Step |
Hyp |
Ref |
Expression |
1 |
|
xlimbr.k |
⊢ Ⅎ 𝑘 𝐹 |
2 |
|
xlimbr.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
xlimbr.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
4 |
|
xlimbr.f |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ* ) |
5 |
|
xlimbr.j |
⊢ 𝐽 = ( ordTop ‘ ≤ ) |
6 |
|
df-xlim |
⊢ ~~>* = ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) |
7 |
6
|
breqi |
⊢ ( 𝐹 ~~>* 𝑃 ↔ 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) 𝑃 ) |
8 |
7
|
a1i |
⊢ ( 𝜑 → ( 𝐹 ~~>* 𝑃 ↔ 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) 𝑃 ) ) |
9 |
|
letopon |
⊢ ( ordTop ‘ ≤ ) ∈ ( TopOn ‘ ℝ* ) |
10 |
9
|
a1i |
⊢ ( 𝜑 → ( ordTop ‘ ≤ ) ∈ ( TopOn ‘ ℝ* ) ) |
11 |
1 10
|
lmbr3 |
⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) 𝑃 ↔ ( 𝐹 ∈ ( ℝ* ↑pm ℂ ) ∧ 𝑃 ∈ ℝ* ∧ ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) ) |
12 |
|
simpr2 |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( ℝ* ↑pm ℂ ) ∧ 𝑃 ∈ ℝ* ∧ ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) → 𝑃 ∈ ℝ* ) |
13 |
5
|
eqcomi |
⊢ ( ordTop ‘ ≤ ) = 𝐽 |
14 |
13
|
raleqi |
⊢ ( ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ↔ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
15 |
3
|
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 ) ) ) |
16 |
15
|
bicomd |
⊢ ( 𝑀 ∈ ℤ → ( ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ↔ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
17 |
16
|
imbi2d |
⊢ ( 𝑀 ∈ ℤ → ( ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ↔ ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) |
18 |
17
|
biimpd |
⊢ ( 𝑀 ∈ ℤ → ( ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) → ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) |
19 |
18
|
ralimdv |
⊢ ( 𝑀 ∈ ℤ → ( ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) → ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) |
20 |
2 19
|
syl |
⊢ ( 𝜑 → ( ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) → ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) |
21 |
20
|
imp |
⊢ ( ( 𝜑 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) → ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
22 |
14 21
|
sylan2b |
⊢ ( ( 𝜑 ∧ ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) → ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
23 |
22
|
3ad2antr3 |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( ℝ* ↑pm ℂ ) ∧ 𝑃 ∈ ℝ* ∧ ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) → ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
24 |
12 23
|
jca |
⊢ ( ( 𝜑 ∧ ( 𝐹 ∈ ( ℝ* ↑pm ℂ ) ∧ 𝑃 ∈ ℝ* ∧ ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) → ( 𝑃 ∈ ℝ* ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) |
25 |
|
cnex |
⊢ ℂ ∈ V |
26 |
25
|
a1i |
⊢ ( 𝜑 → ℂ ∈ V ) |
27 |
10
|
elfvexd |
⊢ ( 𝜑 → ℝ* ∈ V ) |
28 |
3
|
uzsscn2 |
⊢ 𝑍 ⊆ ℂ |
29 |
28
|
a1i |
⊢ ( 𝜑 → 𝑍 ⊆ ℂ ) |
30 |
26 27 29 4
|
fpmd |
⊢ ( 𝜑 → 𝐹 ∈ ( ℝ* ↑pm ℂ ) ) |
31 |
30
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑃 ∈ ℝ* ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) → 𝐹 ∈ ( ℝ* ↑pm ℂ ) ) |
32 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑃 ∈ ℝ* ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) → 𝑃 ∈ ℝ* ) |
33 |
17
|
biimprd |
⊢ ( 𝑀 ∈ ℤ → ( ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) → ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) |
34 |
33
|
ralimdv |
⊢ ( 𝑀 ∈ ℤ → ( ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) → ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) |
35 |
2 34
|
syl |
⊢ ( 𝜑 → ( ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) → ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) |
36 |
35
|
imp |
⊢ ( ( 𝜑 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) → ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
37 |
5
|
raleqi |
⊢ ( ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ↔ ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
38 |
36 37
|
sylib |
⊢ ( ( 𝜑 ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) → ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
39 |
38
|
adantrl |
⊢ ( ( 𝜑 ∧ ( 𝑃 ∈ ℝ* ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) → ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
40 |
31 32 39
|
3jca |
⊢ ( ( 𝜑 ∧ ( 𝑃 ∈ ℝ* ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) → ( 𝐹 ∈ ( ℝ* ↑pm ℂ ) ∧ 𝑃 ∈ ℝ* ∧ ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) |
41 |
24 40
|
impbida |
⊢ ( 𝜑 → ( ( 𝐹 ∈ ( ℝ* ↑pm ℂ ) ∧ 𝑃 ∈ ℝ* ∧ ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ↔ ( 𝑃 ∈ ℝ* ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) ) |
42 |
8 11 41
|
3bitrd |
⊢ ( 𝜑 → ( 𝐹 ~~>* 𝑃 ↔ ( 𝑃 ∈ ℝ* ∧ ∀ 𝑢 ∈ 𝐽 ( 𝑃 ∈ 𝑢 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) ) |