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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminfval3 | Structured version Visualization version GIF version |
Description: Alternate definition of lim inf when the given function is eventually extended real-valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminfval3.x | ⊢ Ⅎ𝑥𝜑 |
liminfval3.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
liminfval3.m | ⊢ (𝜑 → 𝑀 ∈ ℝ) |
liminfval3.b | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) |
Ref | Expression |
---|---|
liminfval3 | ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | liminfval3.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | liminfval3.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | inss1 4087 | . . . . 5 ⊢ (𝐴 ∩ (𝑀[,)+∞)) ⊆ 𝐴 | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝑀[,)+∞)) ⊆ 𝐴) |
5 | 2, 4 | ssexd 5081 | . . 3 ⊢ (𝜑 → (𝐴 ∩ (𝑀[,)+∞)) ∈ V) |
6 | liminfval3.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) | |
7 | 1, 5, 6 | liminfvalxrmpt 41528 | . 2 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒𝐵))) |
8 | liminfval3.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ) | |
9 | eqid 2773 | . . . 4 ⊢ (𝑀[,)+∞) = (𝑀[,)+∞) | |
10 | 8, 9, 2 | liminfresicompt 41522 | . . 3 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵)) = (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵))) |
11 | 10 | eqcomd 2779 | . 2 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = (lim inf‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵))) |
12 | 2, 8, 9 | limsupresicompt 41498 | . . 3 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵)) = (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒𝐵))) |
13 | 12 | xnegeqd 41172 | . 2 ⊢ (𝜑 → -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵)) = -𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒𝐵))) |
14 | 7, 11, 13 | 3eqtr4d 2819 | 1 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 Ⅎwnf 1747 ∈ wcel 2051 Vcvv 3410 ∩ cin 3823 ⊆ wss 3824 ↦ cmpt 5005 ‘cfv 6186 (class class class)co 6975 ℝcr 10333 +∞cpnf 10470 ℝ*cxr 10472 -𝑒cxne 12320 [,)cico 12555 lim supclsp 14687 lim infclsi 41493 |
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 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-pre-sup 10412 |
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 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-isom 6195 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-sup 8700 df-inf 8701 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-nn 11439 df-n0 11707 df-z 11793 df-uz 12058 df-q 12162 df-xneg 12323 df-ico 12559 df-limsup 14688 df-liminf 41494 |
This theorem is referenced by: liminfvaluz 41534 liminf0 41535 limsupval4 41536 |
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