| Step | Hyp | Ref
| Expression |
| 1 | | resimass 45246 |
. . . . . . . . 9
⊢ ((𝐹 ↾ 𝐴) “ (𝑘[,)+∞)) ⊆ (𝐹 “ (𝑘[,)+∞)) |
| 2 | 1 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹 ↾ 𝐴) “ (𝑘[,)+∞)) ⊆ (𝐹 “ (𝑘[,)+∞))) |
| 3 | 2 | ssrind 4244 |
. . . . . . 7
⊢ (𝜑 → (((𝐹 ↾ 𝐴) “ (𝑘[,)+∞)) ∩ ℝ*)
⊆ ((𝐹 “ (𝑘[,)+∞)) ∩
ℝ*)) |
| 4 | | liminfresxr.2 |
. . . . . . . . . . . . 13
⊢ (𝜑 → Fun 𝐹) |
| 5 | 4 | funfnd 6597 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹 Fn dom 𝐹) |
| 6 | | elinel1 4201 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)
→ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) |
| 7 | | fvelima2 6961 |
. . . . . . . . . . . 12
⊢ ((𝐹 Fn dom 𝐹 ∧ 𝑦 ∈ (𝐹 “ (𝑘[,)+∞))) → ∃𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞))(𝐹‘𝑥) = 𝑦) |
| 8 | 5, 6, 7 | syl2an 596 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
→ ∃𝑥 ∈ (dom
𝐹 ∩ (𝑘[,)+∞))(𝐹‘𝑥) = 𝑦) |
| 9 | | elinel1 4201 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞)) → 𝑥 ∈ dom 𝐹) |
| 10 | 9 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞)) ∧ (𝐹‘𝑥) = 𝑦) → 𝑥 ∈ dom 𝐹) |
| 11 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)
∧ (𝐹‘𝑥) = 𝑦) → (𝐹‘𝑥) = 𝑦) |
| 12 | | elinel2 4202 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)
→ 𝑦 ∈
ℝ*) |
| 13 | 12 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)
∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ ℝ*) |
| 14 | 11, 13 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)
∧ (𝐹‘𝑥) = 𝑦) → (𝐹‘𝑥) ∈
ℝ*) |
| 15 | 14 | 3adant2 1132 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞)) ∧ (𝐹‘𝑥) = 𝑦) → (𝐹‘𝑥) ∈
ℝ*) |
| 16 | 10, 15 | jca 511 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞)) ∧ (𝐹‘𝑥) = 𝑦) → (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈
ℝ*)) |
| 17 | 16 | 3adant1l 1177 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞)) ∧ (𝐹‘𝑥) = 𝑦) → (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈
ℝ*)) |
| 18 | | simp1l 1198 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞)) ∧ (𝐹‘𝑥) = 𝑦) → 𝜑) |
| 19 | | elpreima 7078 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ ℝ*) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈
ℝ*))) |
| 20 | 5, 19 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑥 ∈ (◡𝐹 “ ℝ*) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈
ℝ*))) |
| 21 | 18, 20 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞)) ∧ (𝐹‘𝑥) = 𝑦) → (𝑥 ∈ (◡𝐹 “ ℝ*) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈
ℝ*))) |
| 22 | 17, 21 | mpbird 257 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞)) ∧ (𝐹‘𝑥) = 𝑦) → 𝑥 ∈ (◡𝐹 “
ℝ*)) |
| 23 | | liminfresxr.3 |
. . . . . . . . . . . . . . . . 17
⊢ 𝐴 = (◡𝐹 “
ℝ*) |
| 24 | 22, 23 | eleqtrrdi 2852 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞)) ∧ (𝐹‘𝑥) = 𝑦) → 𝑥 ∈ 𝐴) |
| 25 | 24 | 3expa 1119 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞))) ∧ (𝐹‘𝑥) = 𝑦) → 𝑥 ∈ 𝐴) |
| 26 | 25 | fvresd 6926 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞))) ∧ (𝐹‘𝑥) = 𝑦) → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥)) |
| 27 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞))) ∧ (𝐹‘𝑥) = 𝑦) → (𝐹‘𝑥) = 𝑦) |
| 28 | 26, 27 | eqtr2d 2778 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞))) ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 = ((𝐹 ↾ 𝐴)‘𝑥)) |
| 29 | | simplll 775 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞))) ∧ (𝐹‘𝑥) = 𝑦) → 𝜑) |
| 30 | 4 | funresd 6609 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → Fun (𝐹 ↾ 𝐴)) |
| 31 | 29, 30 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞))) ∧ (𝐹‘𝑥) = 𝑦) → Fun (𝐹 ↾ 𝐴)) |
| 32 | 9 | ad2antlr 727 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞))) ∧ (𝐹‘𝑥) = 𝑦) → 𝑥 ∈ dom 𝐹) |
| 33 | 25, 32 | elind 4200 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞))) ∧ (𝐹‘𝑥) = 𝑦) → 𝑥 ∈ (𝐴 ∩ dom 𝐹)) |
| 34 | | dmres 6030 |
. . . . . . . . . . . . . . . 16
⊢ dom
(𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹) |
| 35 | 33, 34 | eleqtrrdi 2852 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞))) ∧ (𝐹‘𝑥) = 𝑦) → 𝑥 ∈ dom (𝐹 ↾ 𝐴)) |
| 36 | 31, 35 | jca 511 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞))) ∧ (𝐹‘𝑥) = 𝑦) → (Fun (𝐹 ↾ 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴))) |
| 37 | | elinel2 4202 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞)) → 𝑥 ∈ (𝑘[,)+∞)) |
| 38 | 37 | ad2antlr 727 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞))) ∧ (𝐹‘𝑥) = 𝑦) → 𝑥 ∈ (𝑘[,)+∞)) |
| 39 | | funfvima 7250 |
. . . . . . . . . . . . . 14
⊢ ((Fun
(𝐹 ↾ 𝐴) ∧ 𝑥 ∈ dom (𝐹 ↾ 𝐴)) → (𝑥 ∈ (𝑘[,)+∞) → ((𝐹 ↾ 𝐴)‘𝑥) ∈ ((𝐹 ↾ 𝐴) “ (𝑘[,)+∞)))) |
| 40 | 36, 38, 39 | sylc 65 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞))) ∧ (𝐹‘𝑥) = 𝑦) → ((𝐹 ↾ 𝐴)‘𝑥) ∈ ((𝐹 ↾ 𝐴) “ (𝑘[,)+∞))) |
| 41 | 28, 40 | eqeltrd 2841 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
∧ 𝑥 ∈ (dom 𝐹 ∩ (𝑘[,)+∞))) ∧ (𝐹‘𝑥) = 𝑦) → 𝑦 ∈ ((𝐹 ↾ 𝐴) “ (𝑘[,)+∞))) |
| 42 | 41 | rexlimdva2 3157 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
→ (∃𝑥 ∈
(dom 𝐹 ∩ (𝑘[,)+∞))(𝐹‘𝑥) = 𝑦 → 𝑦 ∈ ((𝐹 ↾ 𝐴) “ (𝑘[,)+∞)))) |
| 43 | 8, 42 | mpd 15 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*))
→ 𝑦 ∈ ((𝐹 ↾ 𝐴) “ (𝑘[,)+∞))) |
| 44 | 43 | ralrimiva 3146 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)𝑦 ∈ ((𝐹 ↾ 𝐴) “ (𝑘[,)+∞))) |
| 45 | | dfss3 3972 |
. . . . . . . . 9
⊢ (((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)
⊆ ((𝐹 ↾ 𝐴) “ (𝑘[,)+∞)) ↔ ∀𝑦 ∈ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)𝑦 ∈ ((𝐹 ↾ 𝐴) “ (𝑘[,)+∞))) |
| 46 | 44, 45 | sylibr 234 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)
⊆ ((𝐹 ↾ 𝐴) “ (𝑘[,)+∞))) |
| 47 | | inss2 4238 |
. . . . . . . . 9
⊢ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)
⊆ ℝ* |
| 48 | 47 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)
⊆ ℝ*) |
| 49 | 46, 48 | ssind 4241 |
. . . . . . 7
⊢ (𝜑 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)
⊆ (((𝐹 ↾ 𝐴) “ (𝑘[,)+∞)) ∩
ℝ*)) |
| 50 | 3, 49 | eqssd 4001 |
. . . . . 6
⊢ (𝜑 → (((𝐹 ↾ 𝐴) “ (𝑘[,)+∞)) ∩ ℝ*) =
((𝐹 “ (𝑘[,)+∞)) ∩
ℝ*)) |
| 51 | 50 | infeq1d 9517 |
. . . . 5
⊢ (𝜑 → inf((((𝐹 ↾ 𝐴) “ (𝑘[,)+∞)) ∩ ℝ*),
ℝ*, < ) = inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*),
ℝ*, < )) |
| 52 | 51 | mpteq2dv 5244 |
. . . 4
⊢ (𝜑 → (𝑘 ∈ ℝ ↦ inf((((𝐹 ↾ 𝐴) “ (𝑘[,)+∞)) ∩ ℝ*),
ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*),
ℝ*, < ))) |
| 53 | 52 | rneqd 5949 |
. . 3
⊢ (𝜑 → ran (𝑘 ∈ ℝ ↦ inf((((𝐹 ↾ 𝐴) “ (𝑘[,)+∞)) ∩ ℝ*),
ℝ*, < )) = ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*),
ℝ*, < ))) |
| 54 | 53 | supeq1d 9486 |
. 2
⊢ (𝜑 → sup(ran (𝑘 ∈ ℝ ↦
inf((((𝐹 ↾ 𝐴) “ (𝑘[,)+∞)) ∩ ℝ*),
ℝ*, < )), ℝ*, < ) = sup(ran (𝑘 ∈ ℝ ↦
inf(((𝐹 “ (𝑘[,)+∞)) ∩
ℝ*), ℝ*, < )), ℝ*, <
)) |
| 55 | | liminfresxr.1 |
. . . 4
⊢ (𝜑 → 𝐹 ∈ 𝑉) |
| 56 | 55 | resexd 6046 |
. . 3
⊢ (𝜑 → (𝐹 ↾ 𝐴) ∈ V) |
| 57 | | eqid 2737 |
. . . 4
⊢ (𝑘 ∈ ℝ ↦
inf((((𝐹 ↾ 𝐴) “ (𝑘[,)+∞)) ∩ ℝ*),
ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf((((𝐹 ↾ 𝐴) “ (𝑘[,)+∞)) ∩ ℝ*),
ℝ*, < )) |
| 58 | 57 | liminfval 45774 |
. . 3
⊢ ((𝐹 ↾ 𝐴) ∈ V → (lim inf‘(𝐹 ↾ 𝐴)) = sup(ran (𝑘 ∈ ℝ ↦ inf((((𝐹 ↾ 𝐴) “ (𝑘[,)+∞)) ∩ ℝ*),
ℝ*, < )), ℝ*, < )) |
| 59 | 56, 58 | syl 17 |
. 2
⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝐴)) = sup(ran (𝑘 ∈ ℝ ↦ inf((((𝐹 ↾ 𝐴) “ (𝑘[,)+∞)) ∩ ℝ*),
ℝ*, < )), ℝ*, < )) |
| 60 | | eqid 2737 |
. . . 4
⊢ (𝑘 ∈ ℝ ↦
inf(((𝐹 “ (𝑘[,)+∞)) ∩
ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*),
ℝ*, < )) |
| 61 | 60 | liminfval 45774 |
. . 3
⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) = sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*),
ℝ*, < )), ℝ*, < )) |
| 62 | 55, 61 | syl 17 |
. 2
⊢ (𝜑 → (lim inf‘𝐹) = sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*),
ℝ*, < )), ℝ*, < )) |
| 63 | 54, 59, 62 | 3eqtr4d 2787 |
1
⊢ (𝜑 → (lim inf‘(𝐹 ↾ 𝐴)) = (lim inf‘𝐹)) |