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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 7247 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
4 | eqid 2735 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
5 | 4 | liminfval 45715 | . . 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 6758 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*) |
11 | dfss2 3981 | . . . . . . . . . 10 ⊢ ((𝐹 “ (𝑘[,)+∞)) ⊆ ℝ* ↔ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑘[,)+∞))) | |
12 | 10, 11 | sylib 218 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑘[,)+∞))) |
13 | 12 | eqcomd 2741 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 “ (𝑘[,)+∞)) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)) |
14 | 13 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (𝐹 “ (𝑘[,)+∞)) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)) |
15 | 14 | infeq1d 9515 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < ) = inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
16 | 9, 15 | mpteq2da 5246 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ ℝ ↦ inf((𝐹 “ (𝑘[,)+∞)), ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
17 | 8, 16 | eqtr2d 2776 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = 𝐺) |
18 | 17 | rneqd 5952 | . . 3 ⊢ (𝜑 → ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran 𝐺) |
19 | 18 | supeq1d 9484 | . 2 ⊢ (𝜑 → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = sup(ran 𝐺, ℝ*, < )) |
20 | 6, 19 | eqtrd 2775 | 1 ⊢ (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 ↦ cmpt 5231 ran crn 5690 “ cima 5692 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 supcsup 9478 infcinf 9479 ℝcr 11152 +∞cpnf 11290 ℝ*cxr 11292 < clt 11293 [,)cico 13386 lim infclsi 45707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-liminf 45708 |
This theorem is referenced by: liminf10ex 45730 |
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