Step |
Hyp |
Ref |
Expression |
1 |
|
rlimcld2.1 |
⊢ ( 𝜑 → sup ( 𝐴 , ℝ* , < ) = +∞ ) |
2 |
|
rlimcld2.2 |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ⇝𝑟 𝐶 ) |
3 |
|
rlimrege0.4 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
4 |
|
rlimrege0.5 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ ( ℜ ‘ 𝐵 ) ) |
5 |
|
ssrab2 |
⊢ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ⊆ ℂ |
6 |
5
|
a1i |
⊢ ( 𝜑 → { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ⊆ ℂ ) |
7 |
|
eldifi |
⊢ ( 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → 𝑦 ∈ ℂ ) |
8 |
7
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) → 𝑦 ∈ ℂ ) |
9 |
8
|
recld |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) → ( ℜ ‘ 𝑦 ) ∈ ℝ ) |
10 |
|
fveq2 |
⊢ ( 𝑤 = 𝑦 → ( ℜ ‘ 𝑤 ) = ( ℜ ‘ 𝑦 ) ) |
11 |
10
|
breq2d |
⊢ ( 𝑤 = 𝑦 → ( 0 ≤ ( ℜ ‘ 𝑤 ) ↔ 0 ≤ ( ℜ ‘ 𝑦 ) ) ) |
12 |
11
|
notbid |
⊢ ( 𝑤 = 𝑦 → ( ¬ 0 ≤ ( ℜ ‘ 𝑤 ) ↔ ¬ 0 ≤ ( ℜ ‘ 𝑦 ) ) ) |
13 |
|
notrab |
⊢ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) = { 𝑤 ∈ ℂ ∣ ¬ 0 ≤ ( ℜ ‘ 𝑤 ) } |
14 |
12 13
|
elrab2 |
⊢ ( 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ↔ ( 𝑦 ∈ ℂ ∧ ¬ 0 ≤ ( ℜ ‘ 𝑦 ) ) ) |
15 |
14
|
simprbi |
⊢ ( 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → ¬ 0 ≤ ( ℜ ‘ 𝑦 ) ) |
16 |
15
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) → ¬ 0 ≤ ( ℜ ‘ 𝑦 ) ) |
17 |
|
0re |
⊢ 0 ∈ ℝ |
18 |
|
ltnle |
⊢ ( ( ( ℜ ‘ 𝑦 ) ∈ ℝ ∧ 0 ∈ ℝ ) → ( ( ℜ ‘ 𝑦 ) < 0 ↔ ¬ 0 ≤ ( ℜ ‘ 𝑦 ) ) ) |
19 |
9 17 18
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) → ( ( ℜ ‘ 𝑦 ) < 0 ↔ ¬ 0 ≤ ( ℜ ‘ 𝑦 ) ) ) |
20 |
16 19
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) → ( ℜ ‘ 𝑦 ) < 0 ) |
21 |
9 20
|
negelrpd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) → - ( ℜ ‘ 𝑦 ) ∈ ℝ+ ) |
22 |
9
|
renegcld |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) → - ( ℜ ‘ 𝑦 ) ∈ ℝ ) |
23 |
22
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) ∧ 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → - ( ℜ ‘ 𝑦 ) ∈ ℝ ) |
24 |
|
elrabi |
⊢ ( 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } → 𝑧 ∈ ℂ ) |
25 |
24
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) ∧ 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → 𝑧 ∈ ℂ ) |
26 |
8
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) ∧ 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → 𝑦 ∈ ℂ ) |
27 |
25 26
|
subcld |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) ∧ 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → ( 𝑧 − 𝑦 ) ∈ ℂ ) |
28 |
27
|
recld |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) ∧ 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → ( ℜ ‘ ( 𝑧 − 𝑦 ) ) ∈ ℝ ) |
29 |
27
|
abscld |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) ∧ 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → ( abs ‘ ( 𝑧 − 𝑦 ) ) ∈ ℝ ) |
30 |
|
0red |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) ∧ 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → 0 ∈ ℝ ) |
31 |
25
|
recld |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) ∧ 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → ( ℜ ‘ 𝑧 ) ∈ ℝ ) |
32 |
26
|
recld |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) ∧ 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → ( ℜ ‘ 𝑦 ) ∈ ℝ ) |
33 |
|
fveq2 |
⊢ ( 𝑤 = 𝑧 → ( ℜ ‘ 𝑤 ) = ( ℜ ‘ 𝑧 ) ) |
34 |
33
|
breq2d |
⊢ ( 𝑤 = 𝑧 → ( 0 ≤ ( ℜ ‘ 𝑤 ) ↔ 0 ≤ ( ℜ ‘ 𝑧 ) ) ) |
35 |
34
|
elrab |
⊢ ( 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ↔ ( 𝑧 ∈ ℂ ∧ 0 ≤ ( ℜ ‘ 𝑧 ) ) ) |
36 |
35
|
simprbi |
⊢ ( 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } → 0 ≤ ( ℜ ‘ 𝑧 ) ) |
37 |
36
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) ∧ 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → 0 ≤ ( ℜ ‘ 𝑧 ) ) |
38 |
30 31 32 37
|
lesub1dd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) ∧ 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → ( 0 − ( ℜ ‘ 𝑦 ) ) ≤ ( ( ℜ ‘ 𝑧 ) − ( ℜ ‘ 𝑦 ) ) ) |
39 |
|
df-neg |
⊢ - ( ℜ ‘ 𝑦 ) = ( 0 − ( ℜ ‘ 𝑦 ) ) |
40 |
39
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) ∧ 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → - ( ℜ ‘ 𝑦 ) = ( 0 − ( ℜ ‘ 𝑦 ) ) ) |
41 |
25 26
|
resubd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) ∧ 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → ( ℜ ‘ ( 𝑧 − 𝑦 ) ) = ( ( ℜ ‘ 𝑧 ) − ( ℜ ‘ 𝑦 ) ) ) |
42 |
38 40 41
|
3brtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) ∧ 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → - ( ℜ ‘ 𝑦 ) ≤ ( ℜ ‘ ( 𝑧 − 𝑦 ) ) ) |
43 |
27
|
releabsd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) ∧ 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → ( ℜ ‘ ( 𝑧 − 𝑦 ) ) ≤ ( abs ‘ ( 𝑧 − 𝑦 ) ) ) |
44 |
23 28 29 42 43
|
letrd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( ℂ ∖ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) ) ∧ 𝑧 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) → - ( ℜ ‘ 𝑦 ) ≤ ( abs ‘ ( 𝑧 − 𝑦 ) ) ) |
45 |
|
fveq2 |
⊢ ( 𝑤 = 𝐵 → ( ℜ ‘ 𝑤 ) = ( ℜ ‘ 𝐵 ) ) |
46 |
45
|
breq2d |
⊢ ( 𝑤 = 𝐵 → ( 0 ≤ ( ℜ ‘ 𝑤 ) ↔ 0 ≤ ( ℜ ‘ 𝐵 ) ) ) |
47 |
46 3 4
|
elrabd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) |
48 |
1 2 6 21 44 47
|
rlimcld2 |
⊢ ( 𝜑 → 𝐶 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ) |
49 |
|
fveq2 |
⊢ ( 𝑤 = 𝐶 → ( ℜ ‘ 𝑤 ) = ( ℜ ‘ 𝐶 ) ) |
50 |
49
|
breq2d |
⊢ ( 𝑤 = 𝐶 → ( 0 ≤ ( ℜ ‘ 𝑤 ) ↔ 0 ≤ ( ℜ ‘ 𝐶 ) ) ) |
51 |
50
|
elrab |
⊢ ( 𝐶 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } ↔ ( 𝐶 ∈ ℂ ∧ 0 ≤ ( ℜ ‘ 𝐶 ) ) ) |
52 |
51
|
simprbi |
⊢ ( 𝐶 ∈ { 𝑤 ∈ ℂ ∣ 0 ≤ ( ℜ ‘ 𝑤 ) } → 0 ≤ ( ℜ ‘ 𝐶 ) ) |
53 |
48 52
|
syl |
⊢ ( 𝜑 → 0 ≤ ( ℜ ‘ 𝐶 ) ) |