| Step | Hyp | Ref
| Expression |
| 1 | | limsupvaluz2.m |
. . 3
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 2 | | limsupvaluz2.z |
. . 3
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 3 | | limsupvaluz2.f |
. . . 4
⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
| 4 | 3 | frexr 45396 |
. . 3
⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
| 5 | 1, 2, 4 | limsupvaluz 45723 |
. 2
⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
)), ℝ*, < )) |
| 6 | 3 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐹:𝑍⟶ℝ) |
| 7 | 2 | uzssd3 45437 |
. . . . . . . . . 10
⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍) |
| 8 | 7 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) ⊆ 𝑍) |
| 9 | 6, 8 | feqresmpt 6978 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚))) |
| 10 | 9 | rneqd 5949 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚))) |
| 11 | 10 | supeq1d 9486 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )
= sup(ran (𝑚 ∈
(ℤ≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ*, <
)) |
| 12 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑚𝐹 |
| 13 | | limsupvaluz2.r |
. . . . . . . . . . 11
⊢ (𝜑 → (lim sup‘𝐹) ∈
ℝ) |
| 14 | 13 | renepnfd 11312 |
. . . . . . . . . 10
⊢ (𝜑 → (lim sup‘𝐹) ≠
+∞) |
| 15 | 12, 2, 3, 14 | limsupubuz 45728 |
. . . . . . . . 9
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥) |
| 16 | 15 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥) |
| 17 | | ssralv 4052 |
. . . . . . . . . . 11
⊢
((ℤ≥‘𝑛) ⊆ 𝑍 → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥)) |
| 18 | 7, 17 | syl 17 |
. . . . . . . . . 10
⊢ (𝑛 ∈ 𝑍 → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥)) |
| 19 | 18 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥)) |
| 20 | 19 | reximdv 3170 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥)) |
| 21 | 16, 20 | mpd 15 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥) |
| 22 | | nfv 1914 |
. . . . . . . 8
⊢
Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍) |
| 23 | 2 | eluzelz2 45414 |
. . . . . . . . . 10
⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ) |
| 24 | | uzid 12893 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℤ → 𝑛 ∈
(ℤ≥‘𝑛)) |
| 25 | | ne0i 4341 |
. . . . . . . . . 10
⊢ (𝑛 ∈
(ℤ≥‘𝑛) → (ℤ≥‘𝑛) ≠ ∅) |
| 26 | 23, 24, 25 | 3syl 18 |
. . . . . . . . 9
⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅) |
| 27 | 26 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) ≠ ∅) |
| 28 | 6 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝐹:𝑍⟶ℝ) |
| 29 | 8 | sselda 3983 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍) |
| 30 | 28, 29 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑚) ∈ ℝ) |
| 31 | 22, 27, 30 | supxrre3rnmpt 45440 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ*, < ) ∈
ℝ ↔ ∃𝑥
∈ ℝ ∀𝑚
∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥)) |
| 32 | 21, 31 | mpbird 257 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ*, < ) ∈
ℝ) |
| 33 | 11, 32 | eqeltrd 2841 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )
∈ ℝ) |
| 34 | 33 | fmpttd 7135 |
. . . 4
⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
)):𝑍⟶ℝ) |
| 35 | 34 | frnd 6744 |
. . 3
⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
⊆ ℝ) |
| 36 | | nfv 1914 |
. . . 4
⊢
Ⅎ𝑛𝜑 |
| 37 | | eqid 2737 |
. . . 4
⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
= (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾
(ℤ≥‘𝑛)), ℝ*, <
)) |
| 38 | 1, 2 | uzn0d 45436 |
. . . 4
⊢ (𝜑 → 𝑍 ≠ ∅) |
| 39 | 36, 33, 37, 38 | rnmptn0 6264 |
. . 3
⊢ (𝜑 → ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
≠ ∅) |
| 40 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑗𝐹 |
| 41 | 40, 1, 2, 4 | limsupre3uz 45751 |
. . . . . . . 8
⊢ (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔
(∃𝑥 ∈ ℝ
∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)(𝐹‘𝑗) ≤ 𝑥))) |
| 42 | 13, 41 | mpbid 232 |
. . . . . . 7
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)(𝐹‘𝑗) ≤ 𝑥)) |
| 43 | 42 | simpld 494 |
. . . . . 6
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗)) |
| 44 | | simp-4r 784 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ) |
| 45 | 44 | rexrd 11311 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ*) |
| 46 | 4 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → 𝐹:𝑍⟶ℝ*) |
| 47 | 2 | uztrn2 12897 |
. . . . . . . . . . . . 13
⊢ ((𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → 𝑗 ∈ 𝑍) |
| 48 | 47 | 3adant1 1131 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → 𝑗 ∈ 𝑍) |
| 49 | 46, 48 | ffvelcdmd 7105 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) ∈
ℝ*) |
| 50 | 49 | ad5ant134 1369 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ∈
ℝ*) |
| 51 | | rnresss 6035 |
. . . . . . . . . . . . . 14
⊢ ran
(𝐹 ↾
(ℤ≥‘𝑖)) ⊆ ran 𝐹 |
| 52 | 3 | frnd 6744 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
| 53 | 52 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran 𝐹 ⊆ ℝ) |
| 54 | 51, 53 | sstrid 3995 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆
ℝ) |
| 55 | 54 | ssrexr 45443 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆
ℝ*) |
| 56 | 55 | supxrcld 45112 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )
∈ ℝ*) |
| 57 | 56 | ad5ant13 757 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )
∈ ℝ*) |
| 58 | | simpr 484 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ (𝐹‘𝑗)) |
| 59 | 55 | 3adant3 1133 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆
ℝ*) |
| 60 | | fvres 6925 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈
(ℤ≥‘𝑖) → ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗) = (𝐹‘𝑗)) |
| 61 | 60 | eqcomd 2743 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈
(ℤ≥‘𝑖) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗)) |
| 62 | 61 | 3ad2ant3 1136 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗)) |
| 63 | 3 | ffnd 6737 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐹 Fn 𝑍) |
| 64 | 2 | uzssd3 45437 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ 𝑍 → (ℤ≥‘𝑖) ⊆ 𝑍) |
| 65 | | fnssres 6691 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹 Fn 𝑍 ∧ (ℤ≥‘𝑖) ⊆ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑖)) Fn
(ℤ≥‘𝑖)) |
| 66 | 63, 64, 65 | syl2an 596 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑖)) Fn
(ℤ≥‘𝑖)) |
| 67 | | fnfvelrn 7100 |
. . . . . . . . . . . . . 14
⊢ (((𝐹 ↾
(ℤ≥‘𝑖)) Fn (ℤ≥‘𝑖) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ≥‘𝑖))) |
| 68 | 66, 67 | stoic3 1776 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ≥‘𝑖))) |
| 69 | 62, 68 | eqeltrd 2841 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) ∈ ran (𝐹 ↾ (ℤ≥‘𝑖))) |
| 70 | | eqid 2737 |
. . . . . . . . . . . 12
⊢ sup(ran
(𝐹 ↾
(ℤ≥‘𝑖)), ℝ*, < ) = sup(ran
(𝐹 ↾
(ℤ≥‘𝑖)), ℝ*, <
) |
| 71 | 59, 69, 70 | supxrubd 45118 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
)) |
| 72 | 71 | ad5ant134 1369 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
)) |
| 73 | 45, 50, 57, 58, 72 | xrletrd 13204 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
)) |
| 74 | 73 | rexlimdva2 3157 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) → (∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
))) |
| 75 | 74 | ralimdva 3167 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
))) |
| 76 | 75 | reximdva 3168 |
. . . . . 6
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
))) |
| 77 | 43, 76 | mpd 15 |
. . . . 5
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
)) |
| 78 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑖 → (ℤ≥‘𝑛) =
(ℤ≥‘𝑖)) |
| 79 | 78 | reseq2d 5997 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑖 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑖))) |
| 80 | 79 | rneqd 5949 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑖))) |
| 81 | 80 | supeq1d 9486 |
. . . . . . . . 9
⊢ (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )
= sup(ran (𝐹 ↾
(ℤ≥‘𝑖)), ℝ*, <
)) |
| 82 | | eqcom 2744 |
. . . . . . . . 9
⊢ (𝑛 = 𝑖 ↔ 𝑖 = 𝑛) |
| 83 | | eqcom 2744 |
. . . . . . . . 9
⊢ (sup(ran
(𝐹 ↾
(ℤ≥‘𝑛)), ℝ*, < ) = sup(ran
(𝐹 ↾
(ℤ≥‘𝑖)), ℝ*, < ) ↔
sup(ran (𝐹 ↾
(ℤ≥‘𝑖)), ℝ*, < ) = sup(ran
(𝐹 ↾
(ℤ≥‘𝑛)), ℝ*, <
)) |
| 84 | 81, 82, 83 | 3imtr3i 291 |
. . . . . . . 8
⊢ (𝑖 = 𝑛 → sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )
= sup(ran (𝐹 ↾
(ℤ≥‘𝑛)), ℝ*, <
)) |
| 85 | 84 | breq2d 5155 |
. . . . . . 7
⊢ (𝑖 = 𝑛 → (𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )
↔ 𝑥 ≤ sup(ran
(𝐹 ↾
(ℤ≥‘𝑛)), ℝ*, <
))) |
| 86 | 85 | cbvralvw 3237 |
. . . . . 6
⊢
(∀𝑖 ∈
𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )
↔ ∀𝑛 ∈
𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
)) |
| 87 | 86 | rexbii 3094 |
. . . . 5
⊢
(∃𝑥 ∈
ℝ ∀𝑖 ∈
𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )
↔ ∃𝑥 ∈
ℝ ∀𝑛 ∈
𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
)) |
| 88 | 77, 87 | sylib 218 |
. . . 4
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
)) |
| 89 | 36, 33 | rnmptbd2 45256 |
. . . 4
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑛 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )
↔ ∃𝑥 ∈
ℝ ∀𝑦 ∈
ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾
(ℤ≥‘𝑛)), ℝ*, < ))𝑥 ≤ 𝑦)) |
| 90 | 88, 89 | mpbid 232 |
. . 3
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
))𝑥 ≤ 𝑦) |
| 91 | | infxrre 13378 |
. . 3
⊢ ((ran
(𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
⊆ ℝ ∧ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
≠ ∅ ∧ ∃𝑥
∈ ℝ ∀𝑦
∈ ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾
(ℤ≥‘𝑛)), ℝ*, < ))𝑥 ≤ 𝑦) → inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
)), ℝ*, < ) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
)), ℝ, < )) |
| 92 | 35, 39, 90, 91 | syl3anc 1373 |
. 2
⊢ (𝜑 → inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
)), ℝ*, < ) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
)), ℝ, < )) |
| 93 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑛 = 𝑘 → (ℤ≥‘𝑛) =
(ℤ≥‘𝑘)) |
| 94 | 93 | reseq2d 5997 |
. . . . . . . 8
⊢ (𝑛 = 𝑘 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑘))) |
| 95 | 94 | rneqd 5949 |
. . . . . . 7
⊢ (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑘))) |
| 96 | 95 | supeq1d 9486 |
. . . . . 6
⊢ (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )
= sup(ran (𝐹 ↾
(ℤ≥‘𝑘)), ℝ*, <
)) |
| 97 | 96 | cbvmptv 5255 |
. . . . 5
⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
= (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾
(ℤ≥‘𝑘)), ℝ*, <
)) |
| 98 | 97 | rneqi 5948 |
. . . 4
⊢ ran
(𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
= ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾
(ℤ≥‘𝑘)), ℝ*, <
)) |
| 99 | 98 | 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, < ) |
| 100 | 99 | 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, < )) |
| 101 | 5, 92, 100 | 3eqtrd 2781 |
1
⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, <
)), ℝ, < )) |