Step |
Hyp |
Ref |
Expression |
1 |
|
xlimpnfvlem2.k |
⊢ Ⅎ 𝑘 𝜑 |
2 |
|
xlimpnfvlem2.j |
⊢ Ⅎ 𝑗 𝜑 |
3 |
|
xlimpnfvlem2.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
4 |
|
xlimpnfvlem2.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
5 |
|
xlimpnfvlem2.f |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ* ) |
6 |
|
xlimpnfvlem2.g |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑥 < ( 𝐹 ‘ 𝑘 ) ) |
7 |
|
letopon |
⊢ ( ordTop ‘ ≤ ) ∈ ( TopOn ‘ ℝ* ) |
8 |
7
|
a1i |
⊢ ( 𝜑 → ( ordTop ‘ ≤ ) ∈ ( TopOn ‘ ℝ* ) ) |
9 |
8
|
elfvexd |
⊢ ( 𝜑 → ℝ* ∈ V ) |
10 |
|
cnex |
⊢ ℂ ∈ V |
11 |
10
|
a1i |
⊢ ( 𝜑 → ℂ ∈ V ) |
12 |
4
|
uzsscn2 |
⊢ 𝑍 ⊆ ℂ |
13 |
12
|
a1i |
⊢ ( 𝜑 → 𝑍 ⊆ ℂ ) |
14 |
|
elpm2r |
⊢ ( ( ( ℝ* ∈ V ∧ ℂ ∈ V ) ∧ ( 𝐹 : 𝑍 ⟶ ℝ* ∧ 𝑍 ⊆ ℂ ) ) → 𝐹 ∈ ( ℝ* ↑pm ℂ ) ) |
15 |
9 11 5 13 14
|
syl22anc |
⊢ ( 𝜑 → 𝐹 ∈ ( ℝ* ↑pm ℂ ) ) |
16 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
17 |
16
|
a1i |
⊢ ( 𝜑 → +∞ ∈ ℝ* ) |
18 |
|
pnfnei |
⊢ ( ( 𝑢 ∈ ( ordTop ‘ ≤ ) ∧ +∞ ∈ 𝑢 ) → ∃ 𝑥 ∈ ℝ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) |
19 |
18
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ ( ordTop ‘ ≤ ) ) ∧ +∞ ∈ 𝑢 ) → ∃ 𝑥 ∈ ℝ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) |
20 |
|
nfv |
⊢ Ⅎ 𝑗 𝑥 ∈ ℝ |
21 |
2 20
|
nfan |
⊢ Ⅎ 𝑗 ( 𝜑 ∧ 𝑥 ∈ ℝ ) |
22 |
|
nfv |
⊢ Ⅎ 𝑗 ( 𝑥 (,] +∞ ) ⊆ 𝑢 |
23 |
21 22
|
nfan |
⊢ Ⅎ 𝑗 ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) |
24 |
|
simprr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) ∧ ( 𝑗 ∈ 𝑍 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑥 < ( 𝐹 ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑥 < ( 𝐹 ‘ 𝑘 ) ) |
25 |
|
nfv |
⊢ Ⅎ 𝑘 𝑥 ∈ ℝ |
26 |
1 25
|
nfan |
⊢ Ⅎ 𝑘 ( 𝜑 ∧ 𝑥 ∈ ℝ ) |
27 |
|
nfv |
⊢ Ⅎ 𝑘 ( 𝑥 (,] +∞ ) ⊆ 𝑢 |
28 |
26 27
|
nfan |
⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) |
29 |
|
nfv |
⊢ Ⅎ 𝑘 𝑗 ∈ 𝑍 |
30 |
28 29
|
nfan |
⊢ Ⅎ 𝑘 ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) ∧ 𝑗 ∈ 𝑍 ) |
31 |
4
|
uztrn2 |
⊢ ( ( 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ 𝑍 ) |
32 |
31
|
3adant1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ 𝑍 ) |
33 |
5
|
fdmd |
⊢ ( 𝜑 → dom 𝐹 = 𝑍 ) |
34 |
33
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → dom 𝐹 = 𝑍 ) |
35 |
32 34
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝑘 ∈ dom 𝐹 ) |
36 |
35
|
ad5ant134 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → 𝑘 ∈ dom 𝐹 ) |
37 |
36
|
adantl4r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → 𝑘 ∈ dom 𝐹 ) |
38 |
|
simp-4r |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) |
39 |
38
|
adantl4r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) |
40 |
|
simp-4r |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → 𝑥 ∈ ℝ ) |
41 |
|
rexr |
⊢ ( 𝑥 ∈ ℝ → 𝑥 ∈ ℝ* ) |
42 |
40 41
|
syl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → 𝑥 ∈ ℝ* ) |
43 |
16
|
a1i |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → +∞ ∈ ℝ* ) |
44 |
|
simp-4l |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → 𝜑 ) |
45 |
31
|
ad4ant23 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → 𝑘 ∈ 𝑍 ) |
46 |
5
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ* ) |
47 |
44 45 46
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ* ) |
48 |
|
simpr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → 𝑥 < ( 𝐹 ‘ 𝑘 ) ) |
49 |
5
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → 𝐹 : 𝑍 ⟶ ℝ* ) |
50 |
49 32
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ* ) |
51 |
50
|
pnfged |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ≤ +∞ ) |
52 |
51
|
ad5ant134 |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → ( 𝐹 ‘ 𝑘 ) ≤ +∞ ) |
53 |
42 43 47 48 52
|
eliocd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 (,] +∞ ) ) |
54 |
53
|
adantl3r |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 (,] +∞ ) ) |
55 |
39 54
|
sseldd |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) |
56 |
37 55
|
jca |
⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) ∧ 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) |
57 |
56
|
ex |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( 𝑥 < ( 𝐹 ‘ 𝑘 ) → ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
58 |
30 57
|
ralimda |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) ∧ 𝑗 ∈ 𝑍 ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑥 < ( 𝐹 ‘ 𝑘 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
59 |
58
|
adantrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) ∧ ( 𝑗 ∈ 𝑍 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑥 < ( 𝐹 ‘ 𝑘 ) ) ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑥 < ( 𝐹 ‘ 𝑘 ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
60 |
24 59
|
mpd |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) ∧ ( 𝑗 ∈ 𝑍 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑥 < ( 𝐹 ‘ 𝑘 ) ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) |
61 |
60
|
3impb |
⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) ∧ 𝑗 ∈ 𝑍 ∧ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑥 < ( 𝐹 ‘ 𝑘 ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) |
62 |
6
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑥 < ( 𝐹 ‘ 𝑘 ) ) |
63 |
62
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑥 < ( 𝐹 ‘ 𝑘 ) ) |
64 |
23 61 63
|
reximdd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) |
65 |
4
|
rexuz3 |
⊢ ( 𝑀 ∈ ℤ → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ↔ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
66 |
3 65
|
syl |
⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ↔ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
67 |
66
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ↔ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
68 |
64 67
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ ) ∧ ( 𝑥 (,] +∞ ) ⊆ 𝑢 ) → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) |
69 |
68
|
rexlimdva2 |
⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ( 𝑥 (,] +∞ ) ⊆ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
70 |
69
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ ( ordTop ‘ ≤ ) ) ∧ +∞ ∈ 𝑢 ) → ( ∃ 𝑥 ∈ ℝ ( 𝑥 (,] +∞ ) ⊆ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
71 |
19 70
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑢 ∈ ( ordTop ‘ ≤ ) ) ∧ +∞ ∈ 𝑢 ) → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) |
72 |
71
|
ex |
⊢ ( ( 𝜑 ∧ 𝑢 ∈ ( ordTop ‘ ≤ ) ) → ( +∞ ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
73 |
72
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( +∞ ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
74 |
15 17 73
|
3jca |
⊢ ( 𝜑 → ( 𝐹 ∈ ( ℝ* ↑pm ℂ ) ∧ +∞ ∈ ℝ* ∧ ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( +∞ ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) |
75 |
|
nfcv |
⊢ Ⅎ 𝑘 𝐹 |
76 |
75 8
|
lmbr3 |
⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) +∞ ↔ ( 𝐹 ∈ ( ℝ* ↑pm ℂ ) ∧ +∞ ∈ ℝ* ∧ ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( +∞ ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) ) |
77 |
74 76
|
mpbird |
⊢ ( 𝜑 → 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) +∞ ) |
78 |
|
df-xlim |
⊢ ~~>* = ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) |
79 |
78
|
breqi |
⊢ ( 𝐹 ~~>* +∞ ↔ 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) +∞ ) |
80 |
79
|
a1i |
⊢ ( 𝜑 → ( 𝐹 ~~>* +∞ ↔ 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) +∞ ) ) |
81 |
77 80
|
mpbird |
⊢ ( 𝜑 → 𝐹 ~~>* +∞ ) |