Step |
Hyp |
Ref |
Expression |
1 |
|
dvlog2.s |
⊢ 𝑆 = ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) |
2 |
|
cnxmet |
⊢ ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) |
3 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
4 |
|
1xr |
⊢ 1 ∈ ℝ* |
5 |
|
blssm |
⊢ ( ( ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ∧ 1 ∈ ℂ ∧ 1 ∈ ℝ* ) → ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ℂ ) |
6 |
2 3 4 5
|
mp3an |
⊢ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ⊆ ℂ |
7 |
1 6
|
eqsstri |
⊢ 𝑆 ⊆ ℂ |
8 |
7
|
sseli |
⊢ ( 𝑥 ∈ 𝑆 → 𝑥 ∈ ℂ ) |
9 |
|
1red |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → 1 ∈ ℝ ) |
10 |
|
cnmet |
⊢ ( abs ∘ − ) ∈ ( Met ‘ ℂ ) |
11 |
|
mnfxr |
⊢ -∞ ∈ ℝ* |
12 |
|
0re |
⊢ 0 ∈ ℝ |
13 |
|
iocssre |
⊢ ( ( -∞ ∈ ℝ* ∧ 0 ∈ ℝ ) → ( -∞ (,] 0 ) ⊆ ℝ ) |
14 |
11 12 13
|
mp2an |
⊢ ( -∞ (,] 0 ) ⊆ ℝ |
15 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
16 |
14 15
|
sstri |
⊢ ( -∞ (,] 0 ) ⊆ ℂ |
17 |
16
|
sseli |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → 𝑥 ∈ ℂ ) |
18 |
|
metcl |
⊢ ( ( ( abs ∘ − ) ∈ ( Met ‘ ℂ ) ∧ 1 ∈ ℂ ∧ 𝑥 ∈ ℂ ) → ( 1 ( abs ∘ − ) 𝑥 ) ∈ ℝ ) |
19 |
10 3 17 18
|
mp3an12i |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → ( 1 ( abs ∘ − ) 𝑥 ) ∈ ℝ ) |
20 |
|
1m0e1 |
⊢ ( 1 − 0 ) = 1 |
21 |
14
|
sseli |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → 𝑥 ∈ ℝ ) |
22 |
12
|
a1i |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → 0 ∈ ℝ ) |
23 |
|
elioc2 |
⊢ ( ( -∞ ∈ ℝ* ∧ 0 ∈ ℝ ) → ( 𝑥 ∈ ( -∞ (,] 0 ) ↔ ( 𝑥 ∈ ℝ ∧ -∞ < 𝑥 ∧ 𝑥 ≤ 0 ) ) ) |
24 |
11 12 23
|
mp2an |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) ↔ ( 𝑥 ∈ ℝ ∧ -∞ < 𝑥 ∧ 𝑥 ≤ 0 ) ) |
25 |
24
|
simp3bi |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → 𝑥 ≤ 0 ) |
26 |
21 22 9 25
|
lesub2dd |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → ( 1 − 0 ) ≤ ( 1 − 𝑥 ) ) |
27 |
20 26
|
eqbrtrrid |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → 1 ≤ ( 1 − 𝑥 ) ) |
28 |
|
eqid |
⊢ ( abs ∘ − ) = ( abs ∘ − ) |
29 |
28
|
cnmetdval |
⊢ ( ( 1 ∈ ℂ ∧ 𝑥 ∈ ℂ ) → ( 1 ( abs ∘ − ) 𝑥 ) = ( abs ‘ ( 1 − 𝑥 ) ) ) |
30 |
3 17 29
|
sylancr |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → ( 1 ( abs ∘ − ) 𝑥 ) = ( abs ‘ ( 1 − 𝑥 ) ) ) |
31 |
|
0le1 |
⊢ 0 ≤ 1 |
32 |
31
|
a1i |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → 0 ≤ 1 ) |
33 |
21 22 9 25 32
|
letrd |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → 𝑥 ≤ 1 ) |
34 |
21 9 33
|
abssubge0d |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → ( abs ‘ ( 1 − 𝑥 ) ) = ( 1 − 𝑥 ) ) |
35 |
30 34
|
eqtrd |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → ( 1 ( abs ∘ − ) 𝑥 ) = ( 1 − 𝑥 ) ) |
36 |
27 35
|
breqtrrd |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → 1 ≤ ( 1 ( abs ∘ − ) 𝑥 ) ) |
37 |
9 19 36
|
lensymd |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → ¬ ( 1 ( abs ∘ − ) 𝑥 ) < 1 ) |
38 |
2
|
a1i |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ) |
39 |
4
|
a1i |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → 1 ∈ ℝ* ) |
40 |
3
|
a1i |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → 1 ∈ ℂ ) |
41 |
|
elbl2 |
⊢ ( ( ( ( abs ∘ − ) ∈ ( ∞Met ‘ ℂ ) ∧ 1 ∈ ℝ* ) ∧ ( 1 ∈ ℂ ∧ 𝑥 ∈ ℂ ) ) → ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( 1 ( abs ∘ − ) 𝑥 ) < 1 ) ) |
42 |
38 39 40 17 41
|
syl22anc |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ↔ ( 1 ( abs ∘ − ) 𝑥 ) < 1 ) ) |
43 |
37 42
|
mtbird |
⊢ ( 𝑥 ∈ ( -∞ (,] 0 ) → ¬ 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) ) |
44 |
43
|
con2i |
⊢ ( 𝑥 ∈ ( 1 ( ball ‘ ( abs ∘ − ) ) 1 ) → ¬ 𝑥 ∈ ( -∞ (,] 0 ) ) |
45 |
44 1
|
eleq2s |
⊢ ( 𝑥 ∈ 𝑆 → ¬ 𝑥 ∈ ( -∞ (,] 0 ) ) |
46 |
8 45
|
eldifd |
⊢ ( 𝑥 ∈ 𝑆 → 𝑥 ∈ ( ℂ ∖ ( -∞ (,] 0 ) ) ) |
47 |
46
|
ssriv |
⊢ 𝑆 ⊆ ( ℂ ∖ ( -∞ (,] 0 ) ) |