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| Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimmnflimsup2 | Structured version Visualization version GIF version | ||
| Description: A sequence of extended reals converges to -∞ if and only if its superior limit is also -∞. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
| Ref | Expression |
|---|---|
| xlimmnflimsup2.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
| xlimmnflimsup2.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
| xlimmnflimsup2.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
| Ref | Expression |
|---|---|
| xlimmnflimsup2 | ⊢ (𝜑 → (𝐹~~>*-∞ ↔ (lim sup‘𝐹) = -∞)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xlimmnflimsup2.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
| 2 | xlimmnflimsup2.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
| 3 | xlimmnflimsup2.f | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
| 4 | 1, 2, 3 | xlimmnfv 45849 | . 2 ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
| 5 | nfcv 2905 | . . 3 ⊢ Ⅎ𝑗𝐹 | |
| 6 | 5, 1, 2, 3 | limsupmnfuz 45742 | . 2 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
| 7 | 4, 6 | bitr4d 282 | 1 ⊢ (𝜑 → (𝐹~~>*-∞ ↔ (lim sup‘𝐹) = -∞)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 class class class wbr 5143 ⟶wf 6557 ‘cfv 6561 ℝcr 11154 -∞cmnf 11293 ℝ*cxr 11294 ≤ cle 11296 ℤcz 12613 ℤ≥cuz 12878 lim supclsp 15506 ~~>*clsxlim 45833 |
| 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-int 4947 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-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-1o 8506 df-2o 8507 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 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-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fl 13832 df-ceil 13833 df-limsup 15507 df-topgen 17488 df-ordt 17546 df-ps 18611 df-tsr 18612 df-top 22900 df-topon 22917 df-bases 22953 df-lm 23237 df-xlim 45834 |
| This theorem is referenced by: xlimliminflimsup 45877 |
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