| Step | Hyp | Ref
| Expression |
| 1 | | supcnvlimsup.z |
. . 3
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 2 | | supcnvlimsup.m |
. . 3
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 3 | | supcnvlimsup.f |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
| 4 | 3 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐹:𝑍⟶ℝ) |
| 5 | | id 22 |
. . . . . . . . . 10
⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ 𝑍) |
| 6 | 1, 5 | uzssd2 45428 |
. . . . . . . . 9
⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍) |
| 7 | 6 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) ⊆ 𝑍) |
| 8 | 4, 7 | feqresmpt 6978 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚))) |
| 9 | 8 | rneqd 5949 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚))) |
| 10 | 9 | supeq1d 9486 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )
= sup(ran (𝑚 ∈
(ℤ≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ*, <
)) |
| 11 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑚𝐹 |
| 12 | | supcnvlimsup.r |
. . . . . . . . . 10
⊢ (𝜑 → (lim sup‘𝐹) ∈
ℝ) |
| 13 | 12 | renepnfd 11312 |
. . . . . . . . 9
⊢ (𝜑 → (lim sup‘𝐹) ≠
+∞) |
| 14 | 11, 1, 3, 13 | limsupubuz 45728 |
. . . . . . . 8
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥) |
| 15 | 14 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥) |
| 16 | | ssralv 4052 |
. . . . . . . . . 10
⊢
((ℤ≥‘𝑛) ⊆ 𝑍 → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥)) |
| 17 | 6, 16 | syl 17 |
. . . . . . . . 9
⊢ (𝑛 ∈ 𝑍 → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥)) |
| 18 | 17 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥)) |
| 19 | 18 | reximdv 3170 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (∃𝑥 ∈ ℝ ∀𝑚 ∈ 𝑍 (𝐹‘𝑚) ≤ 𝑥 → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥)) |
| 20 | 15, 19 | mpd 15 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑥 ∈ ℝ ∀𝑚 ∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥) |
| 21 | | nfv 1914 |
. . . . . . 7
⊢
Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍) |
| 22 | 1 | eluzelz2 45414 |
. . . . . . . . 9
⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ) |
| 23 | | uzid 12893 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℤ → 𝑛 ∈
(ℤ≥‘𝑛)) |
| 24 | | ne0i 4341 |
. . . . . . . . 9
⊢ (𝑛 ∈
(ℤ≥‘𝑛) → (ℤ≥‘𝑛) ≠ ∅) |
| 25 | 22, 23, 24 | 3syl 18 |
. . . . . . . 8
⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅) |
| 26 | 25 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) ≠ ∅) |
| 27 | 4 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝐹:𝑍⟶ℝ) |
| 28 | 7 | sselda 3983 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍) |
| 29 | 27, 28 | ffvelcdmd 7105 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑚) ∈ ℝ) |
| 30 | 21, 26, 29 | supxrre3rnmpt 45440 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ*, < ) ∈
ℝ ↔ ∃𝑥
∈ ℝ ∀𝑚
∈ (ℤ≥‘𝑛)(𝐹‘𝑚) ≤ 𝑥)) |
| 31 | 20, 30 | mpbird 257 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (𝐹‘𝑚)), ℝ*, < ) ∈
ℝ) |
| 32 | 10, 31 | eqeltrd 2841 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )
∈ ℝ) |
| 33 | 32 | fmpttd 7135 |
. . 3
⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
)):𝑍⟶ℝ) |
| 34 | | eqid 2737 |
. . . . . . . 8
⊢
(ℤ≥‘𝑖) = (ℤ≥‘𝑖) |
| 35 | 1 | eluzelz2 45414 |
. . . . . . . 8
⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ ℤ) |
| 36 | 35 | peano2zd 12725 |
. . . . . . . 8
⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈ ℤ) |
| 37 | 35 | zred 12722 |
. . . . . . . . 9
⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ ℝ) |
| 38 | | lep1 12108 |
. . . . . . . . 9
⊢ (𝑖 ∈ ℝ → 𝑖 ≤ (𝑖 + 1)) |
| 39 | 37, 38 | syl 17 |
. . . . . . . 8
⊢ (𝑖 ∈ 𝑍 → 𝑖 ≤ (𝑖 + 1)) |
| 40 | 34, 35, 36, 39 | eluzd 45420 |
. . . . . . 7
⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈
(ℤ≥‘𝑖)) |
| 41 | | uzss 12901 |
. . . . . . 7
⊢ ((𝑖 + 1) ∈
(ℤ≥‘𝑖) → (ℤ≥‘(𝑖 + 1)) ⊆
(ℤ≥‘𝑖)) |
| 42 | | ssres2 6022 |
. . . . . . 7
⊢
((ℤ≥‘(𝑖 + 1)) ⊆
(ℤ≥‘𝑖) → (𝐹 ↾
(ℤ≥‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ≥‘𝑖))) |
| 43 | | rnss 5950 |
. . . . . . 7
⊢ ((𝐹 ↾
(ℤ≥‘(𝑖 + 1))) ⊆ (𝐹 ↾ (ℤ≥‘𝑖)) → ran (𝐹 ↾
(ℤ≥‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ≥‘𝑖))) |
| 44 | 40, 41, 42, 43 | 4syl 19 |
. . . . . 6
⊢ (𝑖 ∈ 𝑍 → ran (𝐹 ↾
(ℤ≥‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ≥‘𝑖))) |
| 45 | 44 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾
(ℤ≥‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ≥‘𝑖))) |
| 46 | | rnresss 6035 |
. . . . . . . 8
⊢ ran
(𝐹 ↾
(ℤ≥‘𝑖)) ⊆ ran 𝐹 |
| 47 | 46 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ran 𝐹) |
| 48 | 3 | frnd 6744 |
. . . . . . . 8
⊢ (𝜑 → ran 𝐹 ⊆ ℝ) |
| 49 | 48 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran 𝐹 ⊆ ℝ) |
| 50 | 47, 49 | sstrd 3994 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆
ℝ) |
| 51 | | ressxr 11305 |
. . . . . . 7
⊢ ℝ
⊆ ℝ* |
| 52 | 51 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ℝ ⊆
ℝ*) |
| 53 | 50, 52 | sstrd 3994 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆
ℝ*) |
| 54 | | supxrss 13374 |
. . . . 5
⊢ ((ran
(𝐹 ↾
(ℤ≥‘(𝑖 + 1))) ⊆ ran (𝐹 ↾ (ℤ≥‘𝑖)) ∧ ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆ ℝ*)
→ sup(ran (𝐹 ↾
(ℤ≥‘(𝑖 + 1))), ℝ*, < ) ≤
sup(ran (𝐹 ↾
(ℤ≥‘𝑖)), ℝ*, <
)) |
| 55 | 45, 53, 54 | syl2anc 584 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → sup(ran (𝐹 ↾
(ℤ≥‘(𝑖 + 1))), ℝ*, < ) ≤
sup(ran (𝐹 ↾
(ℤ≥‘𝑖)), ℝ*, <
)) |
| 56 | | eqidd 2738 |
. . . . . . 7
⊢ (𝑖 ∈ 𝑍 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
= (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾
(ℤ≥‘𝑛)), ℝ*, <
))) |
| 57 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑛 = (𝑖 + 1) →
(ℤ≥‘𝑛) = (ℤ≥‘(𝑖 + 1))) |
| 58 | 57 | reseq2d 5997 |
. . . . . . . . . 10
⊢ (𝑛 = (𝑖 + 1) → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾
(ℤ≥‘(𝑖 + 1)))) |
| 59 | 58 | rneqd 5949 |
. . . . . . . . 9
⊢ (𝑛 = (𝑖 + 1) → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾
(ℤ≥‘(𝑖 + 1)))) |
| 60 | 59 | supeq1d 9486 |
. . . . . . . 8
⊢ (𝑛 = (𝑖 + 1) → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )
= sup(ran (𝐹 ↾
(ℤ≥‘(𝑖 + 1))), ℝ*, <
)) |
| 61 | 60 | adantl 481 |
. . . . . . 7
⊢ ((𝑖 ∈ 𝑍 ∧ 𝑛 = (𝑖 + 1)) → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )
= sup(ran (𝐹 ↾
(ℤ≥‘(𝑖 + 1))), ℝ*, <
)) |
| 62 | 1 | peano2uzs 12944 |
. . . . . . 7
⊢ (𝑖 ∈ 𝑍 → (𝑖 + 1) ∈ 𝑍) |
| 63 | | xrltso 13183 |
. . . . . . . . 9
⊢ < Or
ℝ* |
| 64 | 63 | supex 9503 |
. . . . . . . 8
⊢ sup(ran
(𝐹 ↾
(ℤ≥‘(𝑖 + 1))), ℝ*, < ) ∈
V |
| 65 | 64 | a1i 11 |
. . . . . . 7
⊢ (𝑖 ∈ 𝑍 → sup(ran (𝐹 ↾
(ℤ≥‘(𝑖 + 1))), ℝ*, < ) ∈
V) |
| 66 | 56, 61, 62, 65 | fvmptd 7023 |
. . . . . 6
⊢ (𝑖 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
))‘(𝑖 + 1)) = sup(ran
(𝐹 ↾
(ℤ≥‘(𝑖 + 1))), ℝ*, <
)) |
| 67 | 66 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
))‘(𝑖 + 1)) = sup(ran
(𝐹 ↾
(ℤ≥‘(𝑖 + 1))), ℝ*, <
)) |
| 68 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑖 → (ℤ≥‘𝑛) =
(ℤ≥‘𝑖)) |
| 69 | 68 | reseq2d 5997 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑖 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑖))) |
| 70 | 69 | rneqd 5949 |
. . . . . . . . 9
⊢ (𝑛 = 𝑖 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑖))) |
| 71 | 70 | supeq1d 9486 |
. . . . . . . 8
⊢ (𝑛 = 𝑖 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )
= sup(ran (𝐹 ↾
(ℤ≥‘𝑖)), ℝ*, <
)) |
| 72 | 71 | adantl 481 |
. . . . . . 7
⊢ ((𝑖 ∈ 𝑍 ∧ 𝑛 = 𝑖) → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )
= sup(ran (𝐹 ↾
(ℤ≥‘𝑖)), ℝ*, <
)) |
| 73 | | id 22 |
. . . . . . 7
⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ 𝑍) |
| 74 | 63 | supex 9503 |
. . . . . . . 8
⊢ sup(ran
(𝐹 ↾
(ℤ≥‘𝑖)), ℝ*, < ) ∈
V |
| 75 | 74 | a1i 11 |
. . . . . . 7
⊢ (𝑖 ∈ 𝑍 → sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )
∈ V) |
| 76 | 56, 72, 73, 75 | fvmptd 7023 |
. . . . . 6
⊢ (𝑖 ∈ 𝑍 → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
))‘𝑖) = sup(ran
(𝐹 ↾
(ℤ≥‘𝑖)), ℝ*, <
)) |
| 77 | 76 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
))‘𝑖) = sup(ran
(𝐹 ↾
(ℤ≥‘𝑖)), ℝ*, <
)) |
| 78 | 67, 77 | breq12d 5156 |
. . . 4
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
))‘(𝑖 + 1)) ≤
((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾
(ℤ≥‘𝑛)), ℝ*, < ))‘𝑖) ↔ sup(ran (𝐹 ↾
(ℤ≥‘(𝑖 + 1))), ℝ*, < ) ≤
sup(ran (𝐹 ↾
(ℤ≥‘𝑖)), ℝ*, <
))) |
| 79 | 55, 78 | mpbird 257 |
. . 3
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
))‘(𝑖 + 1)) ≤
((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾
(ℤ≥‘𝑛)), ℝ*, < ))‘𝑖)) |
| 80 | | nfcv 2905 |
. . . . . . . 8
⊢
Ⅎ𝑗𝐹 |
| 81 | 3 | frexr 45396 |
. . . . . . . 8
⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
| 82 | 80, 2, 1, 81 | limsupre3uz 45751 |
. . . . . . 7
⊢ (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))) |
| 83 | 12, 82 | mpbid 232 |
. . . . . 6
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑖)(𝐹‘𝑗) ≤ 𝑥)) |
| 84 | 83 | simpld 494 |
. . . . 5
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗)) |
| 85 | | simp-4r 784 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ) |
| 86 | 85 | rexrd 11311 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ∈ ℝ*) |
| 87 | 81 | 3ad2ant1 1134 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → 𝐹:𝑍⟶ℝ*) |
| 88 | 1 | uztrn2 12897 |
. . . . . . . . . . . 12
⊢ ((𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → 𝑗 ∈ 𝑍) |
| 89 | 88 | 3adant1 1131 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → 𝑗 ∈ 𝑍) |
| 90 | 87, 89 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) ∈
ℝ*) |
| 91 | 90 | ad5ant134 1369 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ∈
ℝ*) |
| 92 | 53 | supxrcld 45112 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )
∈ ℝ*) |
| 93 | 92 | ad5ant13 757 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, < )
∈ ℝ*) |
| 94 | | simpr 484 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ (𝐹‘𝑗)) |
| 95 | 53 | 3adant3 1133 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → ran (𝐹 ↾ (ℤ≥‘𝑖)) ⊆
ℝ*) |
| 96 | | fvres 6925 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈
(ℤ≥‘𝑖) → ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗) = (𝐹‘𝑗)) |
| 97 | 96 | eqcomd 2743 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈
(ℤ≥‘𝑖) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗)) |
| 98 | 97 | 3ad2ant3 1136 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) = ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗)) |
| 99 | 3 | ffnd 6737 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐹 Fn 𝑍) |
| 100 | 99 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐹 Fn 𝑍) |
| 101 | 1, 73 | uzssd2 45428 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 ∈ 𝑍 → (ℤ≥‘𝑖) ⊆ 𝑍) |
| 102 | 101 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (ℤ≥‘𝑖) ⊆ 𝑍) |
| 103 | | fnssres 6691 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹 Fn 𝑍 ∧ (ℤ≥‘𝑖) ⊆ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑖)) Fn
(ℤ≥‘𝑖)) |
| 104 | 100, 102,
103 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑖)) Fn
(ℤ≥‘𝑖)) |
| 105 | 104 | 3adant3 1133 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹 ↾ (ℤ≥‘𝑖)) Fn
(ℤ≥‘𝑖)) |
| 106 | | simp3 1139 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → 𝑗 ∈ (ℤ≥‘𝑖)) |
| 107 | | fnfvelrn 7100 |
. . . . . . . . . . . . 13
⊢ (((𝐹 ↾
(ℤ≥‘𝑖)) Fn (ℤ≥‘𝑖) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ≥‘𝑖))) |
| 108 | 105, 106,
107 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → ((𝐹 ↾ (ℤ≥‘𝑖))‘𝑗) ∈ ran (𝐹 ↾ (ℤ≥‘𝑖))) |
| 109 | 98, 108 | eqeltrd 2841 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) ∈ ran (𝐹 ↾ (ℤ≥‘𝑖))) |
| 110 | | eqid 2737 |
. . . . . . . . . . 11
⊢ sup(ran
(𝐹 ↾
(ℤ≥‘𝑖)), ℝ*, < ) = sup(ran
(𝐹 ↾
(ℤ≥‘𝑖)), ℝ*, <
) |
| 111 | 95, 109, 110 | supxrubd 45118 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑗) ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
)) |
| 112 | 111 | ad5ant134 1369 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → (𝐹‘𝑗) ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
)) |
| 113 | 86, 91, 93, 94, 112 | xrletrd 13204 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ≥‘𝑖)) ∧ 𝑥 ≤ (𝐹‘𝑗)) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
)) |
| 114 | 113 | rexlimdva2 3157 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ) ∧ 𝑖 ∈ 𝑍) → (∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
))) |
| 115 | 114 | ralimdva 3167 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
))) |
| 116 | 115 | reximdva 3168 |
. . . . 5
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 ∃𝑗 ∈ (ℤ≥‘𝑖)𝑥 ≤ (𝐹‘𝑗) → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
))) |
| 117 | 84, 116 | mpd 15 |
. . . 4
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
)) |
| 118 | | simpl 482 |
. . . . . . 7
⊢ ((𝑦 = 𝑥 ∧ 𝑖 ∈ 𝑍) → 𝑦 = 𝑥) |
| 119 | 76 | adantl 481 |
. . . . . . 7
⊢ ((𝑦 = 𝑥 ∧ 𝑖 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
))‘𝑖) = sup(ran
(𝐹 ↾
(ℤ≥‘𝑖)), ℝ*, <
)) |
| 120 | 118, 119 | breq12d 5156 |
. . . . . 6
⊢ ((𝑦 = 𝑥 ∧ 𝑖 ∈ 𝑍) → (𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
))‘𝑖) ↔ 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
))) |
| 121 | 120 | ralbidva 3176 |
. . . . 5
⊢ (𝑦 = 𝑥 → (∀𝑖 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
))‘𝑖) ↔
∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
))) |
| 122 | 121 | cbvrexvw 3238 |
. . . 4
⊢
(∃𝑦 ∈
ℝ ∀𝑖 ∈
𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
))‘𝑖) ↔
∃𝑥 ∈ ℝ
∀𝑖 ∈ 𝑍 𝑥 ≤ sup(ran (𝐹 ↾ (ℤ≥‘𝑖)), ℝ*, <
)) |
| 123 | 117, 122 | sylibr 234 |
. . 3
⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑖 ∈ 𝑍 𝑦 ≤ ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
))‘𝑖)) |
| 124 | 1, 2, 33, 79, 123 | climinf 45621 |
. 2
⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
⇝ inf(ran (𝑛 ∈
𝑍 ↦ sup(ran (𝐹 ↾
(ℤ≥‘𝑛)), ℝ*, < )), ℝ,
< )) |
| 125 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑛 = 𝑘 → (ℤ≥‘𝑛) =
(ℤ≥‘𝑘)) |
| 126 | 125 | reseq2d 5997 |
. . . . . . 7
⊢ (𝑛 = 𝑘 → (𝐹 ↾ (ℤ≥‘𝑛)) = (𝐹 ↾ (ℤ≥‘𝑘))) |
| 127 | 126 | rneqd 5949 |
. . . . . 6
⊢ (𝑛 = 𝑘 → ran (𝐹 ↾ (ℤ≥‘𝑛)) = ran (𝐹 ↾ (ℤ≥‘𝑘))) |
| 128 | 127 | supeq1d 9486 |
. . . . 5
⊢ (𝑛 = 𝑘 → sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < )
= sup(ran (𝐹 ↾
(ℤ≥‘𝑘)), ℝ*, <
)) |
| 129 | 128 | cbvmptv 5255 |
. . . 4
⊢ (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
= (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾
(ℤ≥‘𝑘)), ℝ*, <
)) |
| 130 | 129 | a1i 11 |
. . 3
⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
= (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾
(ℤ≥‘𝑘)), ℝ*, <
))) |
| 131 | 2, 1, 3, 12 | limsupvaluz2 45753 |
. . . 4
⊢ (𝜑 → (lim sup‘𝐹) = inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
)), ℝ, < )) |
| 132 | 131 | eqcomd 2743 |
. . 3
⊢ (𝜑 → inf(ran (𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, <
)), ℝ, < ) = (lim sup‘𝐹)) |
| 133 | 130, 132 | breq12d 5156 |
. 2
⊢ (𝜑 → ((𝑛 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑛)), ℝ*, < ))
⇝ inf(ran (𝑛 ∈
𝑍 ↦ sup(ran (𝐹 ↾
(ℤ≥‘𝑛)), ℝ*, < )), ℝ,
< ) ↔ (𝑘 ∈
𝑍 ↦ sup(ran (𝐹 ↾
(ℤ≥‘𝑘)), ℝ*, < )) ⇝ (lim
sup‘𝐹))) |
| 134 | 124, 133 | mpbid 232 |
1
⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ sup(ran (𝐹 ↾ (ℤ≥‘𝑘)), ℝ*, < ))
⇝ (lim sup‘𝐹)) |