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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfval4 | Structured version Visualization version GIF version |
Description: Alternate definition of lim inf when the given function is eventually real-valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminfval4.x | ⊢ Ⅎ𝑥𝜑 |
liminfval4.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
liminfval4.m | ⊢ (𝜑 → 𝑀 ∈ ℝ) |
liminfval4.b | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
liminfval4 | ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | liminfval4.x | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
2 | liminfval4.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | inss1 4086 | . . . . . 6 ⊢ (𝐴 ∩ (𝑀[,)+∞)) ⊆ 𝐴 | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴 ∩ (𝑀[,)+∞)) ⊆ 𝐴) |
5 | 2, 4 | ssexd 5080 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝑀[,)+∞)) ∈ V) |
6 | liminfval4.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ) | |
7 | 6 | rexrd 10488 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) |
8 | 1, 5, 7 | liminfvalxrmpt 41532 | . . 3 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒𝐵))) |
9 | 6 | rexnegd 40870 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → -𝑒𝐵 = -𝐵) |
10 | 1, 9 | mpteq2da 5017 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒𝐵) = (𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝐵)) |
11 | 10 | fveq2d 6500 | . . . 4 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒𝐵)) = (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝐵))) |
12 | 11 | xnegeqd 41176 | . . 3 ⊢ (𝜑 → -𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒𝐵)) = -𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝐵))) |
13 | 8, 12 | eqtrd 2807 | . 2 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝐵))) |
14 | liminfval4.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ) | |
15 | eqid 2771 | . . . 4 ⊢ (𝑀[,)+∞) = (𝑀[,)+∞) | |
16 | 14, 15, 2 | liminfresicompt 41526 | . . 3 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵)) = (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵))) |
17 | 16 | eqcomd 2777 | . 2 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = (lim inf‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵))) |
18 | 2, 14, 15 | limsupresicompt 41502 | . . 3 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ -𝐵)) = (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝐵))) |
19 | 18 | xnegeqd 41176 | . 2 ⊢ (𝜑 → -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝐵)) = -𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝐵))) |
20 | 13, 17, 19 | 3eqtr4d 2817 | 1 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 Ⅎwnf 1747 ∈ wcel 2051 Vcvv 3408 ∩ cin 3821 ⊆ wss 3822 ↦ cmpt 5004 ‘cfv 6185 (class class class)co 6974 ℝcr 10332 +∞cpnf 10469 -cneg 10669 -𝑒cxne 12319 [,)cico 12554 lim supclsp 14686 lim infclsi 41497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-sup 8699 df-inf 8700 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-n0 11706 df-z 11792 df-uz 12057 df-q 12161 df-xneg 12322 df-ico 12558 df-limsup 14687 df-liminf 41498 |
This theorem is referenced by: smfliminflem 42569 |
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