Step |
Hyp |
Ref |
Expression |
1 |
|
rlimcxp.1 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 ) |
2 |
|
rlimcxp.2 |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) ⇝𝑟 0 ) |
3 |
|
rlimcxp.3 |
⊢ ( 𝜑 → 𝐶 ∈ ℝ+ ) |
4 |
|
rlimf |
⊢ ( ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) ⇝𝑟 0 → ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) : dom ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) ⟶ ℂ ) |
5 |
2 4
|
syl |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) : dom ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) ⟶ ℂ ) |
6 |
1
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑛 ∈ 𝐴 𝐵 ∈ 𝑉 ) |
7 |
|
dmmptg |
⊢ ( ∀ 𝑛 ∈ 𝐴 𝐵 ∈ 𝑉 → dom ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 ) |
8 |
6 7
|
syl |
⊢ ( 𝜑 → dom ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) = 𝐴 ) |
9 |
8
|
feq2d |
⊢ ( 𝜑 → ( ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) : dom ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) ⟶ ℂ ↔ ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ ) ) |
10 |
5 9
|
mpbid |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ ) |
11 |
|
eqid |
⊢ ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) |
12 |
11
|
fmpt |
⊢ ( ∀ 𝑛 ∈ 𝐴 𝐵 ∈ ℂ ↔ ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ ) |
13 |
10 12
|
sylibr |
⊢ ( 𝜑 → ∀ 𝑛 ∈ 𝐴 𝐵 ∈ ℂ ) |
14 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ∀ 𝑛 ∈ 𝐴 𝐵 ∈ ℂ ) |
15 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝑥 ∈ ℝ+ ) |
16 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝐶 ∈ ℝ+ ) |
17 |
16
|
rprecred |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 1 / 𝐶 ) ∈ ℝ ) |
18 |
15 17
|
rpcxpcld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) ∈ ℝ+ ) |
19 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) ⇝𝑟 0 ) |
20 |
14 18 19
|
rlimi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝐴 ( 𝑦 ≤ 𝑛 → ( abs ‘ ( 𝐵 − 0 ) ) < ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) ) ) |
21 |
1 2
|
rlimmptrcl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
22 |
21
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
23 |
22
|
abscld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( abs ‘ 𝐵 ) ∈ ℝ ) |
24 |
22
|
absge0d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → 0 ≤ ( abs ‘ 𝐵 ) ) |
25 |
18
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) ∈ ℝ+ ) |
26 |
25
|
rpred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) ∈ ℝ ) |
27 |
25
|
rpge0d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → 0 ≤ ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) ) |
28 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → 𝐶 ∈ ℝ+ ) |
29 |
23 24 26 27 28
|
cxplt2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( ( abs ‘ 𝐵 ) < ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) ↔ ( ( abs ‘ 𝐵 ) ↑𝑐 𝐶 ) < ( ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) ↑𝑐 𝐶 ) ) ) |
30 |
22
|
subid1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( 𝐵 − 0 ) = 𝐵 ) |
31 |
30
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( abs ‘ ( 𝐵 − 0 ) ) = ( abs ‘ 𝐵 ) ) |
32 |
31
|
breq1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( ( abs ‘ ( 𝐵 − 0 ) ) < ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) ↔ ( abs ‘ 𝐵 ) < ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) ) ) |
33 |
28
|
rpred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → 𝐶 ∈ ℝ ) |
34 |
|
abscxp2 |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ ) → ( abs ‘ ( 𝐵 ↑𝑐 𝐶 ) ) = ( ( abs ‘ 𝐵 ) ↑𝑐 𝐶 ) ) |
35 |
22 33 34
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( abs ‘ ( 𝐵 ↑𝑐 𝐶 ) ) = ( ( abs ‘ 𝐵 ) ↑𝑐 𝐶 ) ) |
36 |
28
|
rpcnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → 𝐶 ∈ ℂ ) |
37 |
28
|
rpne0d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → 𝐶 ≠ 0 ) |
38 |
36 37
|
recid2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( ( 1 / 𝐶 ) · 𝐶 ) = 1 ) |
39 |
38
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( 𝑥 ↑𝑐 ( ( 1 / 𝐶 ) · 𝐶 ) ) = ( 𝑥 ↑𝑐 1 ) ) |
40 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → 𝑥 ∈ ℝ+ ) |
41 |
17
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( 1 / 𝐶 ) ∈ ℝ ) |
42 |
40 41 36
|
cxpmuld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( 𝑥 ↑𝑐 ( ( 1 / 𝐶 ) · 𝐶 ) ) = ( ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) ↑𝑐 𝐶 ) ) |
43 |
40
|
rpcnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → 𝑥 ∈ ℂ ) |
44 |
43
|
cxp1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( 𝑥 ↑𝑐 1 ) = 𝑥 ) |
45 |
39 42 44
|
3eqtr3rd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → 𝑥 = ( ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) ↑𝑐 𝐶 ) ) |
46 |
35 45
|
breq12d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( ( abs ‘ ( 𝐵 ↑𝑐 𝐶 ) ) < 𝑥 ↔ ( ( abs ‘ 𝐵 ) ↑𝑐 𝐶 ) < ( ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) ↑𝑐 𝐶 ) ) ) |
47 |
29 32 46
|
3bitr4d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( ( abs ‘ ( 𝐵 − 0 ) ) < ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) ↔ ( abs ‘ ( 𝐵 ↑𝑐 𝐶 ) ) < 𝑥 ) ) |
48 |
47
|
biimpd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( ( abs ‘ ( 𝐵 − 0 ) ) < ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) → ( abs ‘ ( 𝐵 ↑𝑐 𝐶 ) ) < 𝑥 ) ) |
49 |
48
|
imim2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ 𝐴 ) → ( ( 𝑦 ≤ 𝑛 → ( abs ‘ ( 𝐵 − 0 ) ) < ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) ) → ( 𝑦 ≤ 𝑛 → ( abs ‘ ( 𝐵 ↑𝑐 𝐶 ) ) < 𝑥 ) ) ) |
50 |
49
|
ralimdva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ∀ 𝑛 ∈ 𝐴 ( 𝑦 ≤ 𝑛 → ( abs ‘ ( 𝐵 − 0 ) ) < ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) ) → ∀ 𝑛 ∈ 𝐴 ( 𝑦 ≤ 𝑛 → ( abs ‘ ( 𝐵 ↑𝑐 𝐶 ) ) < 𝑥 ) ) ) |
51 |
50
|
reximdv |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝐴 ( 𝑦 ≤ 𝑛 → ( abs ‘ ( 𝐵 − 0 ) ) < ( 𝑥 ↑𝑐 ( 1 / 𝐶 ) ) ) → ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝐴 ( 𝑦 ≤ 𝑛 → ( abs ‘ ( 𝐵 ↑𝑐 𝐶 ) ) < 𝑥 ) ) ) |
52 |
20 51
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝐴 ( 𝑦 ≤ 𝑛 → ( abs ‘ ( 𝐵 ↑𝑐 𝐶 ) ) < 𝑥 ) ) |
53 |
52
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝐴 ( 𝑦 ≤ 𝑛 → ( abs ‘ ( 𝐵 ↑𝑐 𝐶 ) ) < 𝑥 ) ) |
54 |
3
|
rpcnd |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
55 |
54
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → 𝐶 ∈ ℂ ) |
56 |
21 55
|
cxpcld |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝐴 ) → ( 𝐵 ↑𝑐 𝐶 ) ∈ ℂ ) |
57 |
56
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑛 ∈ 𝐴 ( 𝐵 ↑𝑐 𝐶 ) ∈ ℂ ) |
58 |
|
rlimss |
⊢ ( ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) ⇝𝑟 0 → dom ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ ) |
59 |
2 58
|
syl |
⊢ ( 𝜑 → dom ( 𝑛 ∈ 𝐴 ↦ 𝐵 ) ⊆ ℝ ) |
60 |
8 59
|
eqsstrrd |
⊢ ( 𝜑 → 𝐴 ⊆ ℝ ) |
61 |
57 60
|
rlim0 |
⊢ ( 𝜑 → ( ( 𝑛 ∈ 𝐴 ↦ ( 𝐵 ↑𝑐 𝐶 ) ) ⇝𝑟 0 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℝ ∀ 𝑛 ∈ 𝐴 ( 𝑦 ≤ 𝑛 → ( abs ‘ ( 𝐵 ↑𝑐 𝐶 ) ) < 𝑥 ) ) ) |
62 |
53 61
|
mpbird |
⊢ ( 𝜑 → ( 𝑛 ∈ 𝐴 ↦ ( 𝐵 ↑𝑐 𝐶 ) ) ⇝𝑟 0 ) |