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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 5311 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ∈ V) |
4 | 0red 11255 | . . . 4 ⊢ (⊤ → 0 ∈ ℝ) | |
5 | noel 4334 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
6 | elinel1 4197 | . . . . . . . 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 45207 | . . 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 6702 | . . 3 ⊢ (𝑥 ∈ ∅ ↦ (∅‘𝑥)) = ∅ | |
15 | 14 | fveq2i 6905 | . 2 ⊢ (lim inf‘(𝑥 ∈ ∅ ↦ (∅‘𝑥))) = (lim inf‘∅) |
16 | mpt0 6702 | . . . . . 6 ⊢ (𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥)) = ∅ | |
17 | 16 | fveq2i 6905 | . . . . 5 ⊢ (lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = (lim sup‘∅) |
18 | limsup0 45111 | . . . . 5 ⊢ (lim sup‘∅) = -∞ | |
19 | 17, 18 | eqtri 2756 | . . . 4 ⊢ (lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = -∞ |
20 | 19 | xnegeqi 44851 | . . 3 ⊢ -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = -𝑒-∞ |
21 | xnegmnf 13229 | . . 3 ⊢ -𝑒-∞ = +∞ | |
22 | 20, 21 | eqtri 2756 | . 2 ⊢ -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = +∞ |
23 | 13, 15, 22 | 3eqtr3i 2764 | 1 ⊢ (lim inf‘∅) = +∞ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ⊤wtru 1534 ∈ wcel 2098 Vcvv 3473 ∩ cin 3948 ∅c0 4326 ↦ cmpt 5235 ‘cfv 6553 (class class class)co 7426 0cc0 11146 +∞cpnf 11283 -∞cmnf 11284 ℝ*cxr 11285 -𝑒cxne 13129 [,)cico 13366 lim supclsp 15454 lim infclsi 45168 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-xneg 13132 df-ico 13370 df-limsup 15455 df-liminf 45169 |
This theorem is referenced by: liminflelimsupcex 45214 |
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