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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limsupgt | Structured version Visualization version GIF version |
Description: Given a sequence of real numbers, there exists an upper part of the sequence that's appxoximated from below by the superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
limsupgt.k | β’ β²ππΉ |
limsupgt.m | β’ (π β π β β€) |
limsupgt.z | β’ π = (β€β₯βπ) |
limsupgt.f | β’ (π β πΉ:πβΆβ) |
limsupgt.r | β’ (π β (lim supβπΉ) β β) |
limsupgt.x | β’ (π β π β β+) |
Ref | Expression |
---|---|
limsupgt | β’ (π β βπ β π βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupgt.m | . . 3 β’ (π β π β β€) | |
2 | limsupgt.z | . . 3 β’ π = (β€β₯βπ) | |
3 | limsupgt.f | . . 3 β’ (π β πΉ:πβΆβ) | |
4 | limsupgt.r | . . 3 β’ (π β (lim supβπΉ) β β) | |
5 | limsupgt.x | . . 3 β’ (π β π β β+) | |
6 | 1, 2, 3, 4, 5 | limsupgtlem 45065 | . 2 β’ (π β βπ β π βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ)) |
7 | limsupgt.k | . . . . . . . . 9 β’ β²ππΉ | |
8 | nfcv 2897 | . . . . . . . . 9 β’ β²ππ | |
9 | 7, 8 | nffv 6895 | . . . . . . . 8 β’ β²π(πΉβπ) |
10 | nfcv 2897 | . . . . . . . 8 β’ β²π β | |
11 | nfcv 2897 | . . . . . . . 8 β’ β²ππ | |
12 | 9, 10, 11 | nfov 7435 | . . . . . . 7 β’ β²π((πΉβπ) β π) |
13 | nfcv 2897 | . . . . . . 7 β’ β²π < | |
14 | nfcv 2897 | . . . . . . . 8 β’ β²πlim sup | |
15 | 14, 7 | nffv 6895 | . . . . . . 7 β’ β²π(lim supβπΉ) |
16 | 12, 13, 15 | nfbr 5188 | . . . . . 6 β’ β²π((πΉβπ) β π) < (lim supβπΉ) |
17 | nfv 1909 | . . . . . 6 β’ β²π((πΉβπ) β π) < (lim supβπΉ) | |
18 | fveq2 6885 | . . . . . . . 8 β’ (π = π β (πΉβπ) = (πΉβπ)) | |
19 | 18 | oveq1d 7420 | . . . . . . 7 β’ (π = π β ((πΉβπ) β π) = ((πΉβπ) β π)) |
20 | 19 | breq1d 5151 | . . . . . 6 β’ (π = π β (((πΉβπ) β π) < (lim supβπΉ) β ((πΉβπ) β π) < (lim supβπΉ))) |
21 | 16, 17, 20 | cbvralw 3297 | . . . . 5 β’ (βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ) β βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ)) |
22 | 21 | a1i 11 | . . . 4 β’ (π = π β (βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ) β βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ))) |
23 | fveq2 6885 | . . . . 5 β’ (π = π β (β€β₯βπ) = (β€β₯βπ)) | |
24 | 23 | raleqdv 3319 | . . . 4 β’ (π = π β (βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ) β βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ))) |
25 | 22, 24 | bitrd 279 | . . 3 β’ (π = π β (βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ) β βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ))) |
26 | 25 | cbvrexvw 3229 | . 2 β’ (βπ β π βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ) β βπ β π βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ)) |
27 | 6, 26 | sylib 217 | 1 β’ (π β βπ β π βπ β (β€β₯βπ)((πΉβπ) β π) < (lim supβπΉ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 β²wnfc 2877 βwral 3055 βwrex 3064 class class class wbr 5141 βΆwf 6533 βcfv 6537 (class class class)co 7405 βcr 11111 < clt 11252 β cmin 11448 β€cz 12562 β€β₯cuz 12826 β+crp 12980 lim supclsp 15420 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-xadd 13099 df-ico 13336 df-fz 13491 df-fzo 13634 df-fl 13763 df-ceil 13764 df-limsup 15421 |
This theorem is referenced by: liminfltlem 45092 liminflimsupclim 45095 |
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