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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > liminf0 | Structured version Visualization version GIF version |
Description: The inferior limit of the empty set. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
liminf0 | ⊢ (lim inf‘∅) = +∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1798 | . . . 4 ⊢ Ⅎ𝑥⊤ | |
2 | 0ex 5308 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ∈ V) |
4 | 0red 11249 | . . . 4 ⊢ (⊤ → 0 ∈ ℝ) | |
5 | noel 4330 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
6 | elinel1 4193 | . . . . . . . 8 ⊢ (𝑥 ∈ (∅ ∩ (0[,)+∞)) → 𝑥 ∈ ∅) | |
7 | 6 | con3i 154 | . . . . . . 7 ⊢ (¬ 𝑥 ∈ ∅ → ¬ 𝑥 ∈ (∅ ∩ (0[,)+∞))) |
8 | 5, 7 | ax-mp 5 | . . . . . 6 ⊢ ¬ 𝑥 ∈ (∅ ∩ (0[,)+∞)) |
9 | pm2.21 123 | . . . . . 6 ⊢ (¬ 𝑥 ∈ (∅ ∩ (0[,)+∞)) → (𝑥 ∈ (∅ ∩ (0[,)+∞)) → (∅‘𝑥) ∈ ℝ*)) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ (𝑥 ∈ (∅ ∩ (0[,)+∞)) → (∅‘𝑥) ∈ ℝ*) |
11 | 10 | adantl 480 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (∅ ∩ (0[,)+∞))) → (∅‘𝑥) ∈ ℝ*) |
12 | 1, 3, 4, 11 | liminfval3 45313 | . . 3 ⊢ (⊤ → (lim inf‘(𝑥 ∈ ∅ ↦ (∅‘𝑥))) = -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥)))) |
13 | 12 | mptru 1540 | . 2 ⊢ (lim inf‘(𝑥 ∈ ∅ ↦ (∅‘𝑥))) = -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) |
14 | mpt0 6698 | . . 3 ⊢ (𝑥 ∈ ∅ ↦ (∅‘𝑥)) = ∅ | |
15 | 14 | fveq2i 6899 | . 2 ⊢ (lim inf‘(𝑥 ∈ ∅ ↦ (∅‘𝑥))) = (lim inf‘∅) |
16 | mpt0 6698 | . . . . . 6 ⊢ (𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥)) = ∅ | |
17 | 16 | fveq2i 6899 | . . . . 5 ⊢ (lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = (lim sup‘∅) |
18 | limsup0 45217 | . . . . 5 ⊢ (lim sup‘∅) = -∞ | |
19 | 17, 18 | eqtri 2753 | . . . 4 ⊢ (lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = -∞ |
20 | 19 | xnegeqi 44957 | . . 3 ⊢ -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = -𝑒-∞ |
21 | xnegmnf 13224 | . . 3 ⊢ -𝑒-∞ = +∞ | |
22 | 20, 21 | eqtri 2753 | . 2 ⊢ -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = +∞ |
23 | 13, 15, 22 | 3eqtr3i 2761 | 1 ⊢ (lim inf‘∅) = +∞ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 Vcvv 3461 ∩ cin 3943 ∅c0 4322 ↦ cmpt 5232 ‘cfv 6549 (class class class)co 7419 0cc0 11140 +∞cpnf 11277 -∞cmnf 11278 ℝ*cxr 11279 -𝑒cxne 13124 [,)cico 13361 lim supclsp 15450 lim infclsi 45274 |
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 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-n0 12506 df-z 12592 df-uz 12856 df-q 12966 df-xneg 13127 df-ico 13365 df-limsup 15451 df-liminf 45275 |
This theorem is referenced by: liminflelimsupcex 45320 |
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