| 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 1804 | . . . 4 ⊢ Ⅎ𝑥⊤ | |
| 2 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → ∅ ∈ V) |
| 4 | 0red 11264 | . . . 4 ⊢ (⊤ → 0 ∈ ℝ) | |
| 5 | noel 4338 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
| 6 | elinel1 4201 | . . . . . . . 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 481 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (∅ ∩ (0[,)+∞))) → (∅‘𝑥) ∈ ℝ*) |
| 12 | 1, 3, 4, 11 | liminfval3 45805 | . . 3 ⊢ (⊤ → (lim inf‘(𝑥 ∈ ∅ ↦ (∅‘𝑥))) = -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥)))) |
| 13 | 12 | mptru 1547 | . 2 ⊢ (lim inf‘(𝑥 ∈ ∅ ↦ (∅‘𝑥))) = -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) |
| 14 | mpt0 6710 | . . 3 ⊢ (𝑥 ∈ ∅ ↦ (∅‘𝑥)) = ∅ | |
| 15 | 14 | fveq2i 6909 | . 2 ⊢ (lim inf‘(𝑥 ∈ ∅ ↦ (∅‘𝑥))) = (lim inf‘∅) |
| 16 | mpt0 6710 | . . . . . 6 ⊢ (𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥)) = ∅ | |
| 17 | 16 | fveq2i 6909 | . . . . 5 ⊢ (lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = (lim sup‘∅) |
| 18 | limsup0 45709 | . . . . 5 ⊢ (lim sup‘∅) = -∞ | |
| 19 | 17, 18 | eqtri 2765 | . . . 4 ⊢ (lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = -∞ |
| 20 | 19 | xnegeqi 45451 | . . 3 ⊢ -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = -𝑒-∞ |
| 21 | xnegmnf 13252 | . . 3 ⊢ -𝑒-∞ = +∞ | |
| 22 | 20, 21 | eqtri 2765 | . 2 ⊢ -𝑒(lim sup‘(𝑥 ∈ ∅ ↦ -𝑒(∅‘𝑥))) = +∞ |
| 23 | 13, 15, 22 | 3eqtr3i 2773 | 1 ⊢ (lim inf‘∅) = +∞ |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ∅c0 4333 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 0cc0 11155 +∞cpnf 11292 -∞cmnf 11293 ℝ*cxr 11294 -𝑒cxne 13151 [,)cico 13389 lim supclsp 15506 lim infclsi 45766 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-xneg 13154 df-ico 13393 df-limsup 15507 df-liminf 45767 |
| This theorem is referenced by: liminflelimsupcex 45812 |
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