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Theorem limsupbnd2 14832
Description: If a sequence is eventually greater than 𝐴, then the limsup is also greater than 𝐴. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 12-Sep-2020.)
Hypotheses
Ref Expression
limsupbnd.1 (𝜑𝐵 ⊆ ℝ)
limsupbnd.2 (𝜑𝐹:𝐵⟶ℝ*)
limsupbnd.3 (𝜑𝐴 ∈ ℝ*)
limsupbnd2.4 (𝜑 → sup(𝐵, ℝ*, < ) = +∞)
limsupbnd2.5 (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)))
Assertion
Ref Expression
limsupbnd2 (𝜑𝐴 ≤ (lim sup‘𝐹))
Distinct variable groups:   𝑗,𝑘,𝐴   𝐵,𝑗,𝑘   𝑗,𝐹,𝑘   𝜑,𝑗,𝑘

Proof of Theorem limsupbnd2
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limsupbnd2.5 . . 3 (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)))
2 limsupbnd2.4 . . . . . . . . 9 (𝜑 → sup(𝐵, ℝ*, < ) = +∞)
3 limsupbnd.1 . . . . . . . . . . 11 (𝜑𝐵 ⊆ ℝ)
4 ressxr 10674 . . . . . . . . . . 11 ℝ ⊆ ℝ*
53, 4sstrdi 3927 . . . . . . . . . 10 (𝜑𝐵 ⊆ ℝ*)
6 supxrunb1 12700 . . . . . . . . . 10 (𝐵 ⊆ ℝ* → (∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗 ↔ sup(𝐵, ℝ*, < ) = +∞))
75, 6syl 17 . . . . . . . . 9 (𝜑 → (∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗 ↔ sup(𝐵, ℝ*, < ) = +∞))
82, 7mpbird 260 . . . . . . . 8 (𝜑 → ∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗)
9 ifcl 4469 . . . . . . . 8 ((𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ) → if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ)
10 breq1 5033 . . . . . . . . . 10 (𝑛 = if(𝑘𝑚, 𝑚, 𝑘) → (𝑛𝑗 ↔ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗))
1110rexbidv 3256 . . . . . . . . 9 (𝑛 = if(𝑘𝑚, 𝑚, 𝑘) → (∃𝑗𝐵 𝑛𝑗 ↔ ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗))
1211rspccva 3570 . . . . . . . 8 ((∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗 ∧ if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ) → ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗)
138, 9, 12syl2an 598 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗)
14 r19.29 3216 . . . . . . . 8 ((∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → ∃𝑗𝐵 ((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗))
15 simplrr 777 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑘 ∈ ℝ)
16 simprl 770 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝑚 ∈ ℝ)
1716adantr 484 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑚 ∈ ℝ)
18 max1 12566 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑘 ≤ if(𝑘𝑚, 𝑚, 𝑘))
1915, 17, 18syl2anc 587 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑘 ≤ if(𝑘𝑚, 𝑚, 𝑘))
2017, 15, 9syl2anc 587 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ)
213adantr 484 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝐵 ⊆ ℝ)
2221sselda 3915 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑗 ∈ ℝ)
23 letr 10723 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℝ ∧ if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ ∧ 𝑗 ∈ ℝ) → ((𝑘 ≤ if(𝑘𝑚, 𝑚, 𝑘) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑘𝑗))
2415, 20, 22, 23syl3anc 1368 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝑘 ≤ if(𝑘𝑚, 𝑚, 𝑘) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑘𝑗))
2519, 24mpand 694 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗𝑘𝑗))
2625imim1d 82 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝑘𝑗𝐴 ≤ (𝐹𝑗)) → (if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗𝐴 ≤ (𝐹𝑗))))
2726impd 414 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ (𝐹𝑗)))
28 max2 12568 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑚 ≤ if(𝑘𝑚, 𝑚, 𝑘))
2915, 17, 28syl2anc 587 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑚 ≤ if(𝑘𝑚, 𝑚, 𝑘))
30 letr 10723 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ℝ ∧ if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ ∧ 𝑗 ∈ ℝ) → ((𝑚 ≤ if(𝑘𝑚, 𝑚, 𝑘) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚𝑗))
3117, 20, 22, 30syl3anc 1368 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝑚 ≤ if(𝑘𝑚, 𝑚, 𝑘) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚𝑗))
3229, 31mpand 694 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗𝑚𝑗))
3332adantld 494 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚𝑗))
34 eqid 2798 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
3534limsupgf 14824 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*
3635ffvelrni 6827 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℝ → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
3736adantl 485 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
3837xrleidd 12533 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))
3938adantrr 716 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))
40 limsupbnd.2 . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝐵⟶ℝ*)
4140adantr 484 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝐹:𝐵⟶ℝ*)
4216, 36syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
4334limsupgle 14826 . . . . . . . . . . . . . . 15 (((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ*) ∧ 𝑚 ∈ ℝ ∧ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ↔ ∀𝑗𝐵 (𝑚𝑗 → (𝐹𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))))
4421, 41, 16, 42, 43syl211anc 1373 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ↔ ∀𝑗𝐵 (𝑚𝑗 → (𝐹𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))))
4539, 44mpbid 235 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∀𝑗𝐵 (𝑚𝑗 → (𝐹𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
4645r19.21bi 3173 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (𝑚𝑗 → (𝐹𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
4733, 46syld 47 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → (𝐹𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
4827, 47jcad 516 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → (𝐴 ≤ (𝐹𝑗) ∧ (𝐹𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))))
49 limsupbnd.3 . . . . . . . . . . . 12 (𝜑𝐴 ∈ ℝ*)
5049ad2antrr 725 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝐴 ∈ ℝ*)
5141ffvelrnda 6828 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (𝐹𝑗) ∈ ℝ*)
5242adantr 484 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
53 xrletr 12539 . . . . . . . . . . 11 ((𝐴 ∈ ℝ* ∧ (𝐹𝑗) ∈ ℝ* ∧ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*) → ((𝐴 ≤ (𝐹𝑗) ∧ (𝐹𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
5450, 51, 52, 53syl3anc 1368 . . . . . . . . . 10 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝐴 ≤ (𝐹𝑗) ∧ (𝐹𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
5548, 54syld 47 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
5655rexlimdva 3243 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (∃𝑗𝐵 ((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
5714, 56syl5 34 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ((∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
5813, 57mpan2d 693 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
5958anassrs 471 . . . . 5 (((𝜑𝑚 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
6059rexlimdva 3243 . . . 4 ((𝜑𝑚 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
6160ralrimdva 3154 . . 3 (𝜑 → (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
621, 61mpd 15 . 2 (𝜑 → ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))
6334limsuple 14827 . . 3 ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ*𝐴 ∈ ℝ*) → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
643, 40, 49, 63syl3anc 1368 . 2 (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
6562, 64mpbird 260 1 (𝜑𝐴 ≤ (lim sup‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  wral 3106  wrex 3107  cin 3880  wss 3881  ifcif 4425   class class class wbr 5030  cmpt 5110  cima 5522  wf 6320  cfv 6324  (class class class)co 7135  supcsup 8888  cr 10525  +∞cpnf 10661  *cxr 10663   < clt 10664  cle 10665  [,)cico 12728  lim supclsp 14819
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  ax-pre-mulgt0 10603  ax-pre-sup 10604
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-rmo 3114  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-sup 8890  df-inf 8891  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-ico 12732  df-limsup 14820
This theorem is referenced by:  caucvgrlem  15021  limsupre  42278
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