Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  xlimpnfxnegmnf Structured version   Visualization version   GIF version

Theorem xlimpnfxnegmnf 42454
 Description: A sequence converges to +∞ if and only if its negation converges to -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.)
Hypotheses
Ref Expression
xlimpnfxnegmnf.1 𝑗𝐹
xlimpnfxnegmnf.2 𝑍 = (ℤ𝑀)
xlimpnfxnegmnf.3 (𝜑𝐹:𝑍⟶ℝ*)
Assertion
Ref Expression
xlimpnfxnegmnf (𝜑 → (∀𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ↔ ∀𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥))
Distinct variable groups:   𝑘,𝐹,𝑥   𝑘,𝑍,𝑥   𝑗,𝑘,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑗,𝑘)   𝐹(𝑗)   𝑀(𝑥,𝑗,𝑘)   𝑍(𝑗)

Proof of Theorem xlimpnfxnegmnf
Dummy variables 𝑖 𝑙 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 5033 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ≤ (𝐹𝑗) ↔ 𝑦 ≤ (𝐹𝑗)))
21rexralbidv 3260 . . . . 5 (𝑥 = 𝑦 → (∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ (𝐹𝑗)))
3 fveq2 6645 . . . . . . . 8 (𝑘 = 𝑖 → (ℤ𝑘) = (ℤ𝑖))
43raleqdv 3364 . . . . . . 7 (𝑘 = 𝑖 → (∀𝑗 ∈ (ℤ𝑘)𝑦 ≤ (𝐹𝑗) ↔ ∀𝑗 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑗)))
5 nfv 1915 . . . . . . . 8 𝑙 𝑦 ≤ (𝐹𝑗)
6 nfcv 2955 . . . . . . . . 9 𝑗𝑦
7 nfcv 2955 . . . . . . . . 9 𝑗
8 xlimpnfxnegmnf.1 . . . . . . . . . 10 𝑗𝐹
9 nfcv 2955 . . . . . . . . . 10 𝑗𝑙
108, 9nffv 6655 . . . . . . . . 9 𝑗(𝐹𝑙)
116, 7, 10nfbr 5077 . . . . . . . 8 𝑗 𝑦 ≤ (𝐹𝑙)
12 fveq2 6645 . . . . . . . . 9 (𝑗 = 𝑙 → (𝐹𝑗) = (𝐹𝑙))
1312breq2d 5042 . . . . . . . 8 (𝑗 = 𝑙 → (𝑦 ≤ (𝐹𝑗) ↔ 𝑦 ≤ (𝐹𝑙)))
145, 11, 13cbvralw 3387 . . . . . . 7 (∀𝑗 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑗) ↔ ∀𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
154, 14syl6bb 290 . . . . . 6 (𝑘 = 𝑖 → (∀𝑗 ∈ (ℤ𝑘)𝑦 ≤ (𝐹𝑗) ↔ ∀𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)))
1615cbvrexvw 3397 . . . . 5 (∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ (𝐹𝑗) ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
172, 16syl6bb 290 . . . 4 (𝑥 = 𝑦 → (∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)))
1817cbvralvw 3396 . . 3 (∀𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ↔ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
1918a1i 11 . 2 (𝜑 → (∀𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ↔ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)))
20 simpll 766 . . . . 5 (((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) ∧ 𝑤 ∈ ℝ) → 𝜑)
21 simpr 488 . . . . 5 (((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) ∧ 𝑤 ∈ ℝ) → 𝑤 ∈ ℝ)
22 xnegrecl 42073 . . . . . . 7 (𝑤 ∈ ℝ → -𝑒𝑤 ∈ ℝ)
23 simpl 486 . . . . . . 7 ((∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ 𝑤 ∈ ℝ) → ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
24 breq1 5033 . . . . . . . . 9 (𝑦 = -𝑒𝑤 → (𝑦 ≤ (𝐹𝑙) ↔ -𝑒𝑤 ≤ (𝐹𝑙)))
2524rexralbidv 3260 . . . . . . . 8 (𝑦 = -𝑒𝑤 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙)))
2625rspcva 3569 . . . . . . 7 ((-𝑒𝑤 ∈ ℝ ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙))
2722, 23, 26syl2an2 685 . . . . . 6 ((∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ 𝑤 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙))
2827adantll 713 . . . . 5 (((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) ∧ 𝑤 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙))
29 simpll 766 . . . . . . . . 9 ((((𝜑𝑤 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → (𝜑𝑤 ∈ ℝ))
30 xlimpnfxnegmnf.2 . . . . . . . . . . 11 𝑍 = (ℤ𝑀)
3130uztrn2 12250 . . . . . . . . . 10 ((𝑖𝑍𝑙 ∈ (ℤ𝑖)) → 𝑙𝑍)
3231adantll 713 . . . . . . . . 9 ((((𝜑𝑤 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → 𝑙𝑍)
33 rexr 10676 . . . . . . . . . . . 12 (𝑤 ∈ ℝ → 𝑤 ∈ ℝ*)
3433ad2antlr 726 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ℝ) ∧ 𝑙𝑍) → 𝑤 ∈ ℝ*)
35 xlimpnfxnegmnf.3 . . . . . . . . . . . . 13 (𝜑𝐹:𝑍⟶ℝ*)
3635ffvelrnda 6828 . . . . . . . . . . . 12 ((𝜑𝑙𝑍) → (𝐹𝑙) ∈ ℝ*)
3736adantlr 714 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ℝ) ∧ 𝑙𝑍) → (𝐹𝑙) ∈ ℝ*)
38 xlenegcon1 42124 . . . . . . . . . . 11 ((𝑤 ∈ ℝ* ∧ (𝐹𝑙) ∈ ℝ*) → (-𝑒𝑤 ≤ (𝐹𝑙) ↔ -𝑒(𝐹𝑙) ≤ 𝑤))
3934, 37, 38syl2anc 587 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℝ) ∧ 𝑙𝑍) → (-𝑒𝑤 ≤ (𝐹𝑙) ↔ -𝑒(𝐹𝑙) ≤ 𝑤))
4039biimpd 232 . . . . . . . . 9 (((𝜑𝑤 ∈ ℝ) ∧ 𝑙𝑍) → (-𝑒𝑤 ≤ (𝐹𝑙) → -𝑒(𝐹𝑙) ≤ 𝑤))
4129, 32, 40syl2anc 587 . . . . . . . 8 ((((𝜑𝑤 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → (-𝑒𝑤 ≤ (𝐹𝑙) → -𝑒(𝐹𝑙) ≤ 𝑤))
4241ralimdva 3144 . . . . . . 7 (((𝜑𝑤 ∈ ℝ) ∧ 𝑖𝑍) → (∀𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙) → ∀𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤))
4342reximdva 3233 . . . . . 6 ((𝜑𝑤 ∈ ℝ) → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤))
4443imp 410 . . . . 5 (((𝜑𝑤 ∈ ℝ) ∧ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙)) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤)
4520, 21, 28, 44syl21anc 836 . . . 4 (((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) ∧ 𝑤 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤)
4645ralrimiva 3149 . . 3 ((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) → ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤)
47 simpll 766 . . . . 5 (((𝜑 ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) ∧ 𝑦 ∈ ℝ) → 𝜑)
48 simpr 488 . . . . 5 (((𝜑 ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
49 xnegrecl 42073 . . . . . . 7 (𝑦 ∈ ℝ → -𝑒𝑦 ∈ ℝ)
50 simpl 486 . . . . . . 7 ((∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤𝑦 ∈ ℝ) → ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤)
51 breq2 5034 . . . . . . . . 9 (𝑤 = -𝑒𝑦 → (-𝑒(𝐹𝑙) ≤ 𝑤 ↔ -𝑒(𝐹𝑙) ≤ -𝑒𝑦))
5251rexralbidv 3260 . . . . . . . 8 (𝑤 = -𝑒𝑦 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤 ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦))
5352rspcva 3569 . . . . . . 7 ((-𝑒𝑦 ∈ ℝ ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦)
5449, 50, 53syl2an2 685 . . . . . 6 ((∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤𝑦 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦)
5554adantll 713 . . . . 5 (((𝜑 ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) ∧ 𝑦 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦)
56 simpll 766 . . . . . . . . 9 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → (𝜑𝑦 ∈ ℝ))
5731adantll 713 . . . . . . . . 9 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → 𝑙𝑍)
58 rexr 10676 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
5958ad2antlr 726 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑙𝑍) → 𝑦 ∈ ℝ*)
6036adantlr 714 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑙𝑍) → (𝐹𝑙) ∈ ℝ*)
61 xleneg 12599 . . . . . . . . . . 11 ((𝑦 ∈ ℝ* ∧ (𝐹𝑙) ∈ ℝ*) → (𝑦 ≤ (𝐹𝑙) ↔ -𝑒(𝐹𝑙) ≤ -𝑒𝑦))
6259, 60, 61syl2anc 587 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑙𝑍) → (𝑦 ≤ (𝐹𝑙) ↔ -𝑒(𝐹𝑙) ≤ -𝑒𝑦))
6362biimprd 251 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑙𝑍) → (-𝑒(𝐹𝑙) ≤ -𝑒𝑦𝑦 ≤ (𝐹𝑙)))
6456, 57, 63syl2anc 587 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → (-𝑒(𝐹𝑙) ≤ -𝑒𝑦𝑦 ≤ (𝐹𝑙)))
6564ralimdva 3144 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ 𝑖𝑍) → (∀𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦 → ∀𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)))
6665reximdva 3233 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦 → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)))
6766imp 410 . . . . 5 (((𝜑𝑦 ∈ ℝ) ∧ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
6847, 48, 55, 67syl21anc 836 . . . 4 (((𝜑 ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) ∧ 𝑦 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
6968ralrimiva 3149 . . 3 ((𝜑 ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) → ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
7046, 69impbida 800 . 2 (𝜑 → (∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤))
71 breq2 5034 . . . . . 6 (𝑤 = 𝑥 → (-𝑒(𝐹𝑙) ≤ 𝑤 ↔ -𝑒(𝐹𝑙) ≤ 𝑥))
7271rexralbidv 3260 . . . . 5 (𝑤 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤 ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑥))
73 fveq2 6645 . . . . . . . 8 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
7473raleqdv 3364 . . . . . . 7 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ𝑘)-𝑒(𝐹𝑙) ≤ 𝑥))
7510nfxneg 42098 . . . . . . . . 9 𝑗-𝑒(𝐹𝑙)
76 nfcv 2955 . . . . . . . . 9 𝑗𝑥
7775, 7, 76nfbr 5077 . . . . . . . 8 𝑗-𝑒(𝐹𝑙) ≤ 𝑥
78 nfv 1915 . . . . . . . 8 𝑙-𝑒(𝐹𝑗) ≤ 𝑥
79 fveq2 6645 . . . . . . . . . 10 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
8079xnegeqd 42072 . . . . . . . . 9 (𝑙 = 𝑗 → -𝑒(𝐹𝑙) = -𝑒(𝐹𝑗))
8180breq1d 5040 . . . . . . . 8 (𝑙 = 𝑗 → (-𝑒(𝐹𝑙) ≤ 𝑥 ↔ -𝑒(𝐹𝑗) ≤ 𝑥))
8277, 78, 81cbvralw 3387 . . . . . . 7 (∀𝑙 ∈ (ℤ𝑘)-𝑒(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥)
8374, 82syl6bb 290 . . . . . 6 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥))
8483cbvrexvw 3397 . . . . 5 (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥)
8572, 84syl6bb 290 . . . 4 (𝑤 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥))
8685cbvralvw 3396 . . 3 (∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤 ↔ ∀𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥)
8786a1i 11 . 2 (𝜑 → (∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤 ↔ ∀𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥))
8819, 70, 873bitrd 308 1 (𝜑 → (∀𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ↔ ∀𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2936  ∀wral 3106  ∃wrex 3107   class class class wbr 5030  ⟶wf 6320  ‘cfv 6324  ℝcr 10525  ℝ*cxr 10663   ≤ cle 10665  ℤ≥cuz 12231  -𝑒cxne 12492 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-z 11970  df-uz 12232  df-xneg 12495 This theorem is referenced by:  liminfpnfuz  42456  xlimpnfxnegmnf2  42498
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