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Theorem xlimpnfxnegmnf 41560
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 4928 . . . . . 6 (𝑥 = 𝑦 → (𝑥 ≤ (𝐹𝑗) ↔ 𝑦 ≤ (𝐹𝑗)))
21rexralbidv 3239 . . . . 5 (𝑥 = 𝑦 → (∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ (𝐹𝑗)))
3 fveq2 6496 . . . . . . . 8 (𝑘 = 𝑖 → (ℤ𝑘) = (ℤ𝑖))
43raleqdv 3348 . . . . . . 7 (𝑘 = 𝑖 → (∀𝑗 ∈ (ℤ𝑘)𝑦 ≤ (𝐹𝑗) ↔ ∀𝑗 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑗)))
5 nfv 1874 . . . . . . . 8 𝑙 𝑦 ≤ (𝐹𝑗)
6 nfcv 2925 . . . . . . . . 9 𝑗𝑦
7 nfcv 2925 . . . . . . . . 9 𝑗
8 xlimpnfxnegmnf.1 . . . . . . . . . 10 𝑗𝐹
9 nfcv 2925 . . . . . . . . . 10 𝑗𝑙
108, 9nffv 6506 . . . . . . . . 9 𝑗(𝐹𝑙)
116, 7, 10nfbr 4972 . . . . . . . 8 𝑗 𝑦 ≤ (𝐹𝑙)
12 fveq2 6496 . . . . . . . . 9 (𝑗 = 𝑙 → (𝐹𝑗) = (𝐹𝑙))
1312breq2d 4937 . . . . . . . 8 (𝑗 = 𝑙 → (𝑦 ≤ (𝐹𝑗) ↔ 𝑦 ≤ (𝐹𝑙)))
145, 11, 13cbvral 3372 . . . . . . 7 (∀𝑗 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑗) ↔ ∀𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
154, 14syl6bb 279 . . . . . 6 (𝑘 = 𝑖 → (∀𝑗 ∈ (ℤ𝑘)𝑦 ≤ (𝐹𝑗) ↔ ∀𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)))
1615cbvrexv 3377 . . . . 5 (∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑦 ≤ (𝐹𝑗) ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
172, 16syl6bb 279 . . . 4 (𝑥 = 𝑦 → (∃𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)))
1817cbvralv 3376 . . 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 755 . . . . 5 (((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) ∧ 𝑤 ∈ ℝ) → 𝜑)
21 simpr 477 . . . . 5 (((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) ∧ 𝑤 ∈ ℝ) → 𝑤 ∈ ℝ)
22 xnegrecl 41177 . . . . . . 7 (𝑤 ∈ ℝ → -𝑒𝑤 ∈ ℝ)
23 simpl 475 . . . . . . 7 ((∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ 𝑤 ∈ ℝ) → ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
24 breq1 4928 . . . . . . . . 9 (𝑦 = -𝑒𝑤 → (𝑦 ≤ (𝐹𝑙) ↔ -𝑒𝑤 ≤ (𝐹𝑙)))
2524rexralbidv 3239 . . . . . . . 8 (𝑦 = -𝑒𝑤 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙)))
2625rspcva 3526 . . . . . . 7 ((-𝑒𝑤 ∈ ℝ ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙))
2722, 23, 26syl2an2 674 . . . . . 6 ((∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ 𝑤 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙))
2827adantll 702 . . . . 5 (((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) ∧ 𝑤 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙))
29 simpll 755 . . . . . . . . 9 ((((𝜑𝑤 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → (𝜑𝑤 ∈ ℝ))
30 xlimpnfxnegmnf.2 . . . . . . . . . . 11 𝑍 = (ℤ𝑀)
3130uztrn2 12074 . . . . . . . . . 10 ((𝑖𝑍𝑙 ∈ (ℤ𝑖)) → 𝑙𝑍)
3231adantll 702 . . . . . . . . 9 ((((𝜑𝑤 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → 𝑙𝑍)
33 rexr 10484 . . . . . . . . . . . 12 (𝑤 ∈ ℝ → 𝑤 ∈ ℝ*)
3433ad2antlr 715 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ℝ) ∧ 𝑙𝑍) → 𝑤 ∈ ℝ*)
35 xlimpnfxnegmnf.3 . . . . . . . . . . . . 13 (𝜑𝐹:𝑍⟶ℝ*)
3635ffvelrnda 6674 . . . . . . . . . . . 12 ((𝜑𝑙𝑍) → (𝐹𝑙) ∈ ℝ*)
3736adantlr 703 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ℝ) ∧ 𝑙𝑍) → (𝐹𝑙) ∈ ℝ*)
38 xlenegcon1 41228 . . . . . . . . . . 11 ((𝑤 ∈ ℝ* ∧ (𝐹𝑙) ∈ ℝ*) → (-𝑒𝑤 ≤ (𝐹𝑙) ↔ -𝑒(𝐹𝑙) ≤ 𝑤))
3934, 37, 38syl2anc 576 . . . . . . . . . 10 (((𝜑𝑤 ∈ ℝ) ∧ 𝑙𝑍) → (-𝑒𝑤 ≤ (𝐹𝑙) ↔ -𝑒(𝐹𝑙) ≤ 𝑤))
4039biimpd 221 . . . . . . . . 9 (((𝜑𝑤 ∈ ℝ) ∧ 𝑙𝑍) → (-𝑒𝑤 ≤ (𝐹𝑙) → -𝑒(𝐹𝑙) ≤ 𝑤))
4129, 32, 40syl2anc 576 . . . . . . . 8 ((((𝜑𝑤 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → (-𝑒𝑤 ≤ (𝐹𝑙) → -𝑒(𝐹𝑙) ≤ 𝑤))
4241ralimdva 3120 . . . . . . 7 (((𝜑𝑤 ∈ ℝ) ∧ 𝑖𝑍) → (∀𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙) → ∀𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤))
4342reximdva 3212 . . . . . 6 ((𝜑𝑤 ∈ ℝ) → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤))
4443imp 398 . . . . 5 (((𝜑𝑤 ∈ ℝ) ∧ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒𝑤 ≤ (𝐹𝑙)) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤)
4520, 21, 28, 44syl21anc 826 . . . 4 (((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) ∧ 𝑤 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤)
4645ralrimiva 3125 . . 3 ((𝜑 ∧ ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)) → ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤)
47 simpll 755 . . . . 5 (((𝜑 ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) ∧ 𝑦 ∈ ℝ) → 𝜑)
48 simpr 477 . . . . 5 (((𝜑 ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
49 xnegrecl 41177 . . . . . . 7 (𝑦 ∈ ℝ → -𝑒𝑦 ∈ ℝ)
50 simpl 475 . . . . . . 7 ((∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤𝑦 ∈ ℝ) → ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤)
51 breq2 4929 . . . . . . . . 9 (𝑤 = -𝑒𝑦 → (-𝑒(𝐹𝑙) ≤ 𝑤 ↔ -𝑒(𝐹𝑙) ≤ -𝑒𝑦))
5251rexralbidv 3239 . . . . . . . 8 (𝑤 = -𝑒𝑦 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤 ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦))
5352rspcva 3526 . . . . . . 7 ((-𝑒𝑦 ∈ ℝ ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦)
5449, 50, 53syl2an2 674 . . . . . 6 ((∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤𝑦 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦)
5554adantll 702 . . . . 5 (((𝜑 ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) ∧ 𝑦 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦)
56 simpll 755 . . . . . . . . 9 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → (𝜑𝑦 ∈ ℝ))
5731adantll 702 . . . . . . . . 9 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → 𝑙𝑍)
58 rexr 10484 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → 𝑦 ∈ ℝ*)
5958ad2antlr 715 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑙𝑍) → 𝑦 ∈ ℝ*)
6036adantlr 703 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ℝ) ∧ 𝑙𝑍) → (𝐹𝑙) ∈ ℝ*)
61 xleneg 12426 . . . . . . . . . . 11 ((𝑦 ∈ ℝ* ∧ (𝐹𝑙) ∈ ℝ*) → (𝑦 ≤ (𝐹𝑙) ↔ -𝑒(𝐹𝑙) ≤ -𝑒𝑦))
6259, 60, 61syl2anc 576 . . . . . . . . . 10 (((𝜑𝑦 ∈ ℝ) ∧ 𝑙𝑍) → (𝑦 ≤ (𝐹𝑙) ↔ -𝑒(𝐹𝑙) ≤ -𝑒𝑦))
6362biimprd 240 . . . . . . . . 9 (((𝜑𝑦 ∈ ℝ) ∧ 𝑙𝑍) → (-𝑒(𝐹𝑙) ≤ -𝑒𝑦𝑦 ≤ (𝐹𝑙)))
6456, 57, 63syl2anc 576 . . . . . . . 8 ((((𝜑𝑦 ∈ ℝ) ∧ 𝑖𝑍) ∧ 𝑙 ∈ (ℤ𝑖)) → (-𝑒(𝐹𝑙) ≤ -𝑒𝑦𝑦 ≤ (𝐹𝑙)))
6564ralimdva 3120 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ 𝑖𝑍) → (∀𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦 → ∀𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)))
6665reximdva 3212 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦 → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙)))
6766imp 398 . . . . 5 (((𝜑𝑦 ∈ ℝ) ∧ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ -𝑒𝑦) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
6847, 48, 55, 67syl21anc 826 . . . 4 (((𝜑 ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) ∧ 𝑦 ∈ ℝ) → ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
6968ralrimiva 3125 . . 3 ((𝜑 ∧ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤) → ∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙))
7046, 69impbida 789 . 2 (𝜑 → (∀𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∀𝑤 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤))
71 breq2 4929 . . . . . 6 (𝑤 = 𝑥 → (-𝑒(𝐹𝑙) ≤ 𝑤 ↔ -𝑒(𝐹𝑙) ≤ 𝑥))
7271rexralbidv 3239 . . . . 5 (𝑤 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤 ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑥))
73 fveq2 6496 . . . . . . . 8 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
7473raleqdv 3348 . . . . . . 7 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ𝑘)-𝑒(𝐹𝑙) ≤ 𝑥))
7510nfxneg 41202 . . . . . . . . 9 𝑗-𝑒(𝐹𝑙)
76 nfcv 2925 . . . . . . . . 9 𝑗𝑥
7775, 7, 76nfbr 4972 . . . . . . . 8 𝑗-𝑒(𝐹𝑙) ≤ 𝑥
78 nfv 1874 . . . . . . . 8 𝑙-𝑒(𝐹𝑗) ≤ 𝑥
79 fveq2 6496 . . . . . . . . . 10 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
8079xnegeqd 41176 . . . . . . . . 9 (𝑙 = 𝑗 → -𝑒(𝐹𝑙) = -𝑒(𝐹𝑗))
8180breq1d 4935 . . . . . . . 8 (𝑙 = 𝑗 → (-𝑒(𝐹𝑙) ≤ 𝑥 ↔ -𝑒(𝐹𝑗) ≤ 𝑥))
8277, 78, 81cbvral 3372 . . . . . . 7 (∀𝑙 ∈ (ℤ𝑘)-𝑒(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥)
8374, 82syl6bb 279 . . . . . 6 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥))
8483cbvrexv 3377 . . . . 5 (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥)
8572, 84syl6bb 279 . . . 4 (𝑤 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)-𝑒(𝐹𝑙) ≤ 𝑤 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)-𝑒(𝐹𝑗) ≤ 𝑥))
8685cbvralv 3376 . . 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 297 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 198  wa 387   = wceq 1508  wcel 2051  wnfc 2909  wral 3081  wrex 3082   class class class wbr 4925  wf 6181  cfv 6185  cr 10332  *cxr 10471  cle 10473  cuz 12056  -𝑒cxne 12319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-nel 3067  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-po 5322  df-so 5323  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-er 8087  df-en 8305  df-dom 8306  df-sdom 8307  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-z 11792  df-uz 12057  df-xneg 12322
This theorem is referenced by:  liminfpnfuz  41562  xlimpnfxnegmnf2  41604
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