Step |
Hyp |
Ref |
Expression |
1 |
|
xlimclim.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
2 |
|
xlimclim.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
3 |
|
xlimclim.f |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ ) |
4 |
|
xlimclim.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
5 |
|
df-xlim |
⊢ ~~>* = ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) |
6 |
5
|
breqi |
⊢ ( 𝐹 ~~>* 𝐴 ↔ 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) 𝐴 ) |
7 |
6
|
a1i |
⊢ ( 𝜑 → ( 𝐹 ~~>* 𝐴 ↔ 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) 𝐴 ) ) |
8 |
|
xrtgioo2 |
⊢ ( topGen ‘ ran (,) ) = ( ( ordTop ‘ ≤ ) ↾t ℝ ) |
9 |
|
reex |
⊢ ℝ ∈ V |
10 |
9
|
a1i |
⊢ ( 𝜑 → ℝ ∈ V ) |
11 |
|
letop |
⊢ ( ordTop ‘ ≤ ) ∈ Top |
12 |
11
|
a1i |
⊢ ( 𝜑 → ( ordTop ‘ ≤ ) ∈ Top ) |
13 |
8 2 10 12 4 1 3
|
lmss |
⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) 𝐴 ↔ 𝐹 ( ⇝𝑡 ‘ ( topGen ‘ ran (,) ) ) 𝐴 ) ) |
14 |
|
eqid |
⊢ ( ⇝𝑡 ‘ ( topGen ‘ ran (,) ) ) = ( ⇝𝑡 ‘ ( topGen ‘ ran (,) ) ) |
15 |
14 2 1 3
|
climreeq |
⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ ( topGen ‘ ran (,) ) ) 𝐴 ↔ 𝐹 ⇝ 𝐴 ) ) |
16 |
7 13 15
|
3bitrd |
⊢ ( 𝜑 → ( 𝐹 ~~>* 𝐴 ↔ 𝐹 ⇝ 𝐴 ) ) |