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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 1767 | . . . 4 ⊢ Ⅎ𝑥⊤ | |
2 | 0ex 5062 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ∈ V) |
4 | 0red 10437 | . . . 4 ⊢ (⊤ → 0 ∈ ℝ) | |
5 | noel 4177 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
6 | elinel1 4054 | . . . . . . . 8 ⊢ (𝑥 ∈ (∅ ∩ (0[,)+∞)) → 𝑥 ∈ ∅) | |
7 | 6 | con3i 152 | . . . . . . 7 ⊢ (¬ 𝑥 ∈ ∅ → ¬ 𝑥 ∈ (∅ ∩ (0[,)+∞))) |
8 | 5, 7 | ax-mp 5 | . . . . . 6 ⊢ ¬ 𝑥 ∈ (∅ ∩ (0[,)+∞)) |
9 | pm2.21 121 | . . . . . 6 ⊢ (¬ 𝑥 ∈ (∅ ∩ (0[,)+∞)) → (𝑥 ∈ (∅ ∩ (0[,)+∞)) → (∅‘𝑥) ∈ ℝ*)) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ (𝑥 ∈ (∅ ∩ (0[,)+∞)) → (∅‘𝑥) ∈ ℝ*) |
11 | 10 | adantl 474 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (∅ ∩ (0[,)+∞))) → (∅‘𝑥) ∈ ℝ*) |
12 | 1, 3, 4, 11 | liminfval3 41502 | . . 3 ⊢ (⊤ → (lim inf‘(𝑥 ∈ ∅ ↦ (∅‘𝑥))) = -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥)))) |
13 | 12 | mptru 1514 | . 2 ⊢ (lim inf‘(𝑥 ∈ ∅ ↦ (∅‘𝑥))) = -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) |
14 | mpt0 6314 | . . 3 ⊢ (𝑥 ∈ ∅ ↦ (∅‘𝑥)) = ∅ | |
15 | 14 | fveq2i 6496 | . 2 ⊢ (lim inf‘(𝑥 ∈ ∅ ↦ (∅‘𝑥))) = (lim inf‘∅) |
16 | mpt0 6314 | . . . . . 6 ⊢ (𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥)) = ∅ | |
17 | 16 | fveq2i 6496 | . . . . 5 ⊢ (lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = (lim sup‘∅) |
18 | limsup0 41406 | . . . . 5 ⊢ (lim sup‘∅) = -∞ | |
19 | 17, 18 | eqtri 2796 | . . . 4 ⊢ (lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = -∞ |
20 | 19 | xnegeqi 41145 | . . 3 ⊢ -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = -𝑒-∞ |
21 | xnegmnf 12414 | . . 3 ⊢ -𝑒-∞ = +∞ | |
22 | 20, 21 | eqtri 2796 | . 2 ⊢ -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = +∞ |
23 | 13, 15, 22 | 3eqtr3i 2804 | 1 ⊢ (lim inf‘∅) = +∞ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1507 ⊤wtru 1508 ∈ wcel 2050 Vcvv 3409 ∩ cin 3822 ∅c0 4172 ↦ cmpt 5002 ‘cfv 6182 (class class class)co 6970 0cc0 10329 +∞cpnf 10465 -∞cmnf 10466 ℝ*cxr 10467 -𝑒cxne 12315 [,)cico 12550 lim supclsp 14682 lim infclsi 41463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-pre-sup 10407 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-er 8083 df-en 8301 df-dom 8302 df-sdom 8303 df-sup 8695 df-inf 8696 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-n0 11702 df-z 11788 df-uz 12053 df-q 12157 df-xneg 12318 df-ico 12554 df-limsup 14683 df-liminf 41464 |
This theorem is referenced by: liminflelimsupcex 41509 |
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