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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupval4 | Structured version Visualization version GIF version |
Description: Alternate definition of lim inf when the given a function is eventually extended real-valued. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
limsupval4.x | ⊢ Ⅎ𝑥𝜑 |
limsupval4.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
limsupval4.m | ⊢ (𝜑 → 𝑀 ∈ ℝ) |
limsupval4.b | ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) |
Ref | Expression |
---|---|
limsupval4 | ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim inf‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7434 | . . . . . . . 8 ⊢ (𝑀[,)+∞) ∈ V | |
2 | 1 | inex2 5308 | . . . . . . 7 ⊢ (𝐴 ∩ (𝑀[,)+∞)) ∈ V |
3 | 2 | mptex 7216 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵) ∈ V |
4 | limsupcl 15413 | . . . . . 6 ⊢ ((𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵) ∈ V → (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵)) ∈ ℝ*) | |
5 | 3, 4 | ax-mp 5 | . . . . 5 ⊢ (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵)) ∈ ℝ* |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵)) ∈ ℝ*) |
7 | 6 | xnegnegd 44603 | . . 3 ⊢ (𝜑 → -𝑒-𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵)) = (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵))) |
8 | 7 | eqcomd 2730 | . 2 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵)) = -𝑒-𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵))) |
9 | limsupval4.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
10 | limsupval4.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ) | |
11 | eqid 2724 | . . 3 ⊢ (𝑀[,)+∞) = (𝑀[,)+∞) | |
12 | 9, 10, 11 | limsupresicompt 44923 | . 2 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵))) |
13 | limsupval4.x | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
14 | limsupval4.b | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) | |
15 | 14 | xnegcld 13275 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → -𝑒𝐵 ∈ ℝ*) |
16 | 13, 9, 10, 15 | liminfval3 44957 | . . . 4 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒-𝑒𝐵))) |
17 | 9, 10, 11 | limsupresicompt 44923 | . . . . . 6 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒-𝑒𝐵)) = (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒-𝑒𝐵))) |
18 | 14 | xnegnegd 44603 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → -𝑒-𝑒𝐵 = 𝐵) |
19 | 13, 18 | mpteq2da 5236 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒-𝑒𝐵) = (𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵)) |
20 | 19 | fveq2d 6885 | . . . . . 6 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒-𝑒𝐵)) = (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵))) |
21 | 17, 20 | eqtrd 2764 | . . . . 5 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒-𝑒𝐵)) = (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵))) |
22 | 21 | xnegeqd 44598 | . . . 4 ⊢ (𝜑 → -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝑒-𝑒𝐵)) = -𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵))) |
23 | 16, 22 | eqtrd 2764 | . . 3 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵)) = -𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵))) |
24 | 23 | xnegeqd 44598 | . 2 ⊢ (𝜑 → -𝑒(lim inf‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵)) = -𝑒-𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵))) |
25 | 8, 12, 24 | 3eqtr4d 2774 | 1 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim inf‘(𝑥 ∈ 𝐴 ↦ -𝑒𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 Ⅎwnf 1777 ∈ wcel 2098 Vcvv 3466 ∩ cin 3939 ↦ cmpt 5221 ‘cfv 6533 (class class class)co 7401 ℝcr 11104 +∞cpnf 11241 ℝ*cxr 11243 -𝑒cxne 13085 [,)cico 13322 lim supclsp 15410 lim infclsi 44918 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-sup 9432 df-inf 9433 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-xneg 13088 df-ico 13326 df-limsup 15411 df-liminf 44919 |
This theorem is referenced by: limsupvaluz3 44965 |
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