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 3833 | . . . . 5 ⊢ (𝐴 ∩ (𝑀[,)+∞)) ⊆ 𝐴 | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝑀[,)+∞)) ⊆ 𝐴) |
5 | 2, 4 | ssexd 4805 | . . 3 ⊢ (𝜑 → (𝐴 ∩ (𝑀[,)+∞)) ∈ V) |
6 | liminfval3.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) | |
7 | 1, 5, 6 | liminfvalxrmpt 40018 | . 2 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒𝐵))) |
8 | liminfval3.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ) | |
9 | eqid 2622 | . . . 4 ⊢ (𝑀[,)+∞) = (𝑀[,)+∞) | |
10 | 8, 9, 2 | liminfresicompt 40012 | . . 3 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵)) = (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵))) |
11 | 10 | eqcomd 2628 | . 2 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = (lim inf‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵))) |
12 | 2, 8, 9 | limsupresicompt 39988 | . . 3 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵)) = (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒𝐵))) |
13 | 12 | xnegeqd 39664 | . 2 ⊢ (𝜑 → -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵)) = -𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒𝐵))) |
14 | 7, 11, 13 | 3eqtr4d 2666 | 1 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 +∞cpnf 10071 ℝ*cxr 10073 -𝑒cxne 11943 [,)cico 12177 lim supclsp 14201 lim infclsi 39983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-xneg 11946 df-ico 12181 df-limsup 14202 df-liminf 39984 |
This theorem is referenced by: liminfvaluz 40024 liminf0 40025 limsupval4 40026 |
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