Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lmdvglim | Structured version Visualization version GIF version |
Description: If a monotonic real number sequence 𝐹 diverges, it converges in the extended real numbers and its limit is plus infinity. (Contributed by Thierry Arnoux, 3-Aug-2017.) |
Ref | Expression |
---|---|
lmdvglim.j | ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
lmdvglim.1 | ⊢ (𝜑 → 𝐹:ℕ⟶(0[,)+∞)) |
lmdvglim.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) |
lmdvglim.3 | ⊢ (𝜑 → ¬ 𝐹 ∈ dom ⇝ ) |
Ref | Expression |
---|---|
lmdvglim | ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)+∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lmdvglim.1 | . . 3 ⊢ (𝜑 → 𝐹:ℕ⟶(0[,)+∞)) | |
2 | lmdvglim.2 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ≤ (𝐹‘(𝑘 + 1))) | |
3 | lmdvglim.3 | . . 3 ⊢ (𝜑 → ¬ 𝐹 ∈ dom ⇝ ) | |
4 | 1, 2, 3 | lmdvg 31641 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < (𝐹‘𝑘)) |
5 | lmdvglim.j | . . 3 ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
6 | icossicc 13049 | . . . 4 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
7 | fss 6581 | . . . 4 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℕ⟶(0[,]+∞)) | |
8 | 1, 6, 7 | sylancl 589 | . . 3 ⊢ (𝜑 → 𝐹:ℕ⟶(0[,]+∞)) |
9 | eqidd 2739 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
10 | 5, 8, 9 | lmxrge0 31640 | . 2 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < (𝐹‘𝑘))) |
11 | 4, 10 | mpbird 260 | 1 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)+∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 ∀wral 3062 ∃wrex 3063 ⊆ wss 3881 class class class wbr 5068 dom cdm 5566 ⟶wf 6394 ‘cfv 6398 (class class class)co 7232 ℝcr 10753 0cc0 10754 1c1 10755 + caddc 10757 +∞cpnf 10889 < clt 10892 ≤ cle 10893 ℕcn 11855 ℤ≥cuz 12463 [,)cico 12962 [,]cicc 12963 ⇝ cli 15070 ↾s cress 16809 TopOpenctopn 16951 ℝ*𝑠cxrs 17030 ⇝𝑡clm 22147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5194 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 ax-cnex 10810 ax-resscn 10811 ax-1cn 10812 ax-icn 10813 ax-addcl 10814 ax-addrcl 10815 ax-mulcl 10816 ax-mulrcl 10817 ax-mulcom 10818 ax-addass 10819 ax-mulass 10820 ax-distr 10821 ax-i2m1 10822 ax-1ne0 10823 ax-1rid 10824 ax-rnegex 10825 ax-rrecex 10826 ax-cnre 10827 ax-pre-lttri 10828 ax-pre-lttrn 10829 ax-pre-ltadd 10830 ax-pre-mulgt0 10831 ax-pre-sup 10832 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-int 4875 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-tr 5177 df-id 5470 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-we 5526 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-pred 6176 df-ord 6234 df-on 6235 df-lim 6236 df-suc 6237 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-riota 7189 df-ov 7235 df-oprab 7236 df-mpo 7237 df-om 7664 df-1st 7780 df-2nd 7781 df-wrecs 8068 df-recs 8129 df-rdg 8167 df-1o 8223 df-er 8412 df-pm 8532 df-en 8648 df-dom 8649 df-sdom 8650 df-fin 8651 df-fi 9052 df-sup 9083 df-pnf 10894 df-mnf 10895 df-xr 10896 df-ltxr 10897 df-le 10898 df-sub 11089 df-neg 11090 df-div 11515 df-nn 11856 df-2 11918 df-3 11919 df-4 11920 df-5 11921 df-6 11922 df-7 11923 df-8 11924 df-9 11925 df-n0 12116 df-z 12202 df-dec 12319 df-uz 12464 df-rp 12612 df-ioo 12964 df-ioc 12965 df-ico 12966 df-icc 12967 df-fz 13121 df-seq 13600 df-exp 13661 df-cj 14687 df-re 14688 df-im 14689 df-sqrt 14823 df-abs 14824 df-clim 15074 df-struct 16725 df-sets 16742 df-slot 16760 df-ndx 16770 df-base 16786 df-ress 16810 df-plusg 16840 df-mulr 16841 df-tset 16846 df-ple 16847 df-ds 16849 df-rest 16952 df-topn 16953 df-topgen 16973 df-ordt 17031 df-xrs 17032 df-ps 18097 df-tsr 18098 df-top 21815 df-topon 21832 df-bases 21867 df-lm 22150 |
This theorem is referenced by: esumcvg 31790 |
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