Step |
Hyp |
Ref |
Expression |
1 |
|
xlimpnfvlem1.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
2 |
|
xlimpnfvlem1.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
3 |
|
xlimpnfvlem1.f |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ* ) |
4 |
|
xlimpnfvlem1.c |
⊢ ( 𝜑 → 𝐹 ~~>* +∞ ) |
5 |
|
xlimpnfvlem1.x |
⊢ ( 𝜑 → 𝑋 ∈ ℝ ) |
6 |
|
iocpnfordt |
⊢ ( 𝑋 (,] +∞ ) ∈ ( ordTop ‘ ≤ ) |
7 |
6
|
a1i |
⊢ ( 𝜑 → ( 𝑋 (,] +∞ ) ∈ ( ordTop ‘ ≤ ) ) |
8 |
|
df-xlim |
⊢ ~~>* = ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) |
9 |
8
|
breqi |
⊢ ( 𝐹 ~~>* +∞ ↔ 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) +∞ ) |
10 |
4 9
|
sylib |
⊢ ( 𝜑 → 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) +∞ ) |
11 |
|
nfcv |
⊢ Ⅎ 𝑘 𝐹 |
12 |
|
letopon |
⊢ ( ordTop ‘ ≤ ) ∈ ( TopOn ‘ ℝ* ) |
13 |
12
|
a1i |
⊢ ( 𝜑 → ( ordTop ‘ ≤ ) ∈ ( TopOn ‘ ℝ* ) ) |
14 |
11 13
|
lmbr3 |
⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) +∞ ↔ ( 𝐹 ∈ ( ℝ* ↑pm ℂ ) ∧ +∞ ∈ ℝ* ∧ ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( +∞ ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) ) |
15 |
10 14
|
mpbid |
⊢ ( 𝜑 → ( 𝐹 ∈ ( ℝ* ↑pm ℂ ) ∧ +∞ ∈ ℝ* ∧ ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( +∞ ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) |
16 |
15
|
simp3d |
⊢ ( 𝜑 → ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( +∞ ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) |
17 |
7 16
|
jca |
⊢ ( 𝜑 → ( ( 𝑋 (,] +∞ ) ∈ ( ordTop ‘ ≤ ) ∧ ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( +∞ ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) ) |
18 |
5
|
rexrd |
⊢ ( 𝜑 → 𝑋 ∈ ℝ* ) |
19 |
15
|
simp2d |
⊢ ( 𝜑 → +∞ ∈ ℝ* ) |
20 |
5
|
ltpnfd |
⊢ ( 𝜑 → 𝑋 < +∞ ) |
21 |
|
ubioc1 |
⊢ ( ( 𝑋 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑋 < +∞ ) → +∞ ∈ ( 𝑋 (,] +∞ ) ) |
22 |
18 19 20 21
|
syl3anc |
⊢ ( 𝜑 → +∞ ∈ ( 𝑋 (,] +∞ ) ) |
23 |
|
eleq2 |
⊢ ( 𝑢 = ( 𝑋 (,] +∞ ) → ( +∞ ∈ 𝑢 ↔ +∞ ∈ ( 𝑋 (,] +∞ ) ) ) |
24 |
|
eleq2 |
⊢ ( 𝑢 = ( 𝑋 (,] +∞ ) → ( ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ↔ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) ) |
25 |
24
|
anbi2d |
⊢ ( 𝑢 = ( 𝑋 (,] +∞ ) → ( ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ↔ ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) ) ) |
26 |
25
|
ralbidv |
⊢ ( 𝑢 = ( 𝑋 (,] +∞ ) → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ↔ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) ) ) |
27 |
26
|
rexbidv |
⊢ ( 𝑢 = ( 𝑋 (,] +∞ ) → ( ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ↔ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) ) ) |
28 |
23 27
|
imbi12d |
⊢ ( 𝑢 = ( 𝑋 (,] +∞ ) → ( ( +∞ ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ↔ ( +∞ ∈ ( 𝑋 (,] +∞ ) → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) ) ) ) |
29 |
28
|
rspcva |
⊢ ( ( ( 𝑋 (,] +∞ ) ∈ ( ordTop ‘ ≤ ) ∧ ∀ 𝑢 ∈ ( ordTop ‘ ≤ ) ( +∞ ∈ 𝑢 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑢 ) ) ) → ( +∞ ∈ ( 𝑋 (,] +∞ ) → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) ) ) |
30 |
17 22 29
|
sylc |
⊢ ( 𝜑 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) ) |
31 |
|
nfv |
⊢ Ⅎ 𝑗 𝜑 |
32 |
|
nfv |
⊢ Ⅎ 𝑘 𝜑 |
33 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) ) → 𝑋 ∈ ℝ* ) |
34 |
3
|
ffdmd |
⊢ ( 𝜑 → 𝐹 : dom 𝐹 ⟶ ℝ* ) |
35 |
34
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ* ) |
36 |
35
|
adantrr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ* ) |
37 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) ) → +∞ ∈ ℝ* ) |
38 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) |
39 |
33 37 38
|
iocgtlbd |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) ) → 𝑋 < ( 𝐹 ‘ 𝑘 ) ) |
40 |
33 36 39
|
xrltled |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) ) → 𝑋 ≤ ( 𝐹 ‘ 𝑘 ) ) |
41 |
40
|
ex |
⊢ ( 𝜑 → ( ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) → 𝑋 ≤ ( 𝐹 ‘ 𝑘 ) ) ) |
42 |
41
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ) → ( ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) → 𝑋 ≤ ( 𝐹 ‘ 𝑘 ) ) ) |
43 |
32 42
|
ralimda |
⊢ ( 𝜑 → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑋 ≤ ( 𝐹 ‘ 𝑘 ) ) ) |
44 |
43
|
a1d |
⊢ ( 𝜑 → ( 𝑗 ∈ ℤ → ( ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) → ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑋 ≤ ( 𝐹 ‘ 𝑘 ) ) ) ) |
45 |
31 44
|
reximdai |
⊢ ( 𝜑 → ( ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝑘 ∈ dom 𝐹 ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑋 (,] +∞ ) ) → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑋 ≤ ( 𝐹 ‘ 𝑘 ) ) ) |
46 |
30 45
|
mpd |
⊢ ( 𝜑 → ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑋 ≤ ( 𝐹 ‘ 𝑘 ) ) |
47 |
2
|
rexuz3 |
⊢ ( 𝑀 ∈ ℤ → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑋 ≤ ( 𝐹 ‘ 𝑘 ) ↔ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑋 ≤ ( 𝐹 ‘ 𝑘 ) ) ) |
48 |
1 47
|
syl |
⊢ ( 𝜑 → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑋 ≤ ( 𝐹 ‘ 𝑘 ) ↔ ∃ 𝑗 ∈ ℤ ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑋 ≤ ( 𝐹 ‘ 𝑘 ) ) ) |
49 |
46 48
|
mpbird |
⊢ ( 𝜑 → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) 𝑋 ≤ ( 𝐹 ‘ 𝑘 ) ) |