Step |
Hyp |
Ref |
Expression |
1 |
|
elioore |
⊢ ( 𝑥 ∈ ( 1 (,) +∞ ) → 𝑥 ∈ ℝ ) |
2 |
|
eliooord |
⊢ ( 𝑥 ∈ ( 1 (,) +∞ ) → ( 1 < 𝑥 ∧ 𝑥 < +∞ ) ) |
3 |
2
|
simpld |
⊢ ( 𝑥 ∈ ( 1 (,) +∞ ) → 1 < 𝑥 ) |
4 |
1 3
|
rplogcld |
⊢ ( 𝑥 ∈ ( 1 (,) +∞ ) → ( log ‘ 𝑥 ) ∈ ℝ+ ) |
5 |
4
|
rprecred |
⊢ ( 𝑥 ∈ ( 1 (,) +∞ ) → ( 1 / ( log ‘ 𝑥 ) ) ∈ ℝ ) |
6 |
5
|
recnd |
⊢ ( 𝑥 ∈ ( 1 (,) +∞ ) → ( 1 / ( log ‘ 𝑥 ) ) ∈ ℂ ) |
7 |
6
|
rgen |
⊢ ∀ 𝑥 ∈ ( 1 (,) +∞ ) ( 1 / ( log ‘ 𝑥 ) ) ∈ ℂ |
8 |
7
|
a1i |
⊢ ( ⊤ → ∀ 𝑥 ∈ ( 1 (,) +∞ ) ( 1 / ( log ‘ 𝑥 ) ) ∈ ℂ ) |
9 |
|
ioossre |
⊢ ( 1 (,) +∞ ) ⊆ ℝ |
10 |
9
|
a1i |
⊢ ( ⊤ → ( 1 (,) +∞ ) ⊆ ℝ ) |
11 |
8 10
|
rlim0lt |
⊢ ( ⊤ → ( ( 𝑥 ∈ ( 1 (,) +∞ ) ↦ ( 1 / ( log ‘ 𝑥 ) ) ) ⇝𝑟 0 ↔ ∀ 𝑦 ∈ ℝ+ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ ( 1 (,) +∞ ) ( 𝑐 < 𝑥 → ( abs ‘ ( 1 / ( log ‘ 𝑥 ) ) ) < 𝑦 ) ) ) |
12 |
11
|
mptru |
⊢ ( ( 𝑥 ∈ ( 1 (,) +∞ ) ↦ ( 1 / ( log ‘ 𝑥 ) ) ) ⇝𝑟 0 ↔ ∀ 𝑦 ∈ ℝ+ ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ ( 1 (,) +∞ ) ( 𝑐 < 𝑥 → ( abs ‘ ( 1 / ( log ‘ 𝑥 ) ) ) < 𝑦 ) ) |
13 |
|
id |
⊢ ( 𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ+ ) |
14 |
13
|
rprecred |
⊢ ( 𝑦 ∈ ℝ+ → ( 1 / 𝑦 ) ∈ ℝ ) |
15 |
14
|
reefcld |
⊢ ( 𝑦 ∈ ℝ+ → ( exp ‘ ( 1 / 𝑦 ) ) ∈ ℝ ) |
16 |
5
|
ad2antlr |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → ( 1 / ( log ‘ 𝑥 ) ) ∈ ℝ ) |
17 |
1
|
ad2antlr |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → 𝑥 ∈ ℝ ) |
18 |
3
|
ad2antlr |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → 1 < 𝑥 ) |
19 |
17 18
|
rplogcld |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → ( log ‘ 𝑥 ) ∈ ℝ+ ) |
20 |
19
|
rpreccld |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → ( 1 / ( log ‘ 𝑥 ) ) ∈ ℝ+ ) |
21 |
20
|
rpge0d |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → 0 ≤ ( 1 / ( log ‘ 𝑥 ) ) ) |
22 |
16 21
|
absidd |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → ( abs ‘ ( 1 / ( log ‘ 𝑥 ) ) ) = ( 1 / ( log ‘ 𝑥 ) ) ) |
23 |
|
simpll |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → 𝑦 ∈ ℝ+ ) |
24 |
4
|
ad2antlr |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → ( log ‘ 𝑥 ) ∈ ℝ+ ) |
25 |
|
simpr |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) |
26 |
|
1rp |
⊢ 1 ∈ ℝ+ |
27 |
26
|
a1i |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → 1 ∈ ℝ+ ) |
28 |
27
|
rpred |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → 1 ∈ ℝ ) |
29 |
28 17 18
|
ltled |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → 1 ≤ 𝑥 ) |
30 |
17 27 29
|
rpgecld |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → 𝑥 ∈ ℝ+ ) |
31 |
30
|
reeflogd |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → ( exp ‘ ( log ‘ 𝑥 ) ) = 𝑥 ) |
32 |
25 31
|
breqtrrd |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → ( exp ‘ ( 1 / 𝑦 ) ) < ( exp ‘ ( log ‘ 𝑥 ) ) ) |
33 |
23
|
rprecred |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → ( 1 / 𝑦 ) ∈ ℝ ) |
34 |
24
|
rpred |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → ( log ‘ 𝑥 ) ∈ ℝ ) |
35 |
|
eflt |
⊢ ( ( ( 1 / 𝑦 ) ∈ ℝ ∧ ( log ‘ 𝑥 ) ∈ ℝ ) → ( ( 1 / 𝑦 ) < ( log ‘ 𝑥 ) ↔ ( exp ‘ ( 1 / 𝑦 ) ) < ( exp ‘ ( log ‘ 𝑥 ) ) ) ) |
36 |
33 34 35
|
syl2anc |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → ( ( 1 / 𝑦 ) < ( log ‘ 𝑥 ) ↔ ( exp ‘ ( 1 / 𝑦 ) ) < ( exp ‘ ( log ‘ 𝑥 ) ) ) ) |
37 |
32 36
|
mpbird |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → ( 1 / 𝑦 ) < ( log ‘ 𝑥 ) ) |
38 |
23 24 37
|
ltrec1d |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → ( 1 / ( log ‘ 𝑥 ) ) < 𝑦 ) |
39 |
22 38
|
eqbrtrd |
⊢ ( ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) ∧ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) → ( abs ‘ ( 1 / ( log ‘ 𝑥 ) ) ) < 𝑦 ) |
40 |
39
|
ex |
⊢ ( ( 𝑦 ∈ ℝ+ ∧ 𝑥 ∈ ( 1 (,) +∞ ) ) → ( ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 → ( abs ‘ ( 1 / ( log ‘ 𝑥 ) ) ) < 𝑦 ) ) |
41 |
40
|
ralrimiva |
⊢ ( 𝑦 ∈ ℝ+ → ∀ 𝑥 ∈ ( 1 (,) +∞ ) ( ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 → ( abs ‘ ( 1 / ( log ‘ 𝑥 ) ) ) < 𝑦 ) ) |
42 |
|
breq1 |
⊢ ( 𝑐 = ( exp ‘ ( 1 / 𝑦 ) ) → ( 𝑐 < 𝑥 ↔ ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 ) ) |
43 |
42
|
rspceaimv |
⊢ ( ( ( exp ‘ ( 1 / 𝑦 ) ) ∈ ℝ ∧ ∀ 𝑥 ∈ ( 1 (,) +∞ ) ( ( exp ‘ ( 1 / 𝑦 ) ) < 𝑥 → ( abs ‘ ( 1 / ( log ‘ 𝑥 ) ) ) < 𝑦 ) ) → ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ ( 1 (,) +∞ ) ( 𝑐 < 𝑥 → ( abs ‘ ( 1 / ( log ‘ 𝑥 ) ) ) < 𝑦 ) ) |
44 |
15 41 43
|
syl2anc |
⊢ ( 𝑦 ∈ ℝ+ → ∃ 𝑐 ∈ ℝ ∀ 𝑥 ∈ ( 1 (,) +∞ ) ( 𝑐 < 𝑥 → ( abs ‘ ( 1 / ( log ‘ 𝑥 ) ) ) < 𝑦 ) ) |
45 |
12 44
|
mprgbir |
⊢ ( 𝑥 ∈ ( 1 (,) +∞ ) ↦ ( 1 / ( log ‘ 𝑥 ) ) ) ⇝𝑟 0 |