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Theorem limsupbnd2 14501
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 10337 . . . . . . . . . . 11 ℝ ⊆ ℝ*
53, 4syl6ss 3773 . . . . . . . . . 10 (𝜑𝐵 ⊆ ℝ*)
6 supxrunb1 12351 . . . . . . . . . 10 (𝐵 ⊆ ℝ* → (∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗 ↔ sup(𝐵, ℝ*, < ) = +∞))
75, 6syl 17 . . . . . . . . 9 (𝜑 → (∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗 ↔ sup(𝐵, ℝ*, < ) = +∞))
82, 7mpbird 248 . . . . . . . 8 (𝜑 → ∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗)
9 ifcl 4287 . . . . . . . 8 ((𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ) → if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ)
10 breq1 4812 . . . . . . . . . 10 (𝑛 = if(𝑘𝑚, 𝑚, 𝑘) → (𝑛𝑗 ↔ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗))
1110rexbidv 3199 . . . . . . . . 9 (𝑛 = if(𝑘𝑚, 𝑚, 𝑘) → (∃𝑗𝐵 𝑛𝑗 ↔ ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗))
1211rspccva 3460 . . . . . . . 8 ((∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗 ∧ if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ) → ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗)
138, 9, 12syl2an 589 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗)
14 r19.29 3219 . . . . . . . 8 ((∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → ∃𝑗𝐵 ((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗))
15 simplrr 796 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑘 ∈ ℝ)
16 simprl 787 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝑚 ∈ ℝ)
1716adantr 472 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑚 ∈ ℝ)
18 max1 12218 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑘 ≤ if(𝑘𝑚, 𝑚, 𝑘))
1915, 17, 18syl2anc 579 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑘 ≤ if(𝑘𝑚, 𝑚, 𝑘))
2017, 15, 9syl2anc 579 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ)
213adantr 472 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝐵 ⊆ ℝ)
2221sselda 3761 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑗 ∈ ℝ)
23 letr 10385 . . . . . . . . . . . . . . 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 1490 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝑘 ≤ if(𝑘𝑚, 𝑚, 𝑘) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑘𝑗))
2519, 24mpand 686 . . . . . . . . . . . . 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 398 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ (𝐹𝑗)))
28 max2 12220 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑚 ≤ if(𝑘𝑚, 𝑚, 𝑘))
2915, 17, 28syl2anc 579 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑚 ≤ if(𝑘𝑚, 𝑚, 𝑘))
30 letr 10385 . . . . . . . . . . . . . . 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 1490 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝑚 ≤ if(𝑘𝑚, 𝑚, 𝑘) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚𝑗))
3229, 31mpand 686 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗𝑚𝑗))
3332adantld 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚𝑗))
34 eqid 2765 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
3534limsupgf 14493 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*
3635ffvelrni 6548 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℝ → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
3736adantl 473 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
3837xrleidd 12185 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))
3938adantrr 708 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . 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 14495 . . . . . . . . . . . . . . 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 1495 . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∀𝑗𝐵 (𝑚𝑗 → (𝐹𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
4645r19.21bi 3079 . . . . . . . . . . . 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 508 . . . . . . . . . 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 717 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝐴 ∈ ℝ*)
5141ffvelrnda 6549 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (𝐹𝑗) ∈ ℝ*)
5242adantr 472 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
53 xrletr 12191 . . . . . . . . . . 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 1490 . . . . . . . . . 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 3178 . . . . . . . 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 685 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
5958anassrs 459 . . . . 5 (((𝜑𝑚 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
6059rexlimdva 3178 . . . 4 ((𝜑𝑚 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
6160ralrimdva 3116 . . 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 14496 . . 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 1490 . 2 (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
6562, 64mpbird 248 1 (𝜑𝐴 ≤ (lim sup‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  cin 3731  wss 3732  ifcif 4243   class class class wbr 4809  cmpt 4888  cima 5280  wf 6064  cfv 6068  (class class class)co 6842  supcsup 8553  cr 10188  +∞cpnf 10325  *cxr 10327   < clt 10328  cle 10329  [,)cico 12379  lim supclsp 14488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-po 5198  df-so 5199  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-sup 8555  df-inf 8556  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-ico 12383  df-limsup 14489
This theorem is referenced by:  caucvgrlem  14690  limsupre  40443
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