Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfval5 | Structured version Visualization version GIF version |
Description: The inferior limit of an infinite sequence 𝐹 of extended real numbers. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
limsupval5.1 | ⊢ Ⅎ𝑘𝜑 |
limsupval5.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
limsupval5.3 | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) |
limsupval5.4 | ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) |
Ref | Expression |
---|---|
liminfval5 | ⊢ (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupval5.3 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ*) | |
2 | limsupval5.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | 1, 2 | fexd 41386 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
4 | eqid 2823 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
5 | 4 | liminfval 42047 | . . 3 ⊢ (𝐹 ∈ V → (lim inf‘𝐹) = sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
6 | 3, 5 | syl 17 | . 2 ⊢ (𝜑 → (lim inf‘𝐹) = sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) |
7 | limsupval5.4 | . . . . . 6 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝜑 → 𝐺 = (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < ))) |
9 | limsupval5.1 | . . . . . 6 ⊢ Ⅎ𝑘𝜑 | |
10 | 1 | fimassd 41505 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*) |
11 | df-ss 3954 | . . . . . . . . . 10 ⊢ ((𝐹 “ (𝑘[,)+∞)) ⊆ ℝ* ↔ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑘[,)+∞))) | |
12 | 10, 11 | sylib 220 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑘[,)+∞))) |
13 | 12 | eqcomd 2829 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 “ (𝑘[,)+∞)) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)) |
14 | 13 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (𝐹 “ (𝑘[,)+∞)) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)) |
15 | 14 | infeq1d 8943 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
16 | 9, 15 | mpteq2da 5162 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
17 | 8, 16 | eqtr2d 2859 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = 𝐺) |
18 | 17 | rneqd 5810 | . . 3 ⊢ (𝜑 → ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran 𝐺) |
19 | 18 | supeq1d 8912 | . 2 ⊢ (𝜑 → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = sup(ran 𝐺, ℝ*, < )) |
20 | 6, 19 | eqtrd 2858 | 1 ⊢ (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 Vcvv 3496 ∩ cin 3937 ⊆ wss 3938 ↦ cmpt 5148 ran crn 5558 “ cima 5560 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 supcsup 8906 infcinf 8907 ℝcr 10538 +∞cpnf 10674 ℝ*cxr 10676 < clt 10677 [,)cico 12743 lim infclsi 42039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-liminf 42040 |
This theorem is referenced by: liminf10ex 42062 |
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