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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimclim2 | Structured version Visualization version GIF version |
Description: Given a sequence of extended reals, it converges to a real number 𝐴 w.r.t. the standard topology on the reals (see climreeq 41770), if and only if it converges to 𝐴 w.r.t. to the standard topology on the extended reals. In order for the first part of the statement to even make sense, the sequence will of course eventually become (and stay) real: showing this, is the key step of the proof. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimclim2.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
xlimclim2.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
xlimclim2.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
xlimclim2.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
xlimclim2 | ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝐹~~>*𝐴) → 𝐹~~>*𝐴) | |
2 | xlimclim2.z | . . . 4 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
3 | xlimclim2.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) | |
4 | 3 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐹~~>*𝐴) → 𝐹:𝑍⟶ℝ*) |
5 | xlimclim2.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
6 | 5 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐹~~>*𝐴) → 𝐴 ∈ ℝ) |
7 | xlimclim2.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
8 | 7 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹~~>*𝐴) → 𝑀 ∈ ℤ) |
9 | 8, 2, 4, 6, 1 | xlimxrre 41988 | . . . 4 ⊢ ((𝜑 ∧ 𝐹~~>*𝐴) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) |
10 | 2, 4, 6, 9 | xlimclim2lem 41996 | . . 3 ⊢ ((𝜑 ∧ 𝐹~~>*𝐴) → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) |
11 | 1, 10 | mpbid 233 | . 2 ⊢ ((𝜑 ∧ 𝐹~~>*𝐴) → 𝐹 ⇝ 𝐴) |
12 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ⇝ 𝐴) → 𝐹 ⇝ 𝐴) | |
13 | 3 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ⇝ 𝐴) → 𝐹:𝑍⟶ℝ*) |
14 | 5 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ⇝ 𝐴) → 𝐴 ∈ ℝ) |
15 | 7 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐹 ⇝ 𝐴) → 𝑀 ∈ ℤ) |
16 | 15, 2, 13, 14, 12 | climxrre 41907 | . . . 4 ⊢ ((𝜑 ∧ 𝐹 ⇝ 𝐴) → ∃𝑗 ∈ 𝑍 (𝐹 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶ℝ) |
17 | 2, 13, 14, 16 | xlimclim2lem 41996 | . . 3 ⊢ ((𝜑 ∧ 𝐹 ⇝ 𝐴) → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) |
18 | 12, 17 | mpbird 258 | . 2 ⊢ ((𝜑 ∧ 𝐹 ⇝ 𝐴) → 𝐹~~>*𝐴) |
19 | 11, 18 | impbida 797 | 1 ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ⟶wf 6344 ‘cfv 6348 ℝcr 10524 ℝ*cxr 10662 ℤcz 11969 ℤ≥cuz 12231 ⇝ cli 14829 ~~>*clsxlim 41975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 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 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fi 8863 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fl 13150 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-rlim 14834 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-mulr 16567 df-starv 16568 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-rest 16684 df-topn 16685 df-topgen 16705 df-ordt 16762 df-ps 17798 df-tsr 17799 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-lm 21765 df-xms 22857 df-ms 22858 df-xlim 41976 |
This theorem is referenced by: climxlim2lem 42002 dfxlim2v 42004 xlimclimdm 42011 |
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