Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfval | Structured version Visualization version GIF version |
Description: The inferior limit of a set 𝐹. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminfval.1 | ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
Ref | Expression |
---|---|
liminfval | ⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-liminf 42999 | . 2 ⊢ lim inf = (𝑥 ∈ V ↦ sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < )) | |
2 | imaeq1 5939 | . . . . . . . 8 ⊢ (𝑥 = 𝐹 → (𝑥 “ (𝑘[,)+∞)) = (𝐹 “ (𝑘[,)+∞))) | |
3 | 2 | ineq1d 4141 | . . . . . . 7 ⊢ (𝑥 = 𝐹 → ((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*) = ((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*)) |
4 | 3 | infeq1d 9118 | . . . . . 6 ⊢ (𝑥 = 𝐹 → inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ) = inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) |
5 | 4 | mpteq2dv 5166 | . . . . 5 ⊢ (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
6 | liminfval.1 | . . . . . 6 ⊢ 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) | |
7 | 6 | a1i 11 | . . . . 5 ⊢ (𝑥 = 𝐹 → 𝐺 = (𝑘 ∈ ℝ ↦ inf(((𝐹 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < ))) |
8 | 5, 7 | eqtr4d 2781 | . . . 4 ⊢ (𝑥 = 𝐹 → (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = 𝐺) |
9 | 8 | rneqd 5822 | . . 3 ⊢ (𝑥 = 𝐹 → ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )) = ran 𝐺) |
10 | 9 | supeq1d 9087 | . 2 ⊢ (𝑥 = 𝐹 → sup(ran (𝑘 ∈ ℝ ↦ inf(((𝑥 “ (𝑘[,)+∞)) ∩ ℝ*), ℝ*, < )), ℝ*, < ) = sup(ran 𝐺, ℝ*, < )) |
11 | elex 3439 | . 2 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) | |
12 | xrltso 12756 | . . . 4 ⊢ < Or ℝ* | |
13 | 12 | supex 9104 | . . 3 ⊢ sup(ran 𝐺, ℝ*, < ) ∈ V |
14 | 13 | a1i 11 | . 2 ⊢ (𝐹 ∈ 𝑉 → sup(ran 𝐺, ℝ*, < ) ∈ V) |
15 | 1, 10, 11, 14 | fvmptd3 6860 | 1 ⊢ (𝐹 ∈ 𝑉 → (lim inf‘𝐹) = sup(ran 𝐺, ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2111 Vcvv 3421 ∩ cin 3880 ↦ cmpt 5150 ran crn 5567 “ cima 5569 ‘cfv 6398 (class class class)co 7232 supcsup 9081 infcinf 9082 ℝcr 10753 +∞cpnf 10889 ℝ*cxr 10891 < clt 10892 [,)cico 12962 lim infclsi 42998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 ax-cnex 10810 ax-resscn 10811 ax-pre-lttri 10828 ax-pre-lttrn 10829 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-rmo 3070 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-br 5069 df-opab 5131 df-mpt 5151 df-id 5470 df-po 5483 df-so 5484 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-er 8412 df-en 8648 df-dom 8649 df-sdom 8650 df-sup 9083 df-inf 9084 df-pnf 10894 df-mnf 10895 df-xr 10896 df-ltxr 10897 df-liminf 42999 |
This theorem is referenced by: liminfcl 43010 liminfvald 43011 liminfval5 43012 liminfresxr 43014 liminfval2 43015 liminfvalxr 43030 |
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