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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 4228 | . . . . . 6 ⊢ (𝐴 ∩ (𝑀[,)+∞)) ⊆ 𝐴 | |
4 | 3 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝐴 ∩ (𝑀[,)+∞)) ⊆ 𝐴) |
5 | 2, 4 | ssexd 5324 | . . . 4 ⊢ (𝜑 → (𝐴 ∩ (𝑀[,)+∞)) ∈ V) |
6 | liminfval4.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ) | |
7 | 6 | rexrd 11271 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → 𝐵 ∈ ℝ*) |
8 | 1, 5, 7 | liminfvalxrmpt 44963 | . . 3 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒𝐵))) |
9 | 6 | rexnegd 44296 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞))) → -𝑒𝐵 = -𝐵) |
10 | 1, 9 | mpteq2da 5246 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒𝐵) = (𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝐵)) |
11 | 10 | fveq2d 6895 | . . . 4 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒𝐵)) = (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝐵))) |
12 | 11 | xnegeqd 44608 | . . 3 ⊢ (𝜑 → -𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝑒𝐵)) = -𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝐵))) |
13 | 8, 12 | eqtrd 2771 | . 2 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝐵))) |
14 | liminfval4.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ) | |
15 | eqid 2731 | . . . 4 ⊢ (𝑀[,)+∞) = (𝑀[,)+∞) | |
16 | 14, 15, 2 | liminfresicompt 44957 | . . 3 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵)) = (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵))) |
17 | 16 | eqcomd 2737 | . 2 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = (lim inf‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ 𝐵))) |
18 | 2, 14, 15 | limsupresicompt 44933 | . . 3 ⊢ (𝜑 → (lim sup‘(𝑥 ∈ 𝐴 ↦ -𝐵)) = (lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝐵))) |
19 | 18 | xnegeqd 44608 | . 2 ⊢ (𝜑 → -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝐵)) = -𝑒(lim sup‘(𝑥 ∈ (𝐴 ∩ (𝑀[,)+∞)) ↦ -𝐵))) |
20 | 13, 17, 19 | 3eqtr4d 2781 | 1 ⊢ (𝜑 → (lim inf‘(𝑥 ∈ 𝐴 ↦ 𝐵)) = -𝑒(lim sup‘(𝑥 ∈ 𝐴 ↦ -𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1784 ∈ wcel 2105 Vcvv 3473 ∩ cin 3947 ⊆ wss 3948 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7412 ℝcr 11115 +∞cpnf 11252 -cneg 11452 -𝑒cxne 13096 [,)cico 13333 lim supclsp 15421 lim infclsi 44928 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9443 df-inf 9444 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-n0 12480 df-z 12566 df-uz 12830 df-q 12940 df-xneg 13099 df-ico 13337 df-limsup 15422 df-liminf 44929 |
This theorem is referenced by: smfliminflem 46007 |
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