| Step | Hyp | Ref
| Expression |
| 1 | | eqid 2737 |
. . 3
⊢ (𝑖 ∈ ℝ ↦
sup(((𝐹 “ (𝑖[,)+∞)) ∩
ℝ*), ℝ*, < )) = (𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*),
ℝ*, < )) |
| 2 | | limsupvaluz.f |
. . . 4
⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
| 3 | | limsupvaluz.z |
. . . . . 6
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 4 | 3 | fvexi 6920 |
. . . . 5
⊢ 𝑍 ∈ V |
| 5 | 4 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝑍 ∈ V) |
| 6 | 2, 5 | fexd 7247 |
. . 3
⊢ (𝜑 → 𝐹 ∈ V) |
| 7 | | uzssre 12900 |
. . . . 5
⊢
(ℤ≥‘𝑀) ⊆ ℝ |
| 8 | 3, 7 | eqsstri 4030 |
. . . 4
⊢ 𝑍 ⊆
ℝ |
| 9 | 8 | a1i 11 |
. . 3
⊢ (𝜑 → 𝑍 ⊆ ℝ) |
| 10 | | limsupvaluz.m |
. . . 4
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 11 | 3 | uzsup 13903 |
. . . 4
⊢ (𝑀 ∈ ℤ → sup(𝑍, ℝ*, < ) =
+∞) |
| 12 | 10, 11 | syl 17 |
. . 3
⊢ (𝜑 → sup(𝑍, ℝ*, < ) =
+∞) |
| 13 | 1, 6, 9, 12 | limsupval2 15516 |
. 2
⊢ (𝜑 → (lim sup‘𝐹) = inf(((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*),
ℝ*, < )) “ 𝑍), ℝ*, <
)) |
| 14 | 9 | mptimass 6091 |
. . . 4
⊢ (𝜑 → ((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*),
ℝ*, < )) “ 𝑍) = ran (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*),
ℝ*, < ))) |
| 15 | | oveq1 7438 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑛 → (𝑖[,)+∞) = (𝑛[,)+∞)) |
| 16 | 15 | imaeq2d 6078 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑛 → (𝐹 “ (𝑖[,)+∞)) = (𝐹 “ (𝑛[,)+∞))) |
| 17 | 16 | ineq1d 4219 |
. . . . . . . . 9
⊢ (𝑖 = 𝑛 → ((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*) =
((𝐹 “ (𝑛[,)+∞)) ∩
ℝ*)) |
| 18 | 17 | supeq1d 9486 |
. . . . . . . 8
⊢ (𝑖 = 𝑛 → sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*),
ℝ*, < ) = sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < )) |
| 19 | 18 | cbvmptv 5255 |
. . . . . . 7
⊢ (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*),
ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < )) |
| 20 | 19 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*),
ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ))) |
| 21 | | fimass 6756 |
. . . . . . . . . . . 12
⊢ (𝐹:𝑍⟶ℝ* → (𝐹 “ (𝑛[,)+∞)) ⊆
ℝ*) |
| 22 | 2, 21 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹 “ (𝑛[,)+∞)) ⊆
ℝ*) |
| 23 | | dfss2 3969 |
. . . . . . . . . . . 12
⊢ ((𝐹 “ (𝑛[,)+∞)) ⊆ ℝ*
↔ ((𝐹 “ (𝑛[,)+∞)) ∩
ℝ*) = (𝐹
“ (𝑛[,)+∞))) |
| 24 | 23 | biimpi 216 |
. . . . . . . . . . 11
⊢ ((𝐹 “ (𝑛[,)+∞)) ⊆ ℝ*
→ ((𝐹 “ (𝑛[,)+∞)) ∩
ℝ*) = (𝐹
“ (𝑛[,)+∞))) |
| 25 | 22, 24 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) =
(𝐹 “ (𝑛[,)+∞))) |
| 26 | 25 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) =
(𝐹 “ (𝑛[,)+∞))) |
| 27 | | df-ima 5698 |
. . . . . . . . . 10
⊢ (𝐹 “ (𝑛[,)+∞)) = ran (𝐹 ↾ (𝑛[,)+∞)) |
| 28 | 27 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 “ (𝑛[,)+∞)) = ran (𝐹 ↾ (𝑛[,)+∞))) |
| 29 | 2 | freld 6742 |
. . . . . . . . . . . . 13
⊢ (𝜑 → Rel 𝐹) |
| 30 | | resindm 6048 |
. . . . . . . . . . . . 13
⊢ (Rel
𝐹 → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞))) |
| 31 | 29, 30 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞))) |
| 32 | 31 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (𝑛[,)+∞))) |
| 33 | | incom 4209 |
. . . . . . . . . . . . . . 15
⊢ ((𝑛[,)+∞) ∩ 𝑍) = (𝑍 ∩ (𝑛[,)+∞)) |
| 34 | 3 | ineq1i 4216 |
. . . . . . . . . . . . . . 15
⊢ (𝑍 ∩ (𝑛[,)+∞)) =
((ℤ≥‘𝑀) ∩ (𝑛[,)+∞)) |
| 35 | 33, 34 | eqtri 2765 |
. . . . . . . . . . . . . 14
⊢ ((𝑛[,)+∞) ∩ 𝑍) =
((ℤ≥‘𝑀) ∩ (𝑛[,)+∞)) |
| 36 | 35 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛[,)+∞) ∩ 𝑍) = ((ℤ≥‘𝑀) ∩ (𝑛[,)+∞))) |
| 37 | 2 | fdmd 6746 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → dom 𝐹 = 𝑍) |
| 38 | 37 | ineq2d 4220 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑛[,)+∞) ∩ dom 𝐹) = ((𝑛[,)+∞) ∩ 𝑍)) |
| 39 | 38 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛[,)+∞) ∩ dom 𝐹) = ((𝑛[,)+∞) ∩ 𝑍)) |
| 40 | 3 | eleq2i 2833 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀)) |
| 41 | 40 | biimpi 216 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑀)) |
| 42 | 41 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (ℤ≥‘𝑀)) |
| 43 | 42 | uzinico2 45575 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) =
((ℤ≥‘𝑀) ∩ (𝑛[,)+∞))) |
| 44 | 36, 39, 43 | 3eqtr4d 2787 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑛[,)+∞) ∩ dom 𝐹) = (ℤ≥‘𝑛)) |
| 45 | 44 | reseq2d 5997 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ ((𝑛[,)+∞) ∩ dom 𝐹)) = (𝐹 ↾ (ℤ≥‘𝑛))) |
| 46 | 32, 45 | eqtr3d 2779 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (𝑛[,)+∞)) = (𝐹 ↾ (ℤ≥‘𝑛))) |
| 47 | 46 | rneqd 5949 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ran (𝐹 ↾ (𝑛[,)+∞)) = ran (𝐹 ↾ (ℤ≥‘𝑛))) |
| 48 | 26, 28, 47 | 3eqtrd 2781 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*) =
ran (𝐹 ↾
(ℤ≥‘𝑛))) |
| 49 | 48 | supeq1d 9486 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < ) = sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
)) |
| 50 | 49 | mpteq2dva 5242 |
. . . . . 6
⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑛[,)+∞)) ∩ ℝ*),
ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
))) |
| 51 | 20, 50 | eqtrd 2777 |
. . . . 5
⊢ (𝜑 → (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*),
ℝ*, < )) = (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
))) |
| 52 | 51 | rneqd 5949 |
. . . 4
⊢ (𝜑 → ran (𝑖 ∈ 𝑍 ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*),
ℝ*, < )) = ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
))) |
| 53 | 14, 52 | eqtrd 2777 |
. . 3
⊢ (𝜑 → ((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*),
ℝ*, < )) “ 𝑍) = ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
))) |
| 54 | 53 | infeq1d 9517 |
. 2
⊢ (𝜑 → inf(((𝑖 ∈ ℝ ↦ sup(((𝐹 “ (𝑖[,)+∞)) ∩ ℝ*),
ℝ*, < )) “ 𝑍), ℝ*, < ) = inf(ran
(𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
)), ℝ*, < )) |
| 55 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑛 = 𝑘 → (ℤ≥‘𝑛) =
(ℤ≥‘𝑘)) |
| 56 | 55 | reseq2d 5997 |
. . . . . . . 8
⊢ (𝑛 = 𝑘 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑘))) |
| 57 | 56 | rneqd 5949 |
. . . . . . 7
⊢ (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑘))) |
| 58 | 57 | supeq1d 9486 |
. . . . . 6
⊢ (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )
= sup(ran (𝐹 ↾
(ℤ≥‘𝑘)), ℝ*, <
)) |
| 59 | 58 | cbvmptv 5255 |
. . . . 5
⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
= (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾
(ℤ≥‘𝑘)), ℝ*, <
)) |
| 60 | 59 | rneqi 5948 |
. . . 4
⊢ ran
(𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
= ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾
(ℤ≥‘𝑘)), ℝ*, <
)) |
| 61 | 60 | infeq1i 9518 |
. . 3
⊢ inf(ran
(𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
)), ℝ*, < ) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, <
)), ℝ*, < ) |
| 62 | 61 | a1i 11 |
. 2
⊢ (𝜑 → inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
)), ℝ*, < ) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, <
)), ℝ*, < )) |
| 63 | 13, 54, 62 | 3eqtrd 2781 |
1
⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, <
)), ℝ*, < )) |