| Step | Hyp | Ref
| Expression |
| 1 | | nfv 1914 |
. . . . 5
⊢
Ⅎ𝑘𝜑 |
| 2 | | reex 11246 |
. . . . . . 7
⊢ ℝ
∈ V |
| 3 | 2 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ℝ ∈
V) |
| 4 | | limsupmnflem.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
| 5 | 3, 4 | ssexd 5324 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ V) |
| 6 | | limsupmnflem.f |
. . . . 5
⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
| 7 | | limsupmnflem.g |
. . . . 5
⊢ 𝐺 = (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, <
)) |
| 8 | 1, 5, 6, 7 | limsupval3 45707 |
. . . 4
⊢ (𝜑 → (lim sup‘𝐹) = inf(ran 𝐺, ℝ*, <
)) |
| 9 | 7 | rneqi 5948 |
. . . . . 6
⊢ ran 𝐺 = ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, <
)) |
| 10 | 9 | infeq1i 9518 |
. . . . 5
⊢ inf(ran
𝐺, ℝ*,
< ) = inf(ran (𝑘 ∈
ℝ ↦ sup((𝐹
“ (𝑘[,)+∞)),
ℝ*, < )), ℝ*, < ) |
| 11 | 10 | a1i 11 |
. . . 4
⊢ (𝜑 → inf(ran 𝐺, ℝ*, < ) = inf(ran
(𝑘 ∈ ℝ ↦
sup((𝐹 “ (𝑘[,)+∞)),
ℝ*, < )), ℝ*, < )) |
| 12 | 8, 11 | eqtrd 2777 |
. . 3
⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ ℝ ↦ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )),
ℝ*, < )) |
| 13 | 12 | eqeq1d 2739 |
. 2
⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ inf(ran
(𝑘 ∈ ℝ ↦
sup((𝐹 “ (𝑘[,)+∞)),
ℝ*, < )), ℝ*, < ) =
-∞)) |
| 14 | | nfv 1914 |
. . 3
⊢
Ⅎ𝑥𝜑 |
| 15 | 6 | fimassd 6757 |
. . . . 5
⊢ (𝜑 → (𝐹 “ (𝑘[,)+∞)) ⊆
ℝ*) |
| 16 | 15 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (𝐹 “ (𝑘[,)+∞)) ⊆
ℝ*) |
| 17 | 16 | supxrcld 45112 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )
∈ ℝ*) |
| 18 | 1, 14, 17 | infxrunb3rnmpt 45439 |
. 2
⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )
≤ 𝑥 ↔ inf(ran
(𝑘 ∈ ℝ ↦
sup((𝐹 “ (𝑘[,)+∞)),
ℝ*, < )), ℝ*, < ) =
-∞)) |
| 19 | 15 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝐹 “ (𝑘[,)+∞)) ⊆
ℝ*) |
| 20 | | ressxr 11305 |
. . . . . . . . 9
⊢ ℝ
⊆ ℝ* |
| 21 | 20 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ℝ ⊆
ℝ*) |
| 22 | 21 | sselda 3983 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ*) |
| 23 | | supxrleub 13368 |
. . . . . . 7
⊢ (((𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*
∧ 𝑥 ∈
ℝ*) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )
≤ 𝑥 ↔ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥)) |
| 24 | 19, 22, 23 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )
≤ 𝑥 ↔ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥)) |
| 25 | 24 | adantr 480 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )
≤ 𝑥 ↔ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥)) |
| 26 | 6 | ffnd 6737 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐹 Fn 𝐴) |
| 27 | 26 | ad3antrrr 730 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝐹 Fn 𝐴) |
| 28 | | simplr 769 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ 𝐴) |
| 29 | 20 | sseli 3979 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℝ → 𝑘 ∈
ℝ*) |
| 30 | 29 | ad3antlr 731 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑘 ∈ ℝ*) |
| 31 | | pnfxr 11315 |
. . . . . . . . . . . . . . 15
⊢ +∞
∈ ℝ* |
| 32 | 31 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → +∞ ∈
ℝ*) |
| 33 | 20 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ℝ ⊆
ℝ*) |
| 34 | 4 | sselda 3983 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ ℝ) |
| 35 | 33, 34 | sseldd 3984 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ ℝ*) |
| 36 | 35 | ad4ant13 751 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ ℝ*) |
| 37 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑘 ≤ 𝑗) |
| 38 | 34 | ltpnfd 13163 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 < +∞) |
| 39 | 38 | ad4ant13 751 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 < +∞) |
| 40 | 30, 32, 36, 37, 39 | elicod 13437 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → 𝑗 ∈ (𝑘[,)+∞)) |
| 41 | 27, 28, 40 | fnfvimad 7254 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ∈ (𝐹 “ (𝑘[,)+∞))) |
| 42 | 41 | adantllr 719 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑘 ∈ ℝ) ∧
∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ∈ (𝐹 “ (𝑘[,)+∞))) |
| 43 | | simpllr 776 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑘 ∈ ℝ) ∧
∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) |
| 44 | | breq1 5146 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝐹‘𝑗) → (𝑦 ≤ 𝑥 ↔ (𝐹‘𝑗) ≤ 𝑥)) |
| 45 | 44 | rspcva 3620 |
. . . . . . . . . . 11
⊢ (((𝐹‘𝑗) ∈ (𝐹 “ (𝑘[,)+∞)) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) → (𝐹‘𝑗) ≤ 𝑥) |
| 46 | 42, 43, 45 | syl2anc 584 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑘 ∈ ℝ) ∧
∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑥) |
| 47 | 46 | adantl4r 755 |
. . . . . . . . 9
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧
∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) ∧ 𝑘 ≤ 𝑗) → (𝐹‘𝑗) ≤ 𝑥) |
| 48 | 47 | ex 412 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧
∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
| 49 | 48 | ralrimiva 3146 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
| 50 | 49 | ex 412 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥 → ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
| 51 | | nfcv 2905 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗𝐹 |
| 52 | 26 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → 𝐹 Fn 𝐴) |
| 53 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) |
| 54 | 51, 52, 53 | fvelimad 6976 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → ∃𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑗) = 𝑦) |
| 55 | 54 | ad4ant14 752 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → ∃𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑗) = 𝑦) |
| 56 | | nfv 1914 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗(𝜑 ∧ 𝑘 ∈ ℝ) |
| 57 | | nfra1 3284 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) |
| 58 | 56, 57 | nfan 1899 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
| 59 | | nfv 1914 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗 𝑦 ≤ 𝑥 |
| 60 | 29 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑘 ∈ ℝ*) |
| 61 | 31 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → +∞ ∈
ℝ*) |
| 62 | | elinel2 4202 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞)) → 𝑗 ∈ (𝑘[,)+∞)) |
| 63 | 62 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑗 ∈ (𝑘[,)+∞)) |
| 64 | 60, 61, 63 | icogelbd 45571 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑘 ∈ ℝ ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑘 ≤ 𝑗) |
| 65 | 64 | adantlr 715 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑘 ∈ ℝ ∧
∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑘 ≤ 𝑗) |
| 66 | | elinel1 4201 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞)) → 𝑗 ∈ 𝐴) |
| 67 | 66 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑗 ∈
𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → 𝑗 ∈ 𝐴) |
| 68 | | rspa 3248 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((∀𝑗 ∈
𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ 𝐴) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
| 69 | 67, 68 | syldan 591 |
. . . . . . . . . . . . . . . . . . 19
⊢
((∀𝑗 ∈
𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
| 70 | 69 | adantll 714 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑘 ∈ ℝ ∧
∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) |
| 71 | 65, 70 | mpd 15 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℝ ∧
∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → (𝐹‘𝑗) ≤ 𝑥) |
| 72 | | id 22 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐹‘𝑗) = 𝑦 → (𝐹‘𝑗) = 𝑦) |
| 73 | 72 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐹‘𝑗) = 𝑦 → 𝑦 = (𝐹‘𝑗)) |
| 74 | 73 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹‘𝑗) ≤ 𝑥 ∧ (𝐹‘𝑗) = 𝑦) → 𝑦 = (𝐹‘𝑗)) |
| 75 | | simpl 482 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹‘𝑗) ≤ 𝑥 ∧ (𝐹‘𝑗) = 𝑦) → (𝐹‘𝑗) ≤ 𝑥) |
| 76 | 74, 75 | eqbrtrd 5165 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐹‘𝑗) ≤ 𝑥 ∧ (𝐹‘𝑗) = 𝑦) → 𝑦 ≤ 𝑥) |
| 77 | 76 | ex 412 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹‘𝑗) ≤ 𝑥 → ((𝐹‘𝑗) = 𝑦 → 𝑦 ≤ 𝑥)) |
| 78 | 71, 77 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℝ ∧
∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → ((𝐹‘𝑗) = 𝑦 → 𝑦 ≤ 𝑥)) |
| 79 | 78 | adantlll 718 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))) → ((𝐹‘𝑗) = 𝑦 → 𝑦 ≤ 𝑥)) |
| 80 | 79 | ex 412 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞)) → ((𝐹‘𝑗) = 𝑦 → 𝑦 ≤ 𝑥))) |
| 81 | 58, 59, 80 | rexlimd 3266 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (∃𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑗) = 𝑦 → 𝑦 ≤ 𝑥)) |
| 82 | 81 | imp 406 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ ∃𝑗 ∈ (𝐴 ∩ (𝑘[,)+∞))(𝐹‘𝑗) = 𝑦) → 𝑦 ≤ 𝑥) |
| 83 | 55, 82 | syldan 591 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → 𝑦 ≤ 𝑥) |
| 84 | 83 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) |
| 85 | 84 | adantllr 719 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥) |
| 86 | 24 | ad2antrr 726 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )
≤ 𝑥 ↔ ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥)) |
| 87 | 85, 86 | mpbird 257 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) ∧ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥)) → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )
≤ 𝑥) |
| 88 | 87 | ex 412 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )
≤ 𝑥)) |
| 89 | 88, 25 | sylibd 239 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥) → ∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥)) |
| 90 | 50, 89 | impbid 212 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (∀𝑦 ∈ (𝐹 “ (𝑘[,)+∞))𝑦 ≤ 𝑥 ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
| 91 | 25, 90 | bitrd 279 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑘 ∈ ℝ) → (sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )
≤ 𝑥 ↔ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
| 92 | 91 | rexbidva 3177 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∃𝑘 ∈ ℝ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )
≤ 𝑥 ↔ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
| 93 | 92 | ralbidva 3176 |
. 2
⊢ (𝜑 → (∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ sup((𝐹 “ (𝑘[,)+∞)), ℝ*, < )
≤ 𝑥 ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |
| 94 | 13, 18, 93 | 3bitr2d 307 |
1
⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔
∀𝑥 ∈ ℝ
∃𝑘 ∈ ℝ
∀𝑗 ∈ 𝐴 (𝑘 ≤ 𝑗 → (𝐹‘𝑗) ≤ 𝑥))) |