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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 5249 | . . 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 45245 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ*) | 
| 6 | 1, 2, 5 | liminfvalxr 45803 | . 2 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)))) | 
| 7 | eqidd 2737 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
| 8 | 7, 4 | fvmpt2d 7028 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) | 
| 9 | 8 | xnegeqd 45453 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → -𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = -𝑒𝐵) | 
| 10 | 3, 9 | mpteq2da 5239 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ -𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)) = (𝑥 ∈ 𝐴 ↦ -𝑒𝐵)) | 
| 11 | 10 | fveq2d 6909 | . . 3 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = (lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) | 
| 12 | 11 | xnegeqd 45453 | . 2 ⊢ (𝜑 → -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) | 
| 13 | 6, 12 | eqtrd 2776 | 1 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2107 ↦ cmpt 5224 ‘cfv 6560 ℝ*cxr 11295 -𝑒cxne 13152 lim supclsp 15507 lim infclsi 45771 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-xneg 13155 df-limsup 15508 df-liminf 45772 | 
| This theorem is referenced by: liminfval4 45809 liminfval3 45810 | 
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