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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 7231 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
4 | eqid 2731 | . . . 4 ⊢ (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
5 | 4 | liminfval 44936 | . . 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 44391 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 “ (𝑘[,)+∞)) ⊆ ℝ*) |
11 | df-ss 3965 | . . . . . . . . . 10 ⊢ ((𝐹 “ (𝑘[,)+∞)) ⊆ ℝ* ↔ ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑘[,)+∞))) | |
12 | 10, 11 | sylib 217 | . . . . . . . . 9 ⊢ (𝜑 → ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*) = (𝐹 “ (𝑘[,)+∞))) |
13 | 12 | eqcomd 2737 | . . . . . . . 8 ⊢ (𝜑 → (𝐹 “ (𝑘[,)+∞)) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)) |
14 | 13 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℝ) → (𝐹 “ (𝑘[,)+∞)) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)) |
15 | 14 | infeq1d 9478 | . . . . . 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 2772 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = 𝐺) |
18 | 17 | rneqd 5937 | . . 3 ⊢ (𝜑 → ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran 𝐺) |
19 | 18 | supeq1d 9447 | . 2 ⊢ (𝜑 → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = sup(ran 𝐺, ℝ*, < )) |
20 | 6, 19 | eqtrd 2771 | 1 ⊢ (𝜑 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1784 ∈ wcel 2105 Vcvv 3473 ∩ cin 3947 ⊆ wss 3948 ↦ cmpt 5231 ran crn 5677 “ cima 5679 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 supcsup 9441 infcinf 9442 ℝcr 11115 +∞cpnf 11252 ℝ*cxr 11254 < clt 11255 [,)cico 13333 lim infclsi 44928 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-pre-lttri 11190 ax-pre-lttrn 11191 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-liminf 44929 |
This theorem is referenced by: liminf10ex 44951 |
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