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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 1807 | . . . 4 ⊢ Ⅎ𝑥⊤ | |
2 | 0ex 5265 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ∈ V) |
4 | 0red 11163 | . . . 4 ⊢ (⊤ → 0 ∈ ℝ) | |
5 | noel 4291 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
6 | elinel1 4156 | . . . . . . . 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 483 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (∅ ∩ (0[,)+∞))) → (∅‘𝑥) ∈ ℝ*) |
12 | 1, 3, 4, 11 | liminfval3 44117 | . . 3 ⊢ (⊤ → (lim inf‘(𝑥 ∈ ∅ ↦ (∅‘𝑥))) = -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥)))) |
13 | 12 | mptru 1549 | . 2 ⊢ (lim inf‘(𝑥 ∈ ∅ ↦ (∅‘𝑥))) = -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) |
14 | mpt0 6644 | . . 3 ⊢ (𝑥 ∈ ∅ ↦ (∅‘𝑥)) = ∅ | |
15 | 14 | fveq2i 6846 | . 2 ⊢ (lim inf‘(𝑥 ∈ ∅ ↦ (∅‘𝑥))) = (lim inf‘∅) |
16 | mpt0 6644 | . . . . . 6 ⊢ (𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥)) = ∅ | |
17 | 16 | fveq2i 6846 | . . . . 5 ⊢ (lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = (lim sup‘∅) |
18 | limsup0 44021 | . . . . 5 ⊢ (lim sup‘∅) = -∞ | |
19 | 17, 18 | eqtri 2761 | . . . 4 ⊢ (lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = -∞ |
20 | 19 | xnegeqi 43761 | . . 3 ⊢ -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = -𝑒-∞ |
21 | xnegmnf 13135 | . . 3 ⊢ -𝑒-∞ = +∞ | |
22 | 20, 21 | eqtri 2761 | . 2 ⊢ -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = +∞ |
23 | 13, 15, 22 | 3eqtr3i 2769 | 1 ⊢ (lim inf‘∅) = +∞ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 Vcvv 3444 ∩ cin 3910 ∅c0 4283 ↦ cmpt 5189 ‘cfv 6497 (class class class)co 7358 0cc0 11056 +∞cpnf 11191 -∞cmnf 11192 ℝ*cxr 11193 -𝑒cxne 13035 [,)cico 13272 lim supclsp 15358 lim infclsi 44078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-q 12879 df-xneg 13038 df-ico 13276 df-limsup 15359 df-liminf 44079 |
This theorem is referenced by: liminflelimsupcex 44124 |
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