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Theorem limsupbnd2 14258
 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 10121 . . . . . . . . . . 11 ℝ ⊆ ℝ*
53, 4syl6ss 3648 . . . . . . . . . 10 (𝜑𝐵 ⊆ ℝ*)
6 supxrunb1 12187 . . . . . . . . . 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 4163 . . . . . . . 8 ((𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ) → if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ)
10 breq1 4688 . . . . . . . . . 10 (𝑛 = if(𝑘𝑚, 𝑚, 𝑘) → (𝑛𝑗 ↔ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗))
1110rexbidv 3081 . . . . . . . . 9 (𝑛 = if(𝑘𝑚, 𝑚, 𝑘) → (∃𝑗𝐵 𝑛𝑗 ↔ ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗))
1211rspccva 3339 . . . . . . . 8 ((∀𝑛 ∈ ℝ ∃𝑗𝐵 𝑛𝑗 ∧ if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ) → ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗)
138, 9, 12syl2an 493 . . . . . . 7 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗)
14 r19.29 3101 . . . . . . . 8 ((∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ ∃𝑗𝐵 if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → ∃𝑗𝐵 ((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗))
15 simplrr 818 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑘 ∈ ℝ)
16 simprl 809 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝑚 ∈ ℝ)
1716adantr 480 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑚 ∈ ℝ)
18 max1 12054 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑘 ≤ if(𝑘𝑚, 𝑚, 𝑘))
1915, 17, 18syl2anc 694 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑘 ≤ if(𝑘𝑚, 𝑚, 𝑘))
2017, 15, 9syl2anc 694 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → if(𝑘𝑚, 𝑚, 𝑘) ∈ ℝ)
213adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝐵 ⊆ ℝ)
2221sselda 3636 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑗 ∈ ℝ)
23 letr 10169 . . . . . . . . . . . . . . 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 1366 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝑘 ≤ if(𝑘𝑚, 𝑚, 𝑘) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑘𝑗))
2519, 24mpand 711 . . . . . . . . . . . . 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 446 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ (𝐹𝑗)))
28 max2 12056 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑚 ≤ if(𝑘𝑚, 𝑚, 𝑘))
2915, 17, 28syl2anc 694 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝑚 ≤ if(𝑘𝑚, 𝑚, 𝑘))
30 letr 10169 . . . . . . . . . . . . . . 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 1366 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝑚 ≤ if(𝑘𝑚, 𝑚, 𝑘) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚𝑗))
3229, 31mpand 711 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗𝑚𝑗))
3332adantld 482 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (((𝑘𝑗𝐴 ≤ (𝐹𝑗)) ∧ if(𝑘𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚𝑗))
34 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))
3534limsupgf 14250 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < )):ℝ⟶ℝ*
3635ffvelrni 6398 . . . . . . . . . . . . . . . . 17 (𝑚 ∈ ℝ → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
3736adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
38 xrleid 12021 . . . . . . . . . . . . . . . 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 753 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 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 14252 . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . 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 2961 . . . . . . . . . . . 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 554 . . . . . . . . . 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 762 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → 𝐴 ∈ ℝ*)
5242ffvelrnda 6399 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → (𝐹𝑗) ∈ ℝ*)
5343adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗𝐵) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*)
54 xrletr 12027 . . . . . . . . . . 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 1366 . . . . . . . . . 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 3060 . . . . . . . 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 710 . . . . . 6 ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
6059anassrs 681 . . . . 5 (((𝜑𝑚 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
6160rexlimdva 3060 . . . 4 ((𝜑𝑚 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗𝐵 (𝑘𝑗𝐴 ≤ (𝐹𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*), ℝ*, < ))‘𝑚)))
6261ralrimdva 2998 . . 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 14253 . . 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 1366 . 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 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   ∩ cin 3606   ⊆ wss 3607  ifcif 4119   class class class wbr 4685   ↦ cmpt 4762   “ cima 5146  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  supcsup 8387  ℝcr 9973  +∞cpnf 10109  ℝ*cxr 10111   < clt 10112   ≤ cle 10113  [,)cico 12215  lim supclsp 14245 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-ico 12219  df-limsup 14246 This theorem is referenced by:  caucvgrlem  14447  limsupre  40191
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