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Theorem limsupbnd2 14143
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 10028 . . . . . . . . . . 11 ℝ ⊆ ℝ*
53, 4syl6ss 3600 . . . . . . . . . 10 (𝜑𝐵 ⊆ ℝ*)
6 supxrunb1 12089 . . . . . . . . . 10 (𝐵 ⊆ ℝ* → (∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗 ↔ sup(𝐵, ℝ*, < ) = +∞))
75, 6syl 17 . . . . . . . . 9 (𝜑 → (∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗 ↔ sup(𝐵, ℝ*, < ) = +∞))
82, 7mpbird 247 . . . . . . . 8 (𝜑 → ∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗)
9 ifcl 4107 . . . . . . . 8 ((𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ) → if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ)
10 breq1 4621 . . . . . . . . . 10 (𝑛 = if(𝑘𝑚, 𝑚, 𝑘) → (𝑛𝑗 ↔ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗))
1110rexbidv 3050 . . . . . . . . 9 (𝑛 = if(𝑘𝑚, 𝑚, 𝑘) → (∃𝑗𝐵 𝑛𝑗 ↔ ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗))
1211rspccva 3299 . . . . . . . 8 ((∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗 ∧ if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ) → ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗)
138, 9, 12syl2an 494 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗)
14 r19.29 3070 . . . . . . . 8 ((∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → ∃𝑗𝐵 ((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗))
15 simplrr 800 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑘 ∈ ℝ)
16 simprl 793 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝑚 ∈ ℝ)
1716adantr 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑚 ∈ ℝ)
18 max1 11958 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑘 ≤ if(𝑘𝑚, 𝑚, 𝑘))
1915, 17, 18syl2anc 692 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑘 ≤ if(𝑘𝑚, 𝑚, 𝑘))
2017, 15, 9syl2anc 692 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ)
213adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝐵 ⊆ ℝ)
2221sselda 3588 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑗 ∈ ℝ)
23 letr 10076 . . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝑘 ≤ if(𝑘𝑚, 𝑚, 𝑘) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑘𝑗))
2519, 24mpand 710 . . . . . . . . . . . . 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 447 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ (𝐹𝑗)))
28 max2 11960 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑚 ≤ if(𝑘𝑚, 𝑚, 𝑘))
2915, 17, 28syl2anc 692 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑚 ≤ if(𝑘𝑚, 𝑚, 𝑘))
30 letr 10076 . . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝑚 ≤ if(𝑘𝑚, 𝑚, 𝑘) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚𝑗))
3229, 31mpand 710 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗𝑚𝑗))
3332adantld 483 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚𝑗))
34 eqid 2626 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
3534limsupgf 14135 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*
3635ffvelrni 6315 . . . . . . . . . . . . . . . . 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*)
38 xrleid 11927 . . . . . . . . . . . . . . . 16 (((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))
3937, 38syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))
4039adantrr 752 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))
41 limsupbnd.2 . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝐵⟶ℝ*)
4241adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝐹:𝐵⟶ℝ*)
4316, 36syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
4434limsupgle 14137 . . . . . . . . . . . . . . 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))))
4521, 42, 16, 43, 44syl211anc 1329 . . . . . . . . . . . . . 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))))
4640, 45mpbid 222 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∀𝑗𝐵 (𝑚𝑗 → (𝐹𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
4746r19.21bi 2932 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (𝑚𝑗 → (𝐹𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
4833, 47syld 47 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → (𝐹𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
4927, 48jcad 555 . . . . . . . . . 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))))
50 limsupbnd.3 . . . . . . . . . . . 12 (𝜑𝐴 ∈ ℝ*)
5150ad2antrr 761 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝐴 ∈ ℝ*)
5242ffvelrnda 6316 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (𝐹𝑗) ∈ ℝ*)
5343adantr 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
54 xrletr 11933 . . . . . . . . . . 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)))
5551, 52, 53, 54syl3anc 1323 . . . . . . . . . 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)))
5649, 55syld 47 . . . . . . . . 9 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
5756rexlimdva 3029 . . . . . . . 8 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (∃𝑗𝐵 ((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
5814, 57syl5 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)))
5913, 58mpan2d 709 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
6059anassrs 679 . . . . 5 (((𝜑𝑚 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
6160rexlimdva 3029 . . . 4 ((𝜑𝑚 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
6261ralrimdva 2968 . . 3 (𝜑 → (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
631, 62mpd 15 . 2 (𝜑 → ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚))
6434limsuple 14138 . . 3 ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ*𝐴 ∈ ℝ*) → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
653, 41, 50, 64syl3anc 1323 . 2 (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
6663, 65mpbird 247 1 (𝜑𝐴 ≤ (lim sup‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  cin 3559  wss 3560  ifcif 4063   class class class wbr 4618  cmpt 4678  cima 5082  wf 5846  cfv 5850  (class class class)co 6605  supcsup 8291  cr 9880  +∞cpnf 10016  *cxr 10018   < clt 10019  cle 10020  [,)cico 12116  lim supclsp 14130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-sup 8293  df-inf 8294  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-ico 12120  df-limsup 14131
This theorem is referenced by:  caucvgrlem  14332  limsupre  39264
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