Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
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 42190 | . 2 ⊢ (𝜑 → (𝐹~~>*-∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
5 | nfcv 2976 | . . 3 ⊢ Ⅎ𝑗𝐹 | |
6 | 5, 1, 2, 3 | limsupmnfuz 42083 | . 2 ⊢ (𝜑 → ((lim sup‘𝐹) = -∞ ↔ ∀𝑥 ∈ ℝ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ≥‘𝑘)(𝐹‘𝑗) ≤ 𝑥)) |
7 | 4, 6 | bitr4d 284 | 1 ⊢ (𝜑 → (𝐹~~>*-∞ ↔ (lim sup‘𝐹) = -∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ∃wrex 3138 class class class wbr 5059 ⟶wf 6344 ‘cfv 6348 ℝcr 10529 -∞cmnf 10666 ℝ*cxr 10667 ≤ cle 10669 ℤcz 11975 ℤ≥cuz 12237 lim supclsp 14820 ~~>*clsxlim 42174 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-pm 8402 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-fi 8868 df-sup 8899 df-inf 8900 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-n0 11892 df-z 11976 df-uz 12238 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fl 13159 df-ceil 13160 df-limsup 14821 df-topgen 16710 df-ordt 16767 df-ps 17803 df-tsr 17804 df-top 21495 df-topon 21512 df-bases 21547 df-lm 21830 df-xlim 42175 |
This theorem is referenced by: xlimliminflimsup 42218 |
Copyright terms: Public domain | W3C validator |