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Theorem xlimpnfxnegmnf 41971
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 5060 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ≤ (𝐹𝑗) ↔ 𝑦 ≤ (𝐹𝑗)))
21rexralbidv 3298 . . . . 5 (𝑥 = 𝑦 → (∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ (𝐹𝑗)))
3 fveq2 6663 . . . . . . . 8 (𝑘 = 𝑖 → (ℤ𝑘) = (ℤ𝑖))
43raleqdv 3413 . . . . . . 7 (𝑘 = 𝑖 → (∀𝑗 ∈ (ℤ𝑘)𝑦 ≤ (𝐹𝑗) ↔ ∀𝑗 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑗)))
5 nfv 1906 . . . . . . . 8 𝑙 𝑦 ≤ (𝐹𝑗)
6 nfcv 2974 . . . . . . . . 9 𝑗𝑦
7 nfcv 2974 . . . . . . . . 9 𝑗
8 xlimpnfxnegmnf.1 . . . . . . . . . 10 𝑗𝐹
9 nfcv 2974 . . . . . . . . . 10 𝑗𝑙
108, 9nffv 6673 . . . . . . . . 9 𝑗(𝐹𝑙)
116, 7, 10nfbr 5104 . . . . . . . 8 𝑗 𝑦 ≤ (𝐹𝑙)
12 fveq2 6663 . . . . . . . . 9 (𝑗 = 𝑙 → (𝐹𝑗) = (𝐹𝑙))
1312breq2d 5069 . . . . . . . 8 (𝑗 = 𝑙 → (𝑦 ≤ (𝐹𝑗) ↔ 𝑦 ≤ (𝐹𝑙)))
145, 11, 13cbvralw 3439 . . . . . . 7 (∀𝑗 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑗) ↔ ∀𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
154, 14syl6bb 288 . . . . . 6 (𝑘 = 𝑖 → (∀𝑗 ∈ (ℤ𝑘)𝑦 ≤ (𝐹𝑗) ↔ ∀𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)))
1615cbvrexvw 3448 . . . . 5 (∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ (𝐹𝑗) ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
172, 16syl6bb 288 . . . 4 (𝑥 = 𝑦 → (∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)))
1817cbvralvw 3447 . . 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 763 . . . . 5 (((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) ∧ 𝑤 ∈ ℝ) → 𝜑)
21 simpr 485 . . . . 5 (((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) ∧ 𝑤 ∈ ℝ) → 𝑤 ∈ ℝ)
22 xnegrecl 41588 . . . . . . 7 (𝑤 ∈ ℝ → -𝑒𝑤 ∈ ℝ)
23 simpl 483 . . . . . . 7 ((∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ 𝑤 ∈ ℝ) → ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
24 breq1 5060 . . . . . . . . 9 (𝑦 = -𝑒𝑤 → (𝑦 ≤ (𝐹𝑙) ↔ -𝑒𝑤 ≤ (𝐹𝑙)))
2524rexralbidv 3298 . . . . . . . 8 (𝑦 = -𝑒𝑤 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙)))
2625rspcva 3618 . . . . . . 7 ((-𝑒𝑤 ∈ ℝ ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙))
2722, 23, 26syl2an2 682 . . . . . 6 ((∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ 𝑤 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙))
2827adantll 710 . . . . 5 (((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) ∧ 𝑤 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙))
29 simpll 763 . . . . . . . . 9 ((((𝜑𝑤 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → (𝜑𝑤 ∈ ℝ))
30 xlimpnfxnegmnf.2 . . . . . . . . . . 11 𝑍 = (ℤ𝑀)
3130uztrn2 12250 . . . . . . . . . 10 ((𝑖𝑍𝑙 ∈ (ℤ𝑖)) → 𝑙𝑍)
3231adantll 710 . . . . . . . . 9 ((((𝜑𝑤 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → 𝑙𝑍)
33 rexr 10675 . . . . . . . . . . . 12 (𝑤 ∈ ℝ → 𝑤 ∈ ℝ*)
3433ad2antlr 723 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ℝ) ∧ 𝑙𝑍) → 𝑤 ∈ ℝ*)
35 xlimpnfxnegmnf.3 . . . . . . . . . . . . 13 (𝜑𝐹:𝑍⟶ℝ*)
3635ffvelrnda 6843 . . . . . . . . . . . 12 ((𝜑𝑙𝑍) → (𝐹𝑙) ∈ ℝ*)
3736adantlr 711 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ℝ) ∧ 𝑙𝑍) → (𝐹𝑙) ∈ ℝ*)
38 xlenegcon1 41639 . . . . . . . . . . 11 ((𝑤 ∈ ℝ* ∧ (𝐹𝑙) ∈ ℝ*) → (-𝑒𝑤 ≤ (𝐹𝑙) ↔ -𝑒(𝐹𝑙) ≤ 𝑤))
3934, 37, 38syl2anc 584 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℝ) ∧ 𝑙𝑍) → (-𝑒𝑤 ≤ (𝐹𝑙) ↔ -𝑒(𝐹𝑙) ≤ 𝑤))
4039biimpd 230 . . . . . . . . 9 (((𝜑𝑤 ∈ ℝ) ∧ 𝑙𝑍) → (-𝑒𝑤 ≤ (𝐹𝑙) → -𝑒(𝐹𝑙) ≤ 𝑤))
4129, 32, 40syl2anc 584 . . . . . . . 8 ((((𝜑𝑤 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → (-𝑒𝑤 ≤ (𝐹𝑙) → -𝑒(𝐹𝑙) ≤ 𝑤))
4241ralimdva 3174 . . . . . . 7 (((𝜑𝑤 ∈ ℝ) ∧ 𝑖𝑍) → (∀𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙) → ∀𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤))
4342reximdva 3271 . . . . . 6 ((𝜑𝑤 ∈ ℝ) → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤))
4443imp 407 . . . . 5 (((𝜑𝑤 ∈ ℝ) ∧ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙)) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤)
4520, 21, 28, 44syl21anc 833 . . . 4 (((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) ∧ 𝑤 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤)
4645ralrimiva 3179 . . 3 ((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) → ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤)
47 simpll 763 . . . . 5 (((𝜑 ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) ∧ 𝑦 ∈ ℝ) → 𝜑)
48 simpr 485 . . . . 5 (((𝜑 ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
49 xnegrecl 41588 . . . . . . 7 (𝑦 ∈ ℝ → -𝑒𝑦 ∈ ℝ)
50 simpl 483 . . . . . . 7 ((∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤𝑦 ∈ ℝ) → ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤)
51 breq2 5061 . . . . . . . . 9 (𝑤 = -𝑒𝑦 → (-𝑒(𝐹𝑙) ≤ 𝑤 ↔ -𝑒(𝐹𝑙) ≤ -𝑒𝑦))
5251rexralbidv 3298 . . . . . . . 8 (𝑤 = -𝑒𝑦 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤 ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦))
5352rspcva 3618 . . . . . . 7 ((-𝑒𝑦 ∈ ℝ ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦)
5449, 50, 53syl2an2 682 . . . . . 6 ((∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤𝑦 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦)
5554adantll 710 . . . . 5 (((𝜑 ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) ∧ 𝑦 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦)
56 simpll 763 . . . . . . . . 9 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → (𝜑𝑦 ∈ ℝ))
5731adantll 710 . . . . . . . . 9 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → 𝑙𝑍)
58 rexr 10675 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
5958ad2antlr 723 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑙𝑍) → 𝑦 ∈ ℝ*)
6036adantlr 711 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑙𝑍) → (𝐹𝑙) ∈ ℝ*)
61 xleneg 12599 . . . . . . . . . . 11 ((𝑦 ∈ ℝ* ∧ (𝐹𝑙) ∈ ℝ*) → (𝑦 ≤ (𝐹𝑙) ↔ -𝑒(𝐹𝑙) ≤ -𝑒𝑦))
6259, 60, 61syl2anc 584 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑙𝑍) → (𝑦 ≤ (𝐹𝑙) ↔ -𝑒(𝐹𝑙) ≤ -𝑒𝑦))
6362biimprd 249 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑙𝑍) → (-𝑒(𝐹𝑙) ≤ -𝑒𝑦𝑦 ≤ (𝐹𝑙)))
6456, 57, 63syl2anc 584 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → (-𝑒(𝐹𝑙) ≤ -𝑒𝑦𝑦 ≤ (𝐹𝑙)))
6564ralimdva 3174 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ 𝑖𝑍) → (∀𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦 → ∀𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)))
6665reximdva 3271 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦 → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)))
6766imp 407 . . . . 5 (((𝜑𝑦 ∈ ℝ) ∧ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
6847, 48, 55, 67syl21anc 833 . . . 4 (((𝜑 ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) ∧ 𝑦 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
6968ralrimiva 3179 . . 3 ((𝜑 ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) → ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
7046, 69impbida 797 . 2 (𝜑 → (∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤))
71 breq2 5061 . . . . . 6 (𝑤 = 𝑥 → (-𝑒(𝐹𝑙) ≤ 𝑤 ↔ -𝑒(𝐹𝑙) ≤ 𝑥))
7271rexralbidv 3298 . . . . 5 (𝑤 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤 ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑥))
73 fveq2 6663 . . . . . . . 8 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
7473raleqdv 3413 . . . . . . 7 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ𝑘)-𝑒(𝐹𝑙) ≤ 𝑥))
7510nfxneg 41613 . . . . . . . . 9 𝑗-𝑒(𝐹𝑙)
76 nfcv 2974 . . . . . . . . 9 𝑗𝑥
7775, 7, 76nfbr 5104 . . . . . . . 8 𝑗-𝑒(𝐹𝑙) ≤ 𝑥
78 nfv 1906 . . . . . . . 8 𝑙-𝑒(𝐹𝑗) ≤ 𝑥
79 fveq2 6663 . . . . . . . . . 10 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
8079xnegeqd 41587 . . . . . . . . 9 (𝑙 = 𝑗 → -𝑒(𝐹𝑙) = -𝑒(𝐹𝑗))
8180breq1d 5067 . . . . . . . 8 (𝑙 = 𝑗 → (-𝑒(𝐹𝑙) ≤ 𝑥 ↔ -𝑒(𝐹𝑗) ≤ 𝑥))
8277, 78, 81cbvralw 3439 . . . . . . 7 (∀𝑙 ∈ (ℤ𝑘)-𝑒(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥)
8374, 82syl6bb 288 . . . . . 6 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥))
8483cbvrexvw 3448 . . . . 5 (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥)
8572, 84syl6bb 288 . . . 4 (𝑤 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥))
8685cbvralvw 3447 . . 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 306 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 207  wa 396   = wceq 1528  wcel 2105  wnfc 2958  wral 3135  wrex 3136   class class class wbr 5057  wf 6344  cfv 6348  cr 10524  *cxr 10662  cle 10664  cuz 12231  -𝑒cxne 12492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-z 11970  df-uz 12232  df-xneg 12495
This theorem is referenced by:  liminfpnfuz  41973  xlimpnfxnegmnf2  42015
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