℝ → 𝑧 ∈ ℂ ) |
59 |
58
|
adantl |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → 𝑧 ∈ ℂ ) |
60 |
59
|
negnegd |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → - - 𝑧 = 𝑧 ) |
61 |
57 60
|
breqtrd |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → - if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ≤ 𝑧 ) |
62 |
61
|
ad5ant23 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → - if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ≤ 𝑧 ) |
63 |
|
rspa |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ∧ 𝑥 ∈ 𝐴 ) → 𝑧 ≤ 𝐵 ) |
64 |
63
|
adantll |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑧 ≤ 𝐵 ) |
65 |
51 52 53 62 64
|
letrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → - if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ≤ 𝐵 ) |
66 |
65
|
adantl3r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → - if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ≤ 𝐵 ) |
67 |
2
|
ad5ant15 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
68 |
|
simp-4r |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → 𝑤 ∈ ℝ ) |
69 |
40
|
ad5ant24 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ∈ ℝ ) |
70 |
|
rspa |
⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≤ 𝑤 ) |
71 |
70
|
ad4ant24 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≤ 𝑤 ) |
72 |
|
max1 |
⊢ ( ( 𝑤 ∈ ℝ ∧ - 𝑧 ∈ ℝ ) → 𝑤 ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ) |
73 |
39 38 72
|
syl2anc |
⊢ ( ( 𝑤 ∈ ℝ ∧ 𝑧 ∈ ℝ ) → 𝑤 ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ) |
74 |
73
|
ad5ant24 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → 𝑤 ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ) |
75 |
67 68 69 71 74
|
letrd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ) |
76 |
75
|
adantlr |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ) |
77 |
66 76
|
jca |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → ( - if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ≤ 𝐵 ∧ 𝐵 ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ) ) |
78 |
2
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
79 |
78
|
3adant2 |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ 𝑧 ∈ ℝ ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
80 |
40
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ 𝑧 ∈ ℝ ) → if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ∈ ℝ ) |
81 |
80
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ 𝑧 ∈ ℝ ∧ 𝑥 ∈ 𝐴 ) → if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ∈ ℝ ) |
82 |
79 81
|
absled |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ 𝑧 ∈ ℝ ∧ 𝑥 ∈ 𝐴 ) → ( ( abs ‘ 𝐵 ) ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ↔ ( - if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ≤ 𝐵 ∧ 𝐵 ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ) ) ) |
83 |
82
|
ad5ant135 |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → ( ( abs ‘ 𝐵 ) ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ↔ ( - if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ≤ 𝐵 ∧ 𝐵 ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ) ) ) |
84 |
77 83
|
mpbird |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ 𝐵 ) ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ) |
85 |
84
|
ex |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) → ( 𝑥 ∈ 𝐴 → ( abs ‘ 𝐵 ) ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ) ) |
86 |
49 85
|
ralrimi |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) → ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ) |
87 |
|
brralrspcev |
⊢ ( ( if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ if ( 𝑤 ≤ - 𝑧 , - 𝑧 , 𝑤 ) ) → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) |
88 |
41 86 87
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ 𝑧 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) |
89 |
88
|
exp31 |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) → ( 𝑧 ∈ ℝ → ( ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) ) ) |
90 |
89
|
exp31 |
⊢ ( 𝜑 → ( 𝑤 ∈ ℝ → ( ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 → ( 𝑧 ∈ ℝ → ( ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) ) ) ) ) |
91 |
90
|
rexlimdv |
⊢ ( 𝜑 → ( ∃ 𝑤 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 → ( 𝑧 ∈ ℝ → ( ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) ) ) ) |
92 |
91
|
imp |
⊢ ( ( 𝜑 ∧ ∃ 𝑤 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) → ( 𝑧 ∈ ℝ → ( ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) ) ) |
93 |
92
|
rexlimdv |
⊢ ( ( 𝜑 ∧ ∃ 𝑤 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) → ( ∃ 𝑧 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) ) |
94 |
93
|
imp |
⊢ ( ( ( 𝜑 ∧ ∃ 𝑤 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ) ∧ ∃ 𝑧 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) |
95 |
94
|
anasss |
⊢ ( ( 𝜑 ∧ ( ∃ 𝑤 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ∧ ∃ 𝑧 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ) → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) |
96 |
95
|
ex |
⊢ ( 𝜑 → ( ( ∃ 𝑤 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ∧ ∃ 𝑧 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ) ) |
97 |
36 96
|
impbid |
⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( abs ‘ 𝐵 ) ≤ 𝑦 ↔ ( ∃ 𝑤 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑤 ∧ ∃ 𝑧 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝐵 ) ) ) |