Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimclim | Structured version Visualization version GIF version |
Description: Given a sequence of reals, it converges to a real number 𝐴 w.r.t. the standard topology on the reals, if and only if it converges to 𝐴 w.r.t. to the standard topology on the extended reals (see climreeq 41884). (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimclim.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
xlimclim.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
xlimclim.f | ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) |
xlimclim.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
xlimclim | ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xlim 42090 | . . . 4 ⊢ ~~>* = (⇝𝑡‘(ordTop‘ ≤ )) | |
2 | 1 | breqi 5063 | . . 3 ⊢ (𝐹~~>*𝐴 ↔ 𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴) |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴)) |
4 | xrtgioo2 41838 | . . 3 ⊢ (topGen‘ran (,)) = ((ordTop‘ ≤ ) ↾t ℝ) | |
5 | xlimclim.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | reex 10620 | . . . 4 ⊢ ℝ ∈ V | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → ℝ ∈ V) |
8 | letop 21806 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ Top | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → (ordTop‘ ≤ ) ∈ Top) |
10 | xlimclim.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
11 | xlimclim.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
12 | xlimclim.f | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ) | |
13 | 4, 5, 7, 9, 10, 11, 12 | lmss 21898 | . 2 ⊢ (𝜑 → (𝐹(⇝𝑡‘(ordTop‘ ≤ ))𝐴 ↔ 𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴)) |
14 | eqid 2819 | . . 3 ⊢ (⇝𝑡‘(topGen‘ran (,))) = (⇝𝑡‘(topGen‘ran (,))) | |
15 | 14, 5, 11, 12 | climreeq 41884 | . 2 ⊢ (𝜑 → (𝐹(⇝𝑡‘(topGen‘ran (,)))𝐴 ↔ 𝐹 ⇝ 𝐴)) |
16 | 3, 13, 15 | 3bitrd 307 | 1 ⊢ (𝜑 → (𝐹~~>*𝐴 ↔ 𝐹 ⇝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1531 ∈ wcel 2108 Vcvv 3493 class class class wbr 5057 ran crn 5549 ⟶wf 6344 ‘cfv 6348 ℝcr 10528 ≤ cle 10668 ℤcz 11973 ℤ≥cuz 12235 (,)cioo 12730 ⇝ cli 14833 topGenctg 16703 ordTopcordt 16764 Topctop 21493 ⇝𝑡clm 21826 ~~>*clsxlim 42089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 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 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fi 8867 df-sup 8898 df-inf 8899 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-ioo 12734 df-ioc 12735 df-ico 12736 df-icc 12737 df-fz 12885 df-fl 13154 df-seq 13362 df-exp 13422 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-rest 16688 df-topn 16689 df-topgen 16709 df-ordt 16766 df-ps 17802 df-tsr 17803 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-cnfld 20538 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-lm 21829 df-xms 22922 df-ms 22923 df-xlim 42090 |
This theorem is referenced by: climxlim 42097 xlimclim2lem 42110 |
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