MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  limsupbnd2 Structured version   Visualization version   GIF version

Theorem limsupbnd2 15409
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 11240 . . . . . . . . . . 11 ℝ ⊆ ℝ*
53, 4sstrdi 3990 . . . . . . . . . 10 (𝜑𝐵 ⊆ ℝ*)
6 supxrunb1 13280 . . . . . . . . . 10 (𝐵 ⊆ ℝ* → (∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗 ↔ sup(𝐵, ℝ*, < ) = +∞))
75, 6syl 17 . . . . . . . . 9 (𝜑 → (∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗 ↔ sup(𝐵, ℝ*, < ) = +∞))
82, 7mpbird 256 . . . . . . . 8 (𝜑 → ∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗)
9 ifcl 4567 . . . . . . . 8 ((𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ) → if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ)
10 breq1 5144 . . . . . . . . . 10 (𝑛 = if(𝑘𝑚, 𝑚, 𝑘) → (𝑛𝑗 ↔ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗))
1110rexbidv 3177 . . . . . . . . 9 (𝑛 = if(𝑘𝑚, 𝑚, 𝑘) → (∃𝑗𝐵 𝑛𝑗 ↔ ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗))
1211rspccva 3608 . . . . . . . 8 ((∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗 ∧ if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ) → ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗)
138, 9, 12syl2an 596 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗)
14 r19.29 3113 . . . . . . . 8 ((∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → ∃𝑗𝐵 ((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗))
15 simplrr 776 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑘 ∈ ℝ)
16 simprl 769 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝑚 ∈ ℝ)
1716adantr 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑚 ∈ ℝ)
18 max1 13146 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑘 ≤ if(𝑘𝑚, 𝑚, 𝑘))
1915, 17, 18syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑘 ≤ if(𝑘𝑚, 𝑚, 𝑘))
2017, 15, 9syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ)
213adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝐵 ⊆ ℝ)
2221sselda 3978 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑗 ∈ ℝ)
23 letr 11290 . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝑘 ≤ if(𝑘𝑚, 𝑚, 𝑘) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑘𝑗))
2519, 24mpand 693 . . . . . . . . . . . . 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 411 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ (𝐹𝑗)))
28 max2 13148 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑚 ≤ if(𝑘𝑚, 𝑚, 𝑘))
2915, 17, 28syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑚 ≤ if(𝑘𝑚, 𝑚, 𝑘))
30 letr 11290 . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝑚 ≤ if(𝑘𝑚, 𝑚, 𝑘) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚𝑗))
3229, 31mpand 693 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗𝑚𝑗))
3332adantld 491 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚𝑗))
34 eqid 2731 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
3534limsupgf 15401 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*
3635ffvelcdmi 7070 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℝ → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
3736adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
3837xrleidd 13113 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))
3938adantrr 715 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 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 15403 . . . . . . . . . . . . . . 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 1376 . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∀𝑗𝐵 (𝑚𝑗 → (𝐹𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
4645r19.21bi 3247 . . . . . . . . . . . 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 513 . . . . . . . . . 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 724 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝐴 ∈ ℝ*)
5141ffvelcdmda 7071 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (𝐹𝑗) ∈ ℝ*)
5242adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
53 xrletr 13119 . . . . . . . . . . 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 1371 . . . . . . . . . 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 3154 . . . . . . . 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 692 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
5958anassrs 468 . . . . 5 (((𝜑𝑚 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
6059rexlimdva 3154 . . . 4 ((𝜑𝑚 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
6160ralrimdva 3153 . . 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 15404 . . 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 1371 . 2 (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
6562, 64mpbird 256 1 (𝜑𝐴 ≤ (lim sup‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3060  wrex 3069  cin 3943  wss 3944  ifcif 4522   class class class wbr 5141  cmpt 5224  cima 5672  wf 6528  cfv 6532  (class class class)co 7393  supcsup 9417  cr 11091  +∞cpnf 11227  *cxr 11229   < clt 11230  cle 11231  [,)cico 13308  lim supclsp 15396
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169  ax-pre-sup 11170
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-er 8686  df-en 8923  df-dom 8924  df-sdom 8925  df-sup 9419  df-inf 9420  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-ico 13312  df-limsup 15397
This theorem is referenced by:  caucvgrlem  15601  limsupre  44128
  Copyright terms: Public domain W3C validator