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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 33388 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < (𝐹‘𝑘)) |
5 | lmdvglim.j | . . 3 ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
6 | icossicc 13409 | . . . 4 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
7 | fss 6724 | . . . 4 ⊢ ((𝐹:ℕ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ (0[,]+∞)) → 𝐹:ℕ⟶(0[,]+∞)) | |
8 | 1, 6, 7 | sylancl 585 | . . 3 ⊢ (𝜑 → 𝐹:ℕ⟶(0[,]+∞)) |
9 | eqidd 2725 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) = (𝐹‘𝑘)) | |
10 | 5, 8, 9 | lmxrge0 33387 | . 2 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 < (𝐹‘𝑘))) |
11 | 4, 10 | mpbird 257 | 1 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)+∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ∃wrex 3062 ⊆ wss 3940 class class class wbr 5138 dom cdm 5666 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ℝcr 11104 0cc0 11105 1c1 11106 + caddc 11108 +∞cpnf 11241 < clt 11244 ≤ cle 11245 ℕcn 12208 ℤ≥cuz 12818 [,)cico 13322 [,]cicc 13323 ⇝ cli 15424 ↾s cress 17171 TopOpenctopn 17365 ℝ*𝑠cxrs 17444 ⇝𝑡clm 23051 |
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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-pm 8818 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fi 9401 df-sup 9432 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-rp 12971 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-tset 17214 df-ple 17215 df-ds 17217 df-rest 17366 df-topn 17367 df-topgen 17387 df-ordt 17445 df-xrs 17446 df-ps 18520 df-tsr 18521 df-top 22717 df-topon 22734 df-bases 22770 df-lm 23054 |
This theorem is referenced by: esumcvg 33539 |
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