| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | limsupbnd2.5 | . . 3
⊢ (𝜑 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗))) | 
| 2 |  | limsupbnd2.4 | . . . . . . . . 9
⊢ (𝜑 → sup(𝐵, ℝ*, < ) =
+∞) | 
| 3 |  | limsupbnd.1 | . . . . . . . . . . 11
⊢ (𝜑 → 𝐵 ⊆ ℝ) | 
| 4 |  | ressxr 11305 | . . . . . . . . . . 11
⊢ ℝ
⊆ ℝ* | 
| 5 | 3, 4 | sstrdi 3996 | . . . . . . . . . 10
⊢ (𝜑 → 𝐵 ⊆
ℝ*) | 
| 6 |  | supxrunb1 13361 | . . . . . . . . . 10
⊢ (𝐵 ⊆ ℝ*
→ (∀𝑛 ∈
ℝ ∃𝑗 ∈
𝐵 𝑛 ≤ 𝑗 ↔ sup(𝐵, ℝ*, < ) =
+∞)) | 
| 7 | 5, 6 | syl 17 | . . . . . . . . 9
⊢ (𝜑 → (∀𝑛 ∈ ℝ ∃𝑗 ∈ 𝐵 𝑛 ≤ 𝑗 ↔ sup(𝐵, ℝ*, < ) =
+∞)) | 
| 8 | 2, 7 | mpbird 257 | . . . . . . . 8
⊢ (𝜑 → ∀𝑛 ∈ ℝ ∃𝑗 ∈ 𝐵 𝑛 ≤ 𝑗) | 
| 9 |  | ifcl 4571 | . . . . . . . 8
⊢ ((𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ) → if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∈ ℝ) | 
| 10 |  | breq1 5146 | . . . . . . . . . 10
⊢ (𝑛 = if(𝑘 ≤ 𝑚, 𝑚, 𝑘) → (𝑛 ≤ 𝑗 ↔ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗)) | 
| 11 | 10 | rexbidv 3179 | . . . . . . . . 9
⊢ (𝑛 = if(𝑘 ≤ 𝑚, 𝑚, 𝑘) → (∃𝑗 ∈ 𝐵 𝑛 ≤ 𝑗 ↔ ∃𝑗 ∈ 𝐵 if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗)) | 
| 12 | 11 | rspccva 3621 | . . . . . . . 8
⊢
((∀𝑛 ∈
ℝ ∃𝑗 ∈
𝐵 𝑛 ≤ 𝑗 ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∈ ℝ) → ∃𝑗 ∈ 𝐵 if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) | 
| 13 | 8, 9, 12 | syl2an 596 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∃𝑗 ∈ 𝐵 if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) | 
| 14 |  | r19.29 3114 | . . . . . . . 8
⊢
((∀𝑗 ∈
𝐵 (𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ ∃𝑗 ∈ 𝐵 if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → ∃𝑗 ∈ 𝐵 ((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗)) | 
| 15 |  | simplrr 778 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑘 ∈ ℝ) | 
| 16 |  | simprl 771 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝑚 ∈ ℝ) | 
| 17 | 16 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑚 ∈ ℝ) | 
| 18 |  | max1 13227 | . . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑘 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘)) | 
| 19 | 15, 17, 18 | syl2anc 584 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑘 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘)) | 
| 20 | 17, 15, 9 | syl2anc 584 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∈ ℝ) | 
| 21 | 3 | adantr 480 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝐵 ⊆ ℝ) | 
| 22 | 21 | sselda 3983 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑗 ∈ ℝ) | 
| 23 |  | letr 11355 | . . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ ℝ ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∈ ℝ ∧ 𝑗 ∈ ℝ) → ((𝑘 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑘 ≤ 𝑗)) | 
| 24 | 15, 20, 22, 23 | syl3anc 1373 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → ((𝑘 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑘 ≤ 𝑗)) | 
| 25 | 19, 24 | mpand 695 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗 → 𝑘 ≤ 𝑗)) | 
| 26 | 25 | imim1d 82 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → ((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) → (if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)))) | 
| 27 | 26 | impd 410 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ (𝐹‘𝑗))) | 
| 28 |  | max2 13229 | . . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ ℝ ∧ 𝑚 ∈ ℝ) → 𝑚 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘)) | 
| 29 | 15, 17, 28 | syl2anc 584 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝑚 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘)) | 
| 30 |  | letr 11355 | . . . . . . . . . . . . . . 15
⊢ ((𝑚 ∈ ℝ ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∈ ℝ ∧ 𝑗 ∈ ℝ) → ((𝑚 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚 ≤ 𝑗)) | 
| 31 | 17, 20, 22, 30 | syl3anc 1373 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → ((𝑚 ≤ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚 ≤ 𝑗)) | 
| 32 | 29, 31 | mpand 695 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗 → 𝑚 ≤ 𝑗)) | 
| 33 | 32 | adantld 490 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝑚 ≤ 𝑗)) | 
| 34 |  | eqid 2737 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℝ ↦
sup(((𝐹 “ (𝑛[,)+∞)) ∩
ℝ*), ℝ*, < )) = (𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < )) | 
| 35 | 34 | limsupgf 15511 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℝ ↦
sup(((𝐹 “ (𝑛[,)+∞)) ∩
ℝ*), ℝ*, <
)):ℝ⟶ℝ* | 
| 36 | 35 | ffvelcdmi 7103 | . . . . . . . . . . . . . . . . 17
⊢ (𝑚 ∈ ℝ → ((𝑛 ∈ ℝ ↦
sup(((𝐹 “ (𝑛[,)+∞)) ∩
ℝ*), ℝ*, < ))‘𝑚) ∈
ℝ*) | 
| 37 | 36 | adantl 481 | . . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚) ∈
ℝ*) | 
| 38 | 37 | xrleidd 13194 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚)) | 
| 39 | 38 | adantrr 717 | . . . . . . . . . . . . . 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
⊢ (𝜑 → 𝐹:𝐵⟶ℝ*) | 
| 41 | 40 | adantr 480 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → 𝐹:𝐵⟶ℝ*) | 
| 42 | 16, 36 | syl 17 | . . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚) ∈
ℝ*) | 
| 43 | 34 | limsupgle 15513 | . . . . . . . . . . . . . . 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) ↔ ∀𝑗 ∈ 𝐵 (𝑚 ≤ 𝑗 → (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚)))) | 
| 44 | 21, 41, 16, 42, 43 | syl211anc 1378 | . . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (((𝑛 ∈ ℝ ↦
sup(((𝐹 “ (𝑛[,)+∞)) ∩
ℝ*), ℝ*, < ))‘𝑚) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚) ↔ ∀𝑗 ∈ 𝐵 (𝑚 ≤ 𝑗 → (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚)))) | 
| 45 | 39, 44 | mpbid 232 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ∀𝑗 ∈ 𝐵 (𝑚 ≤ 𝑗 → (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚))) | 
| 46 | 45 | r19.21bi 3251 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (𝑚 ≤ 𝑗 → (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚))) | 
| 47 | 33, 46 | syld 47 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚))) | 
| 48 | 27, 47 | jcad 512 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → (𝐴 ≤ (𝐹‘𝑗) ∧ (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚)))) | 
| 49 |  | limsupbnd.3 | . . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ∈
ℝ*) | 
| 50 | 49 | ad2antrr 726 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → 𝐴 ∈
ℝ*) | 
| 51 | 41 | ffvelcdmda 7104 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (𝐹‘𝑗) ∈
ℝ*) | 
| 52 | 42 | adantr 480 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚) ∈
ℝ*) | 
| 53 |  | xrletr 13200 | . . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ*
∧ (𝐹‘𝑗) ∈ ℝ*
∧ ((𝑛 ∈ ℝ
↦ sup(((𝐹 “
(𝑛[,)+∞)) ∩
ℝ*), ℝ*, < ))‘𝑚) ∈ ℝ*) → ((𝐴 ≤ (𝐹‘𝑗) ∧ (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚))) | 
| 54 | 50, 51, 52, 53 | syl3anc 1373 | . . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → ((𝐴 ≤ (𝐹‘𝑗) ∧ (𝐹‘𝑗) ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚))) | 
| 55 | 48, 54 | syld 47 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) ∧ 𝑗 ∈ 𝐵) → (((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚))) | 
| 56 | 55 | rexlimdva 3155 | . . . . . . . 8
⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (∃𝑗 ∈ 𝐵 ((𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚))) | 
| 57 | 14, 56 | syl5 34 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → ((∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) ∧ ∃𝑗 ∈ 𝐵 if(𝑘 ≤ 𝑚, 𝑚, 𝑘) ≤ 𝑗) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚))) | 
| 58 | 13, 57 | mpan2d 694 | . . . . . 6
⊢ ((𝜑 ∧ (𝑚 ∈ ℝ ∧ 𝑘 ∈ ℝ)) → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚))) | 
| 59 | 58 | anassrs 467 | . . . . 5
⊢ (((𝜑 ∧ 𝑚 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚))) | 
| 60 | 59 | rexlimdva 3155 | . . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℝ) → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) → 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚))) | 
| 61 | 60 | ralrimdva 3154 | . . 3
⊢ (𝜑 → (∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐵 (𝑘 ≤ 𝑗 → 𝐴 ≤ (𝐹‘𝑗)) → ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚))) | 
| 62 | 1, 61 | mpd 15 | . 2
⊢ (𝜑 → ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚)) | 
| 63 | 34 | limsuple 15514 | . . 3
⊢ ((𝐵 ⊆ ℝ ∧ 𝐹:𝐵⟶ℝ* ∧ 𝐴 ∈ ℝ*)
→ (𝐴 ≤ (lim
sup‘𝐹) ↔
∀𝑚 ∈ ℝ
𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚))) | 
| 64 | 3, 40, 49, 63 | syl3anc 1373 | . 2
⊢ (𝜑 → (𝐴 ≤ (lim sup‘𝐹) ↔ ∀𝑚 ∈ ℝ 𝐴 ≤ ((𝑛 ∈ ℝ ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))‘𝑚))) | 
| 65 | 62, 64 | mpbird 257 | 1
⊢ (𝜑 → 𝐴 ≤ (lim sup‘𝐹)) |