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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfvalxrmpt | Structured version Visualization version GIF version |
Description: Alternate definition of lim inf when 𝐹 is an extended real-valued function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminfvalxrmpt.1 | ⊢ Ⅎ𝑥𝜑 |
liminfvalxrmpt.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
liminfvalxrmpt.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
Ref | Expression |
---|---|
liminfvalxrmpt | ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfmpt1 5155 | . . 3 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | liminfvalxrmpt.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | liminfvalxrmpt.1 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
4 | liminfvalxrmpt.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
5 | 3, 4 | fmptd2f 41381 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ*) |
6 | 1, 2, 5 | liminfvalxr 41940 | . 2 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))) |
7 | eqidd 2819 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
8 | 7, 4 | fvmpt2d 6773 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
9 | 8 | xnegeqd 41587 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = -𝑒𝐵) |
10 | 3, 9 | mpteq2da 5151 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑥 ∈ 𝐴 ↦ -𝑒𝐵)) |
11 | 10 | fveq2d 6667 | . . 3 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = (lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) |
12 | 11 | xnegeqd 41587 | . 2 ⊢ (𝜑 → -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) |
13 | 6, 12 | eqtrd 2853 | 1 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 ↦ cmpt 5137 ‘cfv 6348 ℝ*cxr 10662 -𝑒cxne 12492 lim supclsp 14815 lim infclsi 41908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-xneg 12495 df-limsup 14816 df-liminf 41909 |
This theorem is referenced by: liminfval4 41946 liminfval3 41947 |
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